actes / proceedings cieaem 66 lyon 21-25 juillet / july 2014math.unipa.it/~grim/cieaem...

Actes / Proceedings

CIEAEM 66

Lyon

21-25 juillet / July 2014

Dessin de Victor Bousquet

Editor : Gilles AldonEditor of the Journal : Benedetto Di Paola and Claudio FazioInternational program committee : Gilles Aldon (F), Peter Appelbaum (USA), Françoise Cerquetti-Aberkane

(F), Javier Diez-Palomar (ES), Gail Fitzsimmons (AU), Uwe Gellert (D), Fernando Hitt (Ca), Corinne Hahn (F),François Kalavasis (Gr), Michaela Kaslova (CZ), Corneille Kazadi (Ca), Réjane Monod-Ansaldi (F), Michèle Prieur(F), Cristina Sabena (I), Sophie Soury-Lavergne (F).

Lyon�Quaderni di Ricerca in Didattica (Mathematics)� n. 24, Supplemento n.1, 2014

G.R.I.M. (Department of Mathematics and Computer Science, University of Palermo, Italy) 128

Mathematics and realities



Chapitre 4

Logics when doing (performing)mathematics / Logiques dans les pratiquesmathématiques

4.1 Working group 2 : Logic(s) when doing (performing) mathematics

Ana Serradò Bayés*, Uwe Gellert**

*University of Cadiz, **Frei Universität Berlin

The four sessions of the Working Group 2 were organized to look for an equilibrium of time for the presentation oftwelve papers and time for the discussion among the twenty-seven participants. The papers were grouped in advanceto envision an idea of evolution from the reality of the classroom (session 1), to the reality of the content, the problems,the tasks, the situations (session 2), to the reality of teachers and researchers (session 3) and to more abstract andgeneral theoretical issues (session 4).

With the aim of facilitating the participation of all the members of the working group, di�erent strategies wereused : personal introduction of the participants, discussion about the organisation of the working group and alternativeformats of the �nal report, feedback for clari�cation of central ideas immediately after the presentation of papers, smallgroup discussions on issues that connected individual presentations with suggestions from the discussion document,summaries of group discussions. . .However, it was often di�cult to connect the ideas presented in the papers with theopen questions from the discussion document. In particular, two of the questions in the discussion document, �Whatis role of logic in reasoning ?� and �Is it necessary to include a course on logic in university teacher training ?�, werenot taken up explicitly in the presentations of our working group.

In contrast, two of the questions of the discussion document were discussed in the second session : �What are thelinks with the arguments, evidence ?� and �What are the links between reality and the mathematical object ?� The �rstquestion revealed di�erent cultural conceptions of the word �evidence�, more or less related with the mathematicalconception of proof and demonstration. It became clear that �evidence� is used di�erently in di�erent languages,particularly when the philosophical roots of the concept were put aside. The second question expressed the di�cultiesof relating reality to mathematics, and how the di�erent papers conceive the problems as a lens to see the reality.The mathematical problems introduced during the paper presentation seemed to crystallize the central ideas of therespective papers. This idea was taken up by the �ve members of the team that prepared the �nal report for the lastday of the conference. The discussion of the group was summarized by the title : �Mathematics and realities throughthe lens of the problems.�

During the discussion of the second session, it was furthermore expressed how di�erent epistemological perspectivesconfront reality and mathematical objects. Along all four sessions, di�erent tasks and situations were presented thattried to coordinate reality with mathematical reality from a pedagogical and/or didactical point of view. Thoseattempts drew on : reality, realistic �ction, �ction, fairy tales, mathematical �ction and mathematical reality. Thepresentation of Ana Serradó discussed the question �Can I know which language my friend speaks when I only countthe vowels ?�, which emerged from the reality of her students. The question was intended to propose the students ?

Chapitre 4 Mathematics and realities



own mathematical/statistical questions to be engaged in, to identify the obstacles that emerge in this process and toconjecture about evolutions in the development of hypothetical thinking in an informal modelling process (Serradó,2014). The complexity and possible di�culties of a modelling process were also analysed by Aldon, Durand-Guerrier andRay (2014). Viviane Durand-Guerrier presented two �situations�, drawing on Brousseau's work on didactical situations.The �rst situation was a problem in elementary number theory (�Find all the integers that are sums of at least twoconsecutive numbers !�) which students needed to mathematize �what has been labelled �vertical mathematization� inRealistic Mathematics Education (RME). Durand-Guerrier concluded and discussed after her presentation that thissituation o�ered ample opportunities for the students to work within the abstract world of mathematical symbols. Theimportance of students' work with symbols was underpinned by Daniela Sanna (2014), who presented several problemsthat could facilitate the students' spontaneous use of abstract algebraic symbols. Aldon et al.'s second situation showedthe transformation of a classical problem of regions in a disc, i.e. a mathematical reality to a realistic �ction presentedin the �problem of the artist�. On the basis of this problem, the authors envisage that the alternatives for the horizontalmathematization of the problem will be su�ciently open to encourage discussion in the classroom.

The possibilities and constrictions of students' interpretation of another situation were analysed in the �Fairytale problem� (Pavlopoulou, Patronis and Andrikopoulou, 2014). When clarifying the ideas presented by KalliopiPavlopoulou with the working group, a discussion emerged about how the teacher's action could have transformed themagic logic of a fairy tale in the mathematical logic of a problem. This is a case of �horizontal mathematization�, froma pedagogically designed context to mathematical logic.

The analysis of the logical dimensions in Pavlopoulou et al. distinguishes between the di�erence of solving a �c-tional situation and a mathematical �ctional situation. In the context of classroom teaching, such an analysis oftenconcentrates on epistemological, psychological and didactical issues, without considering a sociological perspectiveon cultural dimensions of logic in society and the mathematics classroom, provided by David Kollosche (2014). Asociological approach towards the use of logic faces the challenge of questioning the power and control executed bygroups of people and the emancipation and subjection of individuals. The questioning of the ideological use of math-ematics, when mapping between the reality of mathematics and the social reality, allows Straehler-Pohl, Gellert andBohlmann (2014) to relate mathematics to the ethical dilemmas of consumerism. After Uwe Gellert's presentation, itwas asked whether we could think that mathematics may provide the solution to that kind of dilemmas. The discussionclari�ed that mathematics does not seem to be of direct help to untangle the dilemma of consumerism. However, inmathematics education at classroom level, mathematics can be used in a way that a deconstruction of ideologicalmessages becomes possible, fostering the emancipation of the students. Meaney's (2014) plenary presentation, whichreferred to the Barbie doll and to Lego constructions, discussed further examples for the importance of mathematicsin deconstructing ideologies. Taken together, the di�erent papers presented the challenges of creating meaningful tasksand situations relating to di�erent epistemological, psychological, didactical or sociological dimensions.

In his plenary presentation, Drijvers (2014) proposed the construction of meaningful tasks on the basis of theRME design principles of guided reinvention, guidance, didactical phenomenology and emergent modelling. PaulineLambrecht (Henry and Lambrecht 2014), in contrast, proposed the construction of didactical engineering as based ona process introduced by Artigue, which distinguishes four phases : preliminary analysis, a priori analysis, a posteriorianalysis, and evaluation. Particular attention was paid to the second phase, a priori analysis, by the presentationof Jarmila Novotná (Nováková and Novotná, 2014). The presentation developed a priori analysis of a mathematicalproblem of the area of a coloured quadrilateral within a triangle. The discussion in the working group focussedon similarities and di�erences with other formats of didactic analysis, such as �lesson studies�, and on the need ofdeveloping a posteriori analyses and evaluation. Teachers' structural di�culties to develop accurate a priori and aposteriori analyses were evidenced in a helicoidally sense of facilitating the development of mathematical contentknowledge. Furthermore, the importance of research for a priori analysis was highlighted.

Meaningful tasks can create tensions between those actors that negotiate the meaning of the tasks (pupils, students,teachers, educators, researchers, [parents]). For instance, tensions can obviously appear between the students and theteacher. Another example is the question of if and how an integrated mathematics curriculum can be bene�cial forchildren's learning as an integrated mathematics curriculum explicitly provides demonstrations of how mathematicsis used in the real world, as presented by Audrey Cooke (2014). Nina Bohlmann (Bohlmann, Straehler-Pohl andGellert, 2014) presented research in the context of in-service teacher education, where the reality of mathematicsword problems created an explicit tension between students and teachers. Accepting word problems as a reality ofthe mathematics classroom (and of assessment), how can all students get access to the code by which these tasks aredesigned Mathias Front and Marie-Line Gardes (2014) presented research about possible tensions between students,teachers, educators and researchers in didactic engineering. In their project, they envisage, among others, a proposition

CHAPITRE 4. WORKING GROUP 2 Mathematics and realities



for organizing the curriculum. The results of their on-going project might provide an approach to the question posedin the discussion document �What kind of teaching could allow the acquisition of reusable logic skills ?� Although thisquestion has been indirectly discussed in the four sessions of the working group, the feeling of the group is that we havenot been discussing the core of the sub-theme 2 as it has been proposed in the discussion document. We think thatentering in a fruitful discussion about the theme of �Logic(s) when doing (performing) mathematics� needs additionaltime for re�ection and discussion about : Which realities in�uence teaching and learning ? How do logic and logicalreasoning shape our reality ? and, How is the learners ? logical thinking developed by mathematical education [alongthe curriculum] ?




4.2 Un projet d'enseignement fondé sur les situations de recherche

Mathias Front*, Marie-Line Gardes**

*ESPE de l'Académie de Lyon, S2HEP, Université Lyon 1.**I3M, Université Montpellier 2.

Résumé : L'ordre culturel en usage dans l'enseignement des mathématiques est en évolution. Ceci entraine une transformation de

l'environnement de l'enseignant qui peut sembler su�sante pour que ses pratiques dans la classe puissent se mettre en accord avec les

recommandations institutionnelles concernant la résolution de problèmes. Mais, ceci peut-il réellement se produire ? Nous présentons, dans

ce texte, les travaux actuels du groupe DREAM-ResCo, qui portent un projet de construction d'ingénieries didactiques ayant pour but de

questionner un enseignement fondé sur les problèmes de recherche et porté par la dimension expérimentale de l'activité mathématique. Ces

ingénieries s'appuient sur des travaux antérieurs sur les situations de recherche et leur intégration dans les ressources d'un enseignant. Elles

visent à proposer des éléments sur la di�usion des compétences travaillées dans ces activités de recherche aux autres cadres de l'activité

mathématique et à construire des propositions d'organisation du curriculum fortement en appui sur ces situations de recherche.

Abstract : The cultural order in use in mathematics education is changing. This leads to a transformation of the teacher's environment.

This may seem su�cient for the rapprochement between classrooms' practices and institutional recommendations for problem solving. But,

can this actually happen ? We present in this text, the current work of DREAM- RESCO team, which carry out a project of teaching

design the aim of whom beeing questionning teaching methods based on problems called �research problem� and, consequently, on the

experimental dimension of mathematics. These engineering are based on previous work on research situations and their integration into

teacher's resources. They aim to provide evidence on how skills developed in these research activities can spread other mathematical

activities ; they also build proposals for teaching organization, strongly based on these research problems.

Introduction

Depuis les années 1970, le type de relation au savoir qui doit vivre dans la classe de mathématiques ne cesse d'êtreau c÷ur des ré�exions des didacticiens, et des mathématiciens qui se sont intéressés à la question. Des propositions dedispositifs visant une évolution de pratiques trop axées sur la seule présentation des savoirs ont vu le jour. Mais forceest de constater que l'évolution est lente. A. Berthè, en 1995, analysait, comme suit, les freins à une transformationdes points de vue :

Jusqu'à ces dernières années, l'ordre de présentation des concepts dans les classes de mathématiques seconformait à des ordres académiques issus de la genèse des savoirs par l'histoire et la culture (phylogenèse).Aujourd'hui les programmes laissent une marge de liberté aux professeurs pour adopter, semble-t-il, unordre compatible avec la genèse des savoirs à partir de la problématique de celui qui apprend (ontogenèse).N'y a-t-il pas cependant un ordre culturel en usage ? Si sa légitimation n'est plus académique, quelle est-elle ? Le décalage entre les di�érents ordres ne pose-t-il pas un problème de gestion aux enseignants ? � siles enseignants faisaient de tels choix, leur charge serait trop lourde et ils se marginaliseraient par rapportà ce qui est attendu par le système �. (Berthè, 1995, p. 84)

Pour reprendre cette di�cile question de la construction par les enseignants d'un ordre de présentation des conceptsnon antagoniste avec le développement des savoirs de leurs élèves, nous envisageons ici la question du point devue des potentialités des situations s'appuyant sur les problèmes de recherche. De nombreuses expériences ont eulieu depuis près de trente ans, tant au collège, qu'à l'école élémentaire et au lycée, tant en France qu'à l'étranger,concernant la mise en ÷uvre de problèmes de recherche en mathématiques dans des contextes de classe (Arsac,Germain & Mante, 1991, Peix & Tisseron, 1998, Schoenfeld, 1999, Grenier et Payan, 2003, Arsac & Mante, 2007,Harskamp & Suhre, 2007, Dias, 2008), dans des contextes interclasse, comme la résolution collaborative de problèmesproposée par le groupe ResCo de l'IREM de Montpellier (Sauter & al., 2008) ou à l'extèrieur de la classe à traversdes expèriences comme MATh.en.JEANS, stages Hippocampe (Tisseron, Feurly-Reynaud & Pontille, 1996, Duchet& Mainguene, 2002, Bressaud, 2012). Les études ont montré que ces expériences permettent des apports en termesd'apprentissage de démarches : développement d'heuristiques, élaboration de conjectures, mobilisation d'outils decontrôle et de validation, etc. Toutefois, bien que de telles situations de recherche continuent à vivre, et malgréles recommandations institutionnelles, elles ne se sont pas généralisées et elles n'in�uent que très lentement sur lespratiques ordinaires. L'accent mis principalement dans l'approche des problèmes de recherche sur le développementde compétences transversales liées au raisonnement, en laissant au second plan les apprentissages sur les notionsmathématiques en jeu, a sans doute été un frein à ce développement. Ce positionnement peut rentrer en con�it avec




certaines contraintes institutionnelles qui pèsent sur les professeurs, en particulier en ce qui concerne l'avancementdans le programme. Par ailleurs, la part importante de la dimension expérimentale, utilisant ou non les technologies,dans le travail de recherche bouscule les représentations des mathématiques que peuvent avoir certains enseignants.Or la dimension expérimentale de l'activité mathématique est centrale dans les situations utilisant des problèmes derecherche. Elle est dé�nie par Durand-Guerrier (2007, p. 17), comme à � le va-et-vient entre les objets que l'on essayede dé�nir et de délimiter et l'élaboration et/ou la mise Ã l'épreuve d'une théorie, le plus souvent locale, visant àrendre compte des propriétés de ces objets à �. Dans le cadre de l'étude d'un objet mathématique, cette dimensionrend compte du travail de découverte, des actions sur l'objet, de l'émission de conjectures, des premières constructionsthéoriques et surtout des allers-retours entre ces di�érentes phases. Cette dimension expérimentale prend en compteà la fois des démarches heuristiques et les concepts mathématiques en jeu, comme l'a montré (Gardes, 2013). Elle estune des dimensions premières dés que l'activité proposée aux élèves s'oriente vers la recherche de problèmes mais ellereste pour l'instant peu familière des pratiques enseignantes. De ce point de vue, l'environnement culturel se révèled'une stabilité redoutable.

Questions de recherche

Depuis 1995, les attentes de l'institution ont continué à évoluer dans le sens d'une intégration de la résolution deproblèmes dans les pratiques. Les travaux en didactique sur les situations de recherche se sont parallèlement a�nés.La place et le rôle des problèmes de recherche dans la classe de mathématiques ont été réinterrogés, notamment ence qui concerne les apprentissages des concepts mathématiques. Il est possible, désormais, d'envisager de proposer desdispositifs d'enseignement fondés sur les situations de recherche qui ne marginalisent pas les enseignants par rapportà ce qui est attendu par le système. Compte-tenu des éléments précédemment décrits, un tel dispositif doit investirdi�érentes dimensions et en particulier : une légitimation renouvelée de l'activité mathématique et des savoirs produitslors de cette activité, la construction d'un nouveau rapport au savoir, le déploiement de nouvelles compétences dansles activités de recherche et leur di�usion aux autres cadres de l'activité mathématique (appropriation et mise en ÷vrede connaissances, de techniques, communication et rédaction de résultats et de preuves, etc.). Le groupe de rechercheDREAM-ResCo, composé d'enseignants-chercheurs en mathématiques, de formateurs d'enseignants et d'enseignants,développe une ingénierie pour la mise en place et l'étude d'un tel dispositif. En appui sur des expérimentations, elledoit apporter des éléments de réponse aux questions suivantes : Comment les savoirs construits lors des situations derecherche prennent-ils leur place dans la progression du temps didactique 1 ? La créativité et l'invention mathématiquedéveloppées dans les problèmes de recherche modi�ent elles l'image des mathématiques chez les élèves (et leur enviede faire des mathématiques), chez les professeurs ? Comment les élèves réinvestissent-ils dans d'autres cadres lescompétences et les connaissances développées dans les activités de recherche de problèmes ? Une progression annuellefondée sur la mise en ÷uvre de situations de recherche en classe est-elle envisageable ?

Cadres théoriques et méthodologie

Au-delà des travaux précédemment évoqués sur les problèmes de recherche, l'ingénierie développée s'appuie surquelques éléments clés. Au niveau épistémologique, le groupe DREAM a produit une ressource dans un cadre théoriqueadapté (Aldon et al., 2010), des analyses �nes de situations parti- culières (Front, 2012 et Gardes, 2013) et égalementun outil qui permet de s'assurer de la proximité des réalisations d'un utilisateur de la ressource aux intentions desauteurs (Aldon, Front & Gardes, (soumis) ). Ces outils utilisent di�érents cadres théoriques : Théorie des SituationsDidactiques (Brousseau, 2004), structuration des milieux d'une situation (Margolinas, 2004), genèse documentaire(Geudet & Trouche, 2008). Au niveau méthodologique, nous avons choisi d'adopter la méthodologie de à � recherche-design à � (Design-Based Research) de Wang et Hanna�n (2005) qui permet, en particulier, d'envisager une démarche,ancrée dans la théorie et dans la pratique, collaborative et participative, qui se développe au niveau d'une communautéde chercheurs et de praticiens et d'une communauté d'apprenants. Dans cette approche les chercheurs sont en étroiterelation avec les praticiens et les interactions, incontournables dans ce type de recherche, sont intégrées dans lathéorisation du processus de recherche. Concernant la communauté de praticiens, il s'agit de proposer des ressourceset des modalités d'action permettant l'élaboration et la mise en ÷uvre d'un projet d'enseignement s'appuyant surdes situations de recherche. L'ingénierie contient également la méthode pour l'étude de cette introduction dans la

1. C'est le temps qui se mesure par les avancées dans l'exposition du savoir. A ne pas confondre avec le temps d'apprentissage, propreà l'élève.




communauté de praticiens. Concernant les apprenants, la méthode prévoit l'étude des actions des praticiens dans lesystème didactique aussi bien du point de vue de la mise en ÷uvre des situations de recherche que de celui de laconstruction des savoirs dans le cadre du projet d'enseignement s'appuyant sur ces situations. La �gure 4.1 représentela répartition des tâches de chaque communauté et leurs interactions dans un cycle de la � recherche-design �.

Figure 4.1 � Répartition des tâches de chaque communauté et interactions

Rappelons que la démarche est par dé�nition itérative comme le précisent Wang et Hanna�n (2005, p.9) :

Design-based research is also characterized by an iterative cycle of design, enactment or implementation,analysis, and redesign.

Sur cet aspect, Basque (2009, p.26) ajoute que � chaque cycle implique aussi une opération continue d'évaluationformative a�n de guider la prise de décision au fur et à mesure du déroulement de la recherche �.

Description du projet envisagé

Concrètement, nous envisageons une expérimentation en trois étapes. La première consiste, d'une part à ques-tionner un ensemble de séquences d'apprentissage fondées sur les problèmes de recherche, travail en partie e�ectuédans la ressource élaborée par le groupe DREAM (Aldon et al., 2010), et d'autre part à tenter de structurer uneprogression annuelle dans une classe pilote dont le professeur est un praticien et membre de l'équipe DREAM-Resco.Les interactions entre les chercheurs et le praticien, lors de cette première étape, ont permis l'élaboration de nouveauxdocuments. Les �gures 4.2, 4.3 et 4.4 en sont des exemples. La �gure 2 propose des liens entre des problèmes issusde la recherche (en jaune), des problèmes proposés par l'enseignant (en violet) et des thèmes mathématiques desprogrammes de collège et lycée en France.

Dans le cadre d'un projet d'enseignement fondé sur les situations de recherche de problèmes, les liens avec lessavoirs mathématiques en jeu dans la recherche sont fondamentaux. Le document suivant (�gure 4.3) envisage lesproblèmes comme autant de n÷uds dans le réseau des savoirs 2.

2. En suivant Conne, nous concevons les problèmes et les savoirs en lien, comme insérés dans des réseaux : � En fait les problèmes nese laissent pas identi�er ni isoler comme cela, ils ne vont jamais seuls, on a toujours a�aire à des chaînes de problèmes s'organisant enréseaux, à l'image des réseaux de savoirs qu'ils représentent � , (Conne, 2004).




Figure 4.2 � Rapprochement des problèmes et des thèmes mathématiques

La �gure 4.4 propose un zoom sur un problème de recherche particulier 3 et les savoirs en lien. L'énoncé de ceproblème est le suivant :

Est-il possible de trouver deux nombres entiers a et b, distincts, tels que 1a + 1

b = 1 ? Et avec 1a + 1

b + 1c ?

Lors de cette première étape de l'expérimentation, les interactions entre les chercheurs et le praticien ont égalementpermis la construction d'un premier cahier des charges. L'objectif était de dégager les premiers éléments autour debonnes pratiques, de gestes professionnels, de suivi des ressources, etc., en s'appuyant sur une méthodologie ré�exive(Gueudet & Trouche, 2008).

En 2014-2015, la deuxième étape de l'expérimentation, doit permettre de s'assurer, de la faisabilité d'une étudeapprofondie à une échelle supèrieure. Nous nous appuierons sur le réseau des Lieux d'éducation Associés 4 et enparticulier le collège-lycée Paul Valery à Sète et le collège-lycée Ampère à Lyon, ainsi que sur d'autres établissementsdont au moins un professeur est enseignant associé à l'IFE ; au total quatre collèges et trois lycées seront impliquésdans la recherche. Lors de cette seconde étape de l'expérimentation, les observations et analyses, dans un contextese rapprochant de celui de la classe ordinaire, doivent aboutir à la rédaction d'un cahier des charges beaucoup plusdétaillé et opérationnel pour la mise en place de la dernière étape de l'expérimentation : en appui sur les ingénieriesprécédemment construites, le suivi d' élèves dans des classes � ordinaires � et le suivi de cohortes d'élèves dans lesclasses des professeurs pilotes et des professeurs du second cercle des LÃ©A, seront conduits.

3. De nombreuses expérimentations en classe ont permis de constater une modi�cation du système de connaissances mathématiquesd'élèves et la possibilité d'institutionnalisation de savoirs, notamment en calcul numérique et en calcul littéral.

4. Les lieux d'éducation associés à l'IFÉ sont des lieux où des équipes de terrain travaillent en collaboration avec des chercheurs. Ilsconstruisent ensemble un projet de recherche qui est signi�ant pour l'ensemble des acteurs impliqués. Le développement des LéA est auc÷ur du projet scienti�que de l'IFÉ.




Figure 4.3 � Problèmes de recherche et n÷uds de savoir, en lien avec les programmes de la classe de quatrième enFrance

Conclusion

Nous avons déjà menés des travaux sur la mise en ÷uvre dans une classe ordinaire de situations de recherche ainsique des expèriences de di�usion de situations de recherche dans le cadre de la formation continue des enseignants dansles académies de Lyon et de Montpellier. Le projet proposé permet maintenant une étude à une échelle supèrieure. Il doitstatuer sur la viabilité des propositions d'organisation du curriculum fortement en appui sur les situations de recherchemais également proposer des éléments sur la di�usion des compétences travaillées dans ces activités de recherche auxautres cadres de l'activité mathématique et ceci sur des e�ectifs d'élèves conséquents. Compte tenu des questionsde recherche posées, des résultats sont attendus tout au long des deux prochaines années d'expérimentation. Ilspermettront d'échanger sur une approche qui doit réintroduire de la consistance mathématique dans les enseignements.

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4.3 7th grades' reactions to an � unusual � mathematical scenario

Kalliopi Pavlopoulou*,Tasos Patronis**, Maria Andrikopoulou**

*School of Applied Mathematical and Physical Sciences, National Technical University of Athens**Department of Mathematics, University of Patras

Résumé : On a donné à des élèves d'un Collège Expérimental à Athènes un scenario qui combine la syntaxe mathématique avec des

signi�cations sociales et une dose d'humour critique. Les élèves devaient décoder un message relatif aux règles de la priorité des opérations

algébriques, à travers d'un texte-adaptation d'une histoire de J. Thurber. Le scénario a introduit les élèves à une activité de caractère

socio-mathématique et plus que la moitié de la classe a réussi à la tache donnée.

Abstract : A scenario combining mathematical syntax with social meaning and a sense of humor was given to classes of 7th grade, in

an experimental school of Athens. The students should decode a message related to priority of algebraic operations, through a text adapted

from a story of J. Thurber. The scenario led the students to a socio-mathematical kind of activity and more than half of them to ful�ll the

given task.

A theoretical distinction

Imaginary stories and tales have been used several times in teaching mathematics. They usually attract attentionof young children, but it does not seem to exist a general agreement about their pedagogical value. Being consid-ered as �magic� contexts, their use has been criticised as o�ering a non realistic mathematical pedagogy (Stree�and,1984/1985). Freudenthal (1982) has also discussed the impact of �magic� contexts on children's thought in relationto the well known phenomenon of �L'Age du Capitaine�, thus indicating that the didactical contract need not beconsidered as the only possible cause of children's �illogical� reaction.

Instead of classifying mathematical pedagogies with respect to their claimed philosophy, we may focus on theiractual teaching-learning process and its relation to the motives and goals of teaching : if these are all situated outsidethe teaching-learning process, then we speak of an instrumental educational frame ; while if at least a part of motivesand goals lies inside the teaching-learning process, we speak of a critical-hermeneutic educational frame since theteaching and learning involves some re�ection and understanding of what is actually learned and why. By leavingaside questions of meaning, instrumental educational frames are perhaps responsible for rote learning and for thewell known responses to meaningless mathematical scenario such as �L'Age du Capitaine� (see, for example, Patronis,1996).

Our experiment

Our scenario (in two versions) borrows some elements from the story The Thirteen Clocks of James Thurber .More speci�cally, the following text was proposed as a scenario to a class of 7th grade from an experimental school inAthens :

A story with Zorn of Zorna and the last Golux on Earth.

�In games and contests time must always be determined. How much time do I need to �nd a thousand sapphires ?�,asked the prince Zorn of Zorna. �If days of your life have to be taken away, you also need to rest somehow� Then thisterrible adventure will take eleven times the days of a week, if you �rst add to them one day and a half and thensubtract a three days weekend.�, said Golux after three minutes of re�ection. �But be aware of the calculations :

SUMAL �the thing we have got from the Duke for doing such work easily �does not do anything more than what youtell it to do !�. � All right�,replied the prince, �whatever I say to SUMAL I am going also to write down, so that I see

what am I asking the SUMAL to do.�. What did Zorn of Zorna write down on his paper ?

This was the �nal form of the text, as result of a pilot research before our main experiment. This scenario iscoherent with Thurber's own attitude and sense of humour ; he addresses his story to both children and adults,critically integrating elements from the past and present. Thus, following the critical humouristic spirit of Thurber,we developed the above Golux sentense as a meta-magic one, rather than a mere rule in some �magic� language game.This means that the text includes (in an open and implicit manner) some ethical social reason of the proposed magicformula : people deserve rest and recovering within the risks and anxiety of today life.

A second, shorter, version was proposed to another class of 7th grades from the same school, as follows :




A story with Zorn of Zorna and the last Golux on Earth.

� In games and contests time must always be determined. How much time do I need to �nd a thousand sapphires ?�,asked the prince Zorn of Zorna. � In order to �nd the required time, subtract twice the number of days of a week from

the number of hours of a full day, and divide the result by two.�, said Golux. �But be aware of the calculations :SUMAL �the thing we have got from the Duke for doing such work easily�does not do anything more than what youtell it to do !�. �All right�, replied the prince, �whatever I say to SUMAL I am going also to write down, so that I see

what am I asking the SUMAL to do.�. What did Zorn of Zorna write down on his paper ?

The aims of our teaching experiment were the following :

a to see how would the students react to an imaginary story as above and whether they would conceive the �rstversion of the scenario as symbolizing a current social situation ;

b to see if the students would address a pragmatic critique to the given mathematical scenario (in both versions) ; and

c to examine in what extent the students would be able to express a part of the text into mathematical operations,without following a linear reading of the text.

The second version (in fact an elusive puzzle) aimed, more speci�cally, to test students' critical competence : wouldthey question the puzzle's unspeci�ed data ?

Reactions and comments from the students

In what concerns the �rst version, some students did a critical reading of the text and sometimes �extended� thestory. These students also did many comments on the scenario of the problem stated :

� The formulation appeared to them as unusual for a mathematical problem and the �nal question was not clearfor them. This question should be formulated as : �How many days needs Zorn to �nd the sapphires ?�, a studentsaid.

� Although they expressed their criticism at the beginning, most of the students have been involved in the scenarioand they attributed to SUMAL not only the role of a calculator but also that of a robot-servant ! So, they wrotethe instructions to SUMAL either by using an algorithmic procedure, by reversing the sequence of the operationsin the verbal formulation : � add 7 with 1,5 and then subtract from the sum the number 3. Finally, multiplicatethe result by 11.� or, by giving instructions refering to needs of daily life : �Prepare the table because I'm hungrynow !�

� There was a questioning about the phrase �subtract a three days weekend� : some students said that it is notnecessary to subtract a three days weekend because �SUMAL works without being tired, so it does not need torelax, so I don't subtract. But, if Zorn has to do this work, he needs a relaxation . . .�.

General results

The rate of absolute success (right decoding and right result) in the �rst version of the scenario was more than50% (14 to 27 students). Also, 3 of the 27 students decoded perfectly the series of operations, but they arrived to afalse result due to multiplication of 11 by a decimal number. Not correct answers were given by 9 to 27 students. Only4 to 27 did a linear reading and 5 students misunderstood the formulation by answering : �11× (7 + 1, 5)− 3 = 90, 5days�.

One third of the students criticized the second version of the scenario as incomplete or incorrect, since there is noinformation about units of time. However, another third produced the result �5� without any comment, while otherschose arbitrary units (e.g., hours).

Qualitative analysis of students' responses

First version of the scenarioArithmetical data in symbolic form do not appear in the text and students are not used to solve such problems

in mathematics. However, most students did not follow a number consideration strategy (Garofalo, 1992), but theyinsisted to understand the scenario.

The most usual type of students' responses show a partial understanding of the text :




a A linear reading of the text, given in verbal form, and linear �translation� in symbolic form : �11×7 = 77days+1.5 =78.5− 3days= 75.5 days �

b Students rarely understand the text as a whole unity, but conceive it in fragments. Some students did the translationstep by step :

� add to them one day and a half → 7 + 1.5 = 8.5� �eleven times the days of a week. . . → 11 × 8, 5 = 93, 5� �subtract a three day weekend → 93, 5− 3 = 90, 5 �

Second version of the scenario We observe four categories of responses in the second version of our scenario.Some students responsed by giving two or more possible answers, so they may belong to more than one of the followingcategories :

1. Rational/Pragmatic critique : about one third of the students red critically the formulation of the problem byjudging it as incorrect or incomplete :�We don't know what �5� is. It could be seconds, minutes, years, days, weeks, years. For this reason it's impossibleto know how much time Zorn of Zorna would need to �nd the sapphires.�

2. The phenomenon of �L'Age du Capitaine�, probably due to the �magic� context of the problem, was observed inabout half of the students. Among them 9 students did successfully the translation to symbolic form, but founda result �5� :[24− (2× 7)] : 2 = 5 or �24− 14 = 10, 10 : 2 = 5�Also, 6 students assigned (arbitrarily) a unit of time to their results (most of them used hours and only one usedminutes).

3. Homogeneous units : Some students were led by the scenario to turn the given (absolute) numbers into their�equivalents� in hours or days, thus obtaining �impossible� solutions :�From hours we cannot subtract days, so if 1 day has 24 hours, 14 days have14 × 24hours = 408 hours (false,due to calculation), 24 - 408 = impossible. . .�

4. Reversing the order of words in the text :[(24× 7)× 2]− 24 = (168× 2)− 24 = 336− 24 = 312.

Conclusions and perspectives

As a �rst result we observed a strong interest of students in the given text. Students' reactions lead us to continuea research with other scenario concerning the teaching of mathematics in Elementary and High School, in interactionwith the teaching of literature. Although some of the students' responses show a non-mathematical involvement in thescenario, the same students performed mathematical calculations and produced mostly logical arithmetical answers.In this sense we have here a complex socio-mathematical activity, similar to that of popularization of mathematics.

It is also interesting that in case of the second version, one third of the students questioned the unspeci�ed data.Moreover, some students were led to turn the given (absolute) numbers into their �equivalents� in hours or days, thuspotentially arriving to negative numbers.

As regards our aims in general we note that more than half of our students were able to �translate� successfully averbal message into mathematical operations, and only a small part followed a linear reading of the text. A systematicwork with texts like this could help students, more generally, in the transformation of natural language to symbolicexpressions and vice versa �a procedure needed in problem solving and meaningful learning of mathematics.

REFERENCES

Freudenthal, H. (1982). 'Fiabilité, Validité et Pertinence - Critères de la Recherche sur l'Enseignment de la Math-ématique', Educational Studies in Mathematics, 13, 395-408.

Garofalo, J. (1992). 'Number-consideration strategies students use to solve word problems', Focus on LearningProblems in Mathematics, 14, 2, 37-50.

Patronis, T. (1996). 'Scene-Setting versus Embodiment : common sense as part of non-institutionalised knowledge',Proceedings of CIEAEM 47.

Stree�and, L. (1984/1985). 'Search for the roots of ratio', Educational Studies in Mathematics, 15 (327-348) and16 (75-94).




4.4 Deconstructing the Filtration of Reality in Word Problems

Nina Bohlmann, Hauke Straehler-Pohl, Uwe Gellert

Frei Universität Berlin

Résumé : La réalité dans la classe de mathématiques est toujours une réalité portée avec un regard mathématique. Quand les

mathématiques sont mis en action, la réalité ne reste pas la même. Cette �ltration de la réalité apparaît comme un mécanisme intégré

aux mathématiques à l'école produit par un principe de recontextualisation. Lorsque les apprenants ne reconnaissent pas ce processus, ils

sont en danger de méconnaître la fonction didactique de la réalité de l'apprentissage des mathématiques et, par conséquent, ne sont pas en

mesure de produire un texte légitime dans le contexte de la classe. Nous soutenons pour faire face au dilemme de la �ltration de la réalité

comme une nécessité structurelle des mathématiques à l'école et de prendre ce dilemme comme un point de départ pour trouver des moyens

de production pour aller de l'avant. Une stratégie visant à o�rir l'égalité des chances pour l'apprentissage des mathématiques est d'élargir

la prise de conscience de la �ltration de la réalité et de la transformer en un outil pour les étudiants, qui restent habituellement dans le

non-dit et donc la compréhension des règles est réservée à quelques-uns, qui peuvent utiliser cet outil pour les placer dans une position

d'apprenants � mathématiquement capables �. Lors de l'élaboration d'une stratégie pour rendre le processus de �ltration de la réalité

visible, nous avons été inspirés par le travail de De Freitas (2008), et nous avons fait participer les élèves à une activité de déconstruction

des réalités di�érentes dans un ensemble de trois activités mathématiquement similaires, mais dans des contextes totalement di�érents -

les problèmes de mot. Nous discutons de cette activité et nous examinons de manière critique la façon dont la �ltration de la réalité peut

être rendu visible pour les étudiants.

Abstract : Reality in the mathematics classroom is always a reality under a mathematical gaze. When mathematics is brought into

action, the reality does not remain the same. This reality �ltration appears as an in-built mechanism of school mathematics as it is produced

by the recontextualisation principle. When learners of mathematics do not recognize this process, they are in danger of misunderstanding

the didactical function of reality for the learning of mathematics and, thus, are not able to produce legitimate text. We argue for facing the

dilemma of reality �ltration as a structural necessity of school mathematics and for taking this dilemma as a point of departure for �nding

productive ways forward. One strategy to provide equal opportunities for mathematics learning is to broaden the awareness of reality

�ltration and to convert it into a tool for the students, that usually remains in the unspoken and hence reserved to just some, who can use

this selectively distributed tool to position themselves as 'mathematically able' learners. In developing a strategy for making the process

of reality �ltration visible, we were inspired by the work of De Freitas (2008), and we involved students in an activity of deconstructing

the di�erent realities in a set of three -mathematically similar, contextually totally di�erent- word problems. We discuss this activity and

critically examine how reality �ltration can be made visible to the students.

Introduction

Where school mathematics is taught by modelling and applications, by context tasks and mathematics in everydaycontext, a mathematical gaze is thrown on reality. This particular gaze brings about a requirement for recognitionof some selected elements of reality and a suppression of all others. Experiences of life very seldom provide adequatemeans for such recognition. In this way, the gaze decontextualizes reality and recontextualizes its selected elements inpedagogic practice. As Bernstein (2000) argues, the recontextualising principle as a key organizer of school mathemat-ics practice is hardly known to all those learners of school mathematics whose orientation to meaning can be describedas contextual rather than as decontextual, as concrete rather than as abstract. When learners of mathematics do notrecognize that 'in the classroom' reality is always a reality under a mathematical gaze, they are in danger of misun-derstanding the didactical function of reality for the learning of mathematics and, thus, not able to produce legitimatetext (e.g., Gorgorió, Planas and Vilella 2002). In this way, context problems bear the danger of marginalizing stu-dents' experiences of life, of mythologizing the relation between mathematics and life, and of jeopardizing disciplinarymathematical learning (Dowling 1998). Therefore, it has been argued (e.g., Bourne 2004, Jablonka and Gellert 2012,Cooper and Dunne 2000) that it is necessary 'and possible' to make the recontextualisation principle, to some extent,visible also for those learners, who do not enter schools already equipped with the prevalent recognition rules. Buthow can this be achieved ?

Reality �ltration

As argued above, reality in the mathematics classroom is a reality under a mathematical gaze. When modellingand application tasks are designed, this happens under the primacy of fostering the learning of mathematics. When




mathematics is brought into action, the reality does not remain the same. Skovsmose discusses an experience witha teaching project 'Family support in a Micro-Society' where students are �rst involved in formulating principlesaccording to how they want to distribute child bene�ts among families (cf. Skovsmose, 1994, ch. 9). He reports how'[t]he ethical principle, which might have guided the initial considerations [of the students], becomes substituted bythe technical administration of the system' (Skovsmose 2008, p. 166f.). He calls the repression of the ethical principleby means of mathematisation an 'ethical �ltration' (p. 167). For Skovsmose, a critical approach to mathematicseducation addresses this phenomenon 'as a general feature of bringing mathematics into action' (p. 167). According toDe Freitas (2008, p. 87), even politically delicate contexts, such as working with statistics of homeless people, genderand racial bias, run the risk to become reduced to 'extremely inadequate and often unethical representations of the'real' experience of those under study' in the mathematics classroom.

Similar to the ethical �ltration e�ect caused by mathematisation, a reality �ltration e�ect is produced by therecontextualisation principle. The subordination of reality to the learning of mathematics is an inherent characteristicof the structure of school mathematics. There seems to be reason to talk about the impossibility of real-life problemsin mathematics classrooms (Gerofsky 2010, Lundin 2012). Reality �ltration appears as an in-built mechanism of schoolmathematics. We argue for facing the dilemma of reality �ltration as a structural necessity of school mathematics andfor taking this dilemma as a point of departure for �nding productive ways forward.

Context variation : making reality �ltration visible

Accepting that the recontextualisation of 'reality' in school mathematics and the e�ect of reality �ltration arestructural necessities of mathematics education, we conclude that one strategy to provide equal opportunities formathematics learning is to broaden the awareness of reality �ltration and to convert it into a tool for the students.A tool that usually remains in the unspoken and hence remains reserved to just some, who can use this selectivelydistributed tool to position themselves as 'mathematically able' learners.

In developing a strategy for making the process of reality �ltration visible, we were inspired by De Freitas (2008)who introduced future mathematics teachers to an activity of re-writing textbook problems, making them aware of andcritical about the 'real' as it was presented in the problems, by shifting contexts of context tasks while maintaining themathematical structure unchanged. We transposed this activity in order to make it applicable for the work with sixth-grade students (11'12 years old). De Freitas' focus is on the potential of this activity for fostering critical awareness. Inour case, the critical awareness that we aim to foster is not directed towards political issues, but con�ned to a criticalawareness of the implicit principles of school mathematics. However, as we have argued above, this can be seen as apolitical issue in itself.

Context variation at work

In a professional development workshop, we involved mathematics teachers, which are used to work with under-privileged learners, in an activity of context variation. Together with the teachers, we developed a set of three wordproblems that can be considered as variations of reality concerning one abstract problem of proportional reasoning.

Context A : Road crossing

The phases of a tra�c light are such that with every green phase, 14 people can cross the street. For the �nal ofthe UEFA Euro 2010, 269 people want to cross the central crossing at Brandenburg Gate to reach the central publicviewing on time for the kick-o�. How many green phases are needed, so that all supporters can cross the road'

Context B : The lift

A sign in a lift at an o�ce block says : This lift can carry up to 14 people. In the morning rush, 269 people wantto go up in this lift. How many times must it go up ? (This was the original problem, on which the context variationwas based. It is discussed in Cooper and Dunne 2000)

Context C : The cable car

In the morning, �ve coaches reach the valley station of the cable car that supports the ski runs. 269 skiers leavethe coaches in order to take the cable car to the mountain station. A sign at the cable car says : This cable car cancarry up to 14 people. How many cable cars are needed at least to bring all skiers to the top'

One of the teachers from the professional development workshop designed a group-work activity in which di�erentgroups of students worked on the di�erent contexts. The groups prepared posters to present and justify their solutions.After all presentations had been �nished, the teacher started a whole-class discussion, shifting the focus on the relation




between the word-problem's gaze on reality and the students' knowledge of these realities. In our paper presentation,we will discuss excerpts from the whole-class discussion. The research question that guides our analysis is : How doteachers transfer the context variation activity into the classroom and how do their students react ? In the remainderof this paper, we summarise our preliminary �ndings.

Findings

From the presentations of the student-groups, we can conclude that in each group were at least some students thatwere able to execute a reality �ltration and solve the problem in a way that was desired by the teacher. Also, thegroups worked according to the 'goldilocks principle' (Gates and Vistro-Yu 2003, p. 53) 'not too much, but not toolittle realistic considerations' in their solutions and all produced the desired solution of 20. However, one has to becareful to conclude from the fact, that all groups (as a collective) realised a reality �ltration, to the assumption thatthis process has been visible for all students within each group. Kotsopoulos (2010) demonstrates how group work canbe 'not collaborative' particularly for those students who are 'perceived as 'not getting it' or holding back the group'sprogress' (p. 133).

The whole-class discussion of contexts A and B can be seen as good examples of how reality �ltration can be madevisible to all students. Once the teacher had opened the space to formulate a critique on the problems' gaze on reality,a wide variety of students showed up and articulated a whole range of realistic considerations that, however, have tobe suppressed, if one wants to come to a mathematical solution. The teacher makes explicit that this gaze is no naturalforce, but has a social location, namely 'people who have invented these problems in these situations' (Teacher). Thestudents can, then, take part in an exercise of deconstructing the inventors' gaze. The cascades of student-utteranceswithout any interference of the teacher are not regular characteristics of German mathematic classrooms (Begehr2004) and hence can be interpreted as a sign for a remarkable dissolution of power within this exercise. It seems thata space in the mathematics classroom has been opened in which students could yield their knowledge about their livesin a legitimate and appreciated way.

This process collapsed dramatically during the discussion of the third context, which had initially been designedas the context with the least frictions between the real and the mathematical. The students' contributions, each ofthem pointing towards the same deconstruction that had been appreciated before, was followed by a refutation fromthe side of the teacher. Instead of valuing the students' consideration, she started to narrow the scope of reality downto the particular situation, where the reality has to follow the rules of the word-problem.

We concede that our own initial conception of the method of context variation within the professional developmentworkshop might have played a role in this �nal development that undermines our own aims : We proposed to designthree problems with a gradual progression concerning the degree of realistic (im)possibility. The teacher seemed to havepicked this up and put particular emphasis on it, as she formulated the aim of �nding out, which of the problems wasthe most realistic for the whole-class discussion. Finally, it appeared that she acted under a pressure to demonstratethat the �nal context was the most realistic and hence was not letting the students proceed in their exercise ofdeconstructing the gaze of school mathematics.

REFERENCES

Begehr, Astrid (2004) Teilnahme und Teilhabe am Mathematikunterricht : Eine Analyse von Schülerpartizipation.[Passive, Active, and Interactive Participation in Mathematics Classrooms : An Analysis of Participation]. Berlin :Freie Universität Berlin.

Bernstein, B. (2000). Pedagogy, symbolic control and identity : theory, research, critique. Lanham : Rowman andLittle�eld.

Bourne, J. (2004). Framing talk : towards a 'radical visible pedagogy'. In J. Muller, B. Davies & A. Morais (Eds.),Reading Bernstein, researching Bernstein (pp. 61-74). London : RoutledgeFalmer.

Cooper, B., & Dunne, M. (2000). Assessing children's mathematical knowledge : social class, sex and problem-solving. Buckingham : Open University Press.

De Freitas, E. (2008). Critical mathematics education : recognizing the ethical dimension of problem solving.International Electronic Journal of Mathematics Education, 3(2), 79-95.

Dowling, P. (1998). The sociology of mathematics education : mathematical myths/pedagogic texts. London : Rout-ledgeFalmer.




Gates, P., & Vistro-Yu, C.P. (2003). Is mathematics for all ? In A.J. Bishop & al. (Eds) Second internationalhandbook of mathematics education (pp. 31-74). Dordrecht : Kluwer.

Gerofsky, S. (2010). The impossibility of 'real-life' word problems (according to Bakhtin, Lacan, Zizek and Bau-drillard). Discourse : Studies in the Cultural Politics of Education, 31(1), 61-73.

Gorgorió, N., Planas, N., & Vilella, X. (2002). Immigrant children learning mathematics in mainstream schools. InG. de Abreu, A.J. Bishop & N.C. Presmeg (Eds.), Transitions between contexts of mathematical practices (pp. 23-52).Dordrecht : Kluwer.

Jablonka, E., & Gellert, U. (2012). Potential, pitfalls, and discriminations : curriculum conceptions revisited. InO. Skovsmose & B. Greer (Eds.), Opening the cage : critique and politics of mathematics education (pp. 287-307).Rotterdam : Sense.

Kotsopoulos, D. (2010). When collaborative is not collaborative : supporting student learning through self-surveillance,International Journal of Educational Research, 49(4-5), 129-140

Lundin, S. (2012). Hating school, loving mathematics : on the ideological function of critique and reform in math-ematics education. Educational Studies in Mathematics, 80(2), 73-85.

Skovsmose, O. (2004). Towards a philosophy of critical mathematics education. Dordrecht : Kluwer.Skovsmose, O. (2008). Mathematics education in a knowledge market : developing functional and critical com-

petencies. In E. de Freitas & K. Nolan (Eds.), Opening the research text : critical insights and in(ter)ventions intomathematics education (pp. 159-174). New York : Springer.




4.5 Des problèmes pour favoriser la dévolution du processus de mathé-matisation : un exemple en théorie des nombres et une �ction réaliste

Gilles Aldon*, Viviane Durand-Guerrier**, Benoit Ray***

*IFÉ - École Normale Supérieure de Lyon, France**Université de Montpellier 2, France***Lycée français de Tunis, Tunisie

Résumé : La modélisation est un processus complexe et souvent di�cile pour peu que les phénomènes étudiés soient issus de situations

réelles. C'est pourtant un travail important du mathématicien qui mérite d'être abordé dans l'enseignement des mathématiques. Dans le

processus de recherche de problèmes dans l'enseignement, nous avons testé à la fois des énoncés posés au sein des mathématiques et d'autres

posés en dehors des mathématiques quali�ées de � �ctions réalistes �(pour lesquelles un problème inséré dans un contexte �ctionnel permet

la mise en ÷uvre d'un processus de modélisation par les élèves). Nous faisons l'hypothèse que la résolution de problèmes relevant de l'une ou

l'autre de ces deux catégories participe à la fois au développement de compétences meta-mathématiques et de connaissances mathématiques.

Abstract : Modeling is a complex and di�cult process as long as the studied phenomena come from real situations. However it's

an important part of the work of mathematicians that needs to be addressed in the teaching of mathematics. We have experienced two

di�erent types of problems, the �rst within mathematics and the second in form of � realistic �ctions �, that is to say, a problem inserted

in a �ctional context allowing a modeling process. We hypothesize that the solving of both types of problem is involved in construction of

meta-mathematical skills as well as mathematical knowledge.

Introduction

Depuis de nombreuses années les équipes DREAM (Démarches de recherche pour l'enseignement et l'apprentissagedes mathématiques, IREM de Lyon) et Resco (Résolution collaborative de problèmes, IREM de Montpellier) travaillentconjointement à l'introduction de problèmes de recherches dans le cours de mathématiques. Le travail de l'équipeDREAM s'appuie sur l'ensemble des travaux développés autour du problème ouvert au sein de l'IREM de Lyon depuisplus de vingt ans, ainsi que sur les travaux de recherche développés au LEPS sur � la dimension expérimentale desmathématiques dans la perspective de leur apprentissage � (Dias & Durand-Guerrier, 2005) ; l'équipe DREAM aproduit un cédérom, EXPRIME (Aldon & al. 2010) présentant, dans le cadre des recherches du groupe, sept situationsde recherche pour la classe. Le dispositif de résolution collaborative de problèmes de l'équipe ResCo (Sauter & al.2008) repose sur des échanges entre des classes qui cherchent à résoudre le même problème, posé sous une forme nonmathématique. Pendant cinq semaines, les élèves échangent des questions, des réponses, des idées, des procédures etdes conjectures. Les deux premières semaines sont consacrées à l'exploration du problème et aux premières pistes versune mathématisation. Une relance recentre les recherches sur un problème mathématique commun, travaillé pendantles deux semaines suivantes. La session se termine par la rédaction d'un compte-rendu individuel de la recherche quiva alimenter le débat de clôture de la cinquième semaine. La spéci�cité de ce dispositif conduit à proposer des énoncésoriginaux répondant à un certain nombre de contraintes et que nous appelons � �ctions réalistes �. Nous présentonsbrièvement nos hypothèses et le cadre de notre travail.

Les hypothèses et le cadre de notre travail

A la suite de nombreux travaux de didactique des mathématiques conduits depuis les années 1980, les préconisationsinstitutionnelles en France mettent en avant depuis plusieurs années la nécessité de mettre la résolution de problèmeau c÷ur de l'activité mathématiques et de proposer en classe des problèmes issues d'autres domaines disciplinaires oude la vie courante pour permettre aux élèves de donner du sens aux mathématiques étudiées. S'inscrivant dans cettedynamique, et prenant acte du constat de la di�culté d'une mise en ÷uvre généralisée par les enseignants de telles pra-tiques, nous développons des situations au sens de Brousseau (2004) et proposons des formation visant à accompagnerles enseignants à mettre en place des activités de résolution de problèmes en classe (Aldon & Durand-Guerrier, 2009).Un des enjeux des situations que nous élaborons est de proposer un véritable travail de mathématisation, c'est à direune dévolution du choix des outils qui pourront être utilisés pour avancer dans la résolution des problèmes. Deux typesde situations ont été élaborées et testées : d'une part des situations pour lesquelles le questionnement est interne auxmathématiques (Gardes, 2013) et d'autre part des situations posées sous une forme non mathématique en laissant à




la responsabilité des élèves une phase de mathématisation préalable à la résolution mathématique du problème. Danscette communication, nous nous intéressons plus particulièrement au processus de mathématisation dans le cas dedeux situations. La première situation est issue du cédérom EXPRIME (Aldon & al. 2010) qui présente 7 situationsmathématiques, leurs analyses et leurs déclinaisons à di�érents niveaux d'enseignement. La deuxième situation est unexemple de �ction réaliste développée dans le cadre du dispositif de résolution collaborative de problème.

Un exemple en théorie élémentaire des nombres

Le premier exemple, tiré du cédérom Exprime, concerne le problème mathématique suivant :

Trouver tous les nombres entiers qui sont la somme d'au moins deux nombresentiers naturels consécutifs.

L'énoncé du problème en théorie élémentaire des nombres

Un objectif de notre travail est de repérer les connaissances mathématiques qui peuvent être mobilisées à di�érentsniveaux de classe dans une situation didactique (Brousseau 2004) élaborée pour permettre aux élèves de rentrerdans une véritable recherche s'appuyant sur des connaissances naturalisées et permettant de construire de nouvellesconnaissances dans le va et vient entre l'expérience sur des objets en cours de construction et les théories sous-jacentes

Dans ce qui suit, nous montrons di�érentes approches possibles et les connaissances mobilisées ou à mobiliser pourprolonger le raisonnement.

Une expérimentation numérique peut rapidement conduire à la conjecture que les puissances de 2 ne seront pasatteintes ; cette approche met en ÷uvre des connaissances sur les calculs de sommes d'entiers tant d'un point de vue ducalcul mental que du calcul ré�échi. L'utilisation de la calculatrice peut faciliter les contrôles. Cette approche est trèsféconde pour trouver la conjecture mais insu�sante pour la démontrer. La mathématisation du problème passe alorspar le choix d'autres outils. Une résolution algébrique cherchant à construire une formule explicite de la somme dedeux entiers consécutifs, de trois entiers, etc. conduit à trouver expérimentalement la forme de chacune de ces sommeset réciproquement de montrer que tous les nombres de cette forme sont atteints ; par exemple, avec deux nombresconsécutifs, on atteint tous les impairs :

n+ n+ 1 = 2n+ 1 est impairsi p est impair, p = 2n+ 1 = n+ (n+ 1).Avec trois nombres consécutifs :(n− 1) + n+ (n+ 1) = 3n et les multiples de 3 sont atteintssi p est un multiple de 3, alors p = 3n = (n− 1) + n+ (n+ 1)S'engager dans cette stratégie conduit d'une part à dégager des sous-problèmes, à se poser le problème de la

démonstration, de la preuve, de la réciproque, à observer des invariants et/ou des relations de récurrence conjecturéeset construites sur les résultats de l'expérience. Les connaissances mathématiques en jeu et qui sont travaillées sontnombreuses : l'utilisation de la lettre pour désigner un nombre, des éléments de calcul algébrique, les décompositionsdes nombres entiers naturels, les entiers pairs, impairs et leur caractérisation � algébrique �, opérationnelle opposée àla caractérisation numérique, la divisibilité et les multiples d'un nombre, là encore regardé d'une façon algébrique. Cesconnaissances a�ermies par les aller-retour avec l'expérience sont alors su�santes pour construire une démonstrationde la conjecture, pour peu que la formule de Gauss de la somme des n premiers entiers naturels : Sn=1+2+3+. . .+nsoit connue (nous donnons la preuve correspondante en annexe 1).

Cette analyse (incomplète, puisqu'on pourrait aussi analyser les apports de technologies, calculatrices, tableurs,etc. dans la recherche) met en évidence la nécessaire mathématisation du problème pour aller au-delà des simplesconstats. Cette mathématisation peut utiliser des outils mathématiques et des compétences variées et le problèmeconduit les élèves à � manipuler �leurs connaissances et à les approfondir.

Un exemple de �ction réaliste

Pourquoi des �ctions réalistes ?

Comme nous l'avons dit en introduction, le dispositif de résolution collaborative de problème présente l'originalitéde faire travailler des classes en réseau sur une durée de cinq semaines en mettant en place dans une première phaseun jeu de questions réponses entre deux ou trois classes comme préliminaire à un travail de mathématisation. Le




travail mathématique est alors engagé par le biais d'une relance envoyée aux élèves par un enseignant chercheurmembre du groupe. De ce fait, les problèmes qui sont proposés chaque année ne sont pas posés directement sous formemathématique, si bien qu'une fois envisagé un problème mathématique ou une famille de problème mathématiquepour la session de formation annuelle, se pose la question de sa contextualisation en prenant en compte plusieurscontraintes :

1. La contextualisation doit dissimuler su�samment le problème mathématique pour qu'il ne soit pas identi�éimmédiatement par les élèves (autrement dit, on ne va pas se contenter d'un simple habillage du problème).

2. Une exploration par un moteur de recherche classique ne doit pas conduire immédiatement au problème ; leproblème doit donc être énoncé sous une forme originale.

3. Le problème doit pouvoir être posé avec pro�t à des élèves dès le début du collège jusqu'à la �n du lycée 5.

4. Le champ des possibles pour la mathématisation du problème est su�samment ouvert pour que le jeu desquestions réponses soit riche et pertinent ; autrement dit, di�érents problèmes mathématiques peuvent émerger,que les élèves peuvent tous envisager selon le niveau de classe.

5. La relance doit pouvoir être comprise par les élèves comme répondant à certaines de leurs questions et comme�xant de manière non arti�cielle certains des éléments du problème.

La prise en compte de l'ensemble de ces contraintes et l'observation naturaliste de notre manière de travailler àcette élaboration nous a conduits à introduire la notion de � �ction réaliste �pour décrire des situations répondantautant que faire se peut à l'ensemble de ces caractéristiques.

Présentation du problème de l'artiste

L'exemple du problème proposé par ResCo en 2009-2010 va nous permettre d'illustrer ce qui précède. Le problèmemathématique initial est le problème classique du nombre de régions dans un disque :

On place n points sur un cercle. Combien de régions détermine-t-on à l'intérieurde ce cercle en joignant les points deux à deux ?

L'énoncé du problème mathématique initial

Plusieurs contextes possibles ont été envisagés pour aboutir à une �ction réaliste. Finalement, c'est l'idée de laréalisation d'une ÷uvre contemporaine qui a été retenue et qui a permis de faire émerger le � problème de l'artiste�sous la forme suivante, comme il a été présenté aux élèves.

Un artiste contemporain veut réaliser une ÷uvre sur un support rond, en plan-tant des clous sur le pourtour et en tendant des �ls entre les clous. Il se proposede peindre chaque zone d'une couleur di�érente. De combien de couleurs aura-t-il besoin ?

L'énoncé du problème de l'artiste

Le problème de l'artiste, une �ction réaliste ?

Le problème de l'artiste a été construit pour remplir les contraintes identi�ées ci-dessus ; sa mise en ÷uvre dansdeux sessions de résolution collaborative (novembre-décembre 2009 (16 classes de la 6ème à la Terminale) et en janvier-février 2010 (22 classes de la 6ème à la seconde)) a con�rmé que c'était bien le cas. En e�et, les élèves ne reconnaissentpas immédiatement le problème mathématique sous-jacent (critère 1) ; nous nous sommes assurés que la recherche surInternet ne renvoyait pas à ce problème (critère 2) ; il peut être proposé de la 6ème à la Terminale (critère 3) voiremême au-delà. En outre, le champ des possibles pour la mathématisation est su�samment ouvert et le jeu de questionsréponses est consistant (critère 4) ; par exemple : a) le support est un disque, ou un autre objet rond (sphère, toreetc.) - b) on choisit ou pas de néglige la taille des objets (les clous sont assimilés à des points, les �ls tendus sont dessegments de droite) ou pas - c) chaque clou est relié à tous les autres, ou pas - d) les clous sont placés de manièrerégulière sur le bord du disque ou non - e) il existe ou il n'existe pas de points communs à plus de deux cordes etc.

Selon les valeurs prises par les variables indiquées, on aboutit à des problèmes mathématiques di�érents ; toutesces variables ont donné lieu à des questions pertinentes d'élèves ; en prenant en compte les contenus de ces échanges,

5. C'est à dire pour des élèves de 11 ans (début du collège) à 18 ans (�n du lycée).




la relance (voir annexe 3) a �xé les valeurs de ces variables a�n de conduire au problème classique de régionnement dudisque (on cherche le nombre maximal de zones), ceci de manière non arti�cielle (critère 5). L'analyse des échanges dequestions et de réponses est détaillée dans Ray (2013) ; elle montre qu'une dévolution du processus de mathématisationa e�ectivement eu lieu lors de cette première phase de recherche : sans qu'un modèle unique ne soit imposé, un travailsur la représentation des objets réels a fait émerger plusieurs candidats-modèles (même s'ils restent incomplets et pourla plupart implicites).

Conclusion

Les deux types de situation que nous avons présentés correspondent à deux approches complémentaires du jeuqui se joue dans la confrontation à un problème susceptible de relever d'un traitement mathématique. Dans les deuxcas, le choix des problèmes et l'organisation du travail des élèves sont contrôlés minutieusement a�n de permettre ladévolution du processus de mathématisation. Les expérimentations faites tant avec les problèmes de EXPRIME ou les�ctions réalistes élaborées par ResCo tendant à montrer que le rapport des élèves aux mathématiques évolue pour serapprocher de ce qui est préconisé par l'institution en accord avec les résultats des travaux internationaux de rechercheen éducation mathématique sur la résolution de problèmes. Pour prolonger ce travail, des questions concernant lesapprentissages des mathématiques se posent encore, notamment : les problèmes de recherche qui développent une formed'acquisition des savoirs font ils progresser les élèves dans les autres domaines de l'activité mathématique ? Commentles élèves réinvestissent-ils dans d'autres cadres les compétences et les connaissances développées ?

REFERENCES

Aldon, G., Durand-Guerrier, V. (2011). Exprime, une ressource pour les professeurs. In Kuzniak, A., Sokhna,M. (Eds) Actes du colloque Espace mathématique francophone : Enseignement des mathématiques et développement :enjeux de société et de formation, Université Cheikh Anta Diop de Dakar

Aldon, G., Cahuet, P.-Y., Durand-Guerrier, V., Front, M., Krieger, D., Mizony, M., & Tardy, C. (2010). Expéri-menter des problèmes de recherche innovants en mathématiques à l'école. Cédérom, INRP.

Brousseau, G. (2004). Théories des situations didactiques. La pensée sauvage, Grenoble.Dias T., Durand-Guerrier V. (2005) Expérimenter pour apprendre en mathématiques, Repères IREM, 60, pp. 61-78Gardes M-L. (2013) Étude de processus de recherche de chercheurs, élèves et étudiants, engagés dans la recherche

d'un problème non résolu en théorie des nombres. Thèse de doctorat. Université de Lyon 1.Ray, B. (2013) Les �ctions réalistes : un outil pour favoriser la dévolution du processus de modélisation mathé-

matique. Une étude de cas dans le cadre de la résolution collaborative de problème. Mémoire de Master 2 RechercheHistoire, Philosophie et didactique des Sciences, Universités Lyon 1 et Montpellier 2.

Sauter M., Combes M.-C., De Crozals, A., Droniou J., Lacage M., Saumade H., Théret D., (2008) Une communautéd'enseignants pour une recherche collaborative de problèmes,Repères IREM. 72., 25-45

Annexe1 : la preuve du problème de théorie élémentaire des nombres

N étant un entier naturel, on cherche s'il existe deux entiers naturels a et b tels que

N = Sa+b−1 − Sa−1

Ce qui conduit à2N = (a+ b)2 − a− b− a2 + a = b(2a+ b− 1)

On peut alors raisonner sur la parité de l'entier b.Si b est pair : 2a+ b− 1 est impair.Si b est impair : 2a+ b− 1 est pair.Par conséquent, les deux entiers b et 2a+ b− 1 ne sont pas de même parité et comme leur produit est égal à 2N ,

cela entraîne que N possède un facteur premier impair : N n'est donc pas une puissance de 2.Réciproquement, 2N est le produit d'un nombre impair et d'un nombre pair et 2N = b(2a+ b− 1)si i < p alors il su�t de poser b = i et p = 2a+ b− 1 , soit a = (p− b+ 1)/2si i > p alors il su�t de poser b = p et i = 2a+ b− 1, soit a = (i− b+ 1)/2




La conjecture est ainsi complètement démontrée, et cette démonstration donne un procédé pratique pour déterminera et b entiers naturels tels que N = a+ (a+ 1) + (a+ 2) + · · ·+ (a+ b− 1).

Annexe 2 : La relance du problème de l'artiste

Montpellier, le 24 janvier 2010Le problème de l'ArtistePistes pour poursuivre la rechercheBonjour à tous et à toutes,Dans toutes les classes, vous avez déjà bien travaillé sur le problème de l'Artiste que nous vous avons proposé et

plusieurs pistes possibles ont été envisagées.On voudrait pouvoir donner une réponse précise à l'Artiste a�n de l'aider à faire ses choix pour réaliser son ÷uvre.Pour cela, on se propose de traiter mathématiquement le Problème de l'Artiste.Dans ce but, je vous propose de considérer que :

1. Le nombre de couleurs est le nombre de zones.

2. On cherche une solution générale, c'est-à-dire qu'on cherche le nombre maximum de zones en fonction du nombrede clous.

3. Le support de l'÷uvre est un disque et les clous sont répartis sur sa circonférence.

4. La taille du support est su�sante pour que l'on puisse négliger la taille des clous et l'épaisseur des �ls. Parconséquent, on assimile les clous à des points, et les �ls tendus à des segments de droite.

Je vous souhaite à tous et à toutes une très bonne poursuite de la recherche.Viviane DURAND-GUERRIERUniversité Montpellier 2Département de Mathématiques




4.6 Proposition d'ingénierie pour l'étude de la proportionnalité par con-frontation à la non-proportionnalité via des manipulations

Valérie Henry, Pauline Lambrecht

UNamur et CREM (Belgique)

Résumé : La thèse de Pauline Lambrecht se base sur une séquence destinée à favoriser l'apprentissage de la proportionnalité et

ce, par l'introduction de manipulations et la confrontation à la non-proportionnalité. Ainsi, l'observation de la variation du volume d'un

cylindre en fonction de sa hauteur dans un premier temps et en fonction de son diamètre dans un second temps amène les élèves à construire

les caractéristiques d'un phénomène proportionnel par comparaison avec un phénomène qui ne l'est pas. Le cadre de la recherche ainsi que

quelques éléments de la validation interne (analyses a priori et a posteriori) sont abordés dans ce texte.

Abstract : Pauline Lambrecht's thesis is based on a sequence intended to encourage proportionality's learning and this, by the

introduction of manipulations and comparison to non-proportionality. Thus, the observation of the volume's variation of a cylinder as a

function of its height in a �rst time and according to its diameter in a second time leads students to build the characteristics of proportional

phenomenon compared with a phenomenon which is not. The research framework and some elements of the internal validation (a priori

and a posteriori analysis) are discussed in this text.

Présentation du problème

Cet article présente un travail de thèse en cours à l'UNamur en Belgique. L'ingénierie développée dans ce cadreest en lien avec une recherche menée au Centre de Recherche sur l'Enseignement des Mathématiques (CREM) deNivelles en Belgique. Ces trois dernières années, une équipe de chercheurs a mis au point des activités, appelées Math& Manips, intégrant des manipulations destinées à diverses tranches d'âge de l'enseignement (2 ans et demi à 18 ans)[9]. L'intérêt de cette recherche est de présenter aux enseignants l'apport d'une activité expérimentale dans le processusde construction des savoirs mathématiques.

Ces séquences d'apprentissage présentent une forte composante a-didactique et visent à provoquer chez les élèves dela curiosité par des expérimentations dont les résultats semblent en contradiction avec leurs connaissances antérieures.Elles doivent amener les élèves à entrer dans un processus de questionnement visant à faire émerger un modèle quicorrespond au mieux à la réalité de la situation.

La séquence présentée dans cet article est l'une de ces activités. Elle est destinée aux élèves du début du secondaire(12-13 ans) et propose de confronter une situation de proportionnalité à une autre qui ne l'est pas.

Dans cette séquence, le passage de l'expérimental aux modèles mathématiques se fait dans di�érents contextesa�n de favoriser le passage d'un registre de représentation sémiotique à un autre. En e�et, l'activité expérimentaledébouche nécessairement sur un relevé d'informations qui doivent être traitées de diverses manières. Les résultats sontdécrits dans le langage courant, intégrés dans des tableaux de nombres et interprétés sous forme de graphiques.

Articles publiés sur le sujet

De nombreux documents ayant trait à l'apprentissage de la proportionnalité ont été écrits ces dernières annéeset ont nourri notre ré�exion. Ainsi, l'ouvrage de Boisnard, Houdebine, Julo, Kerb÷uf et Merri [2] décrit ladi�culté de cette notion et présente notamment une classi�cation des problèmes de proportionnalité. Dans la publi-cation de Nowak, Tran et Zucchetta [13], on retrouve également di�érentes procédures liées à la proportionnalité.Hersant étudie dans sa thèse [10] les problèmes de proportionnalité ainsi que l'utilisation d'un logiciel pour aiderà l'apprentissage de la proportionnalité. Une étude belge menée par Géron, Stegen & Daro[8] s'attache à l'en-seignement de la proportionnalité. Les auteurs ont rassemblé dans ce document des données de diverses recherches endidactique des mathématiques et classent également les problèmes de proportionnalité suivant une typologie proposéepar Vergnaud (proportionnalité simple et directe, proportionnalité simple composée et proportionnalité multiple).

Plusieurs articles de revues ont également traité la proportionnalité. Citons entre autres ceux de Comin [4] et deDupuis et Pluvinage [6] dans Recherches en Didactique des Mathématiques et celui de Hoyles, Noss et Pozzi [11]dans Journal for Research in Mathematics Education par exemple. Notons qu'un groupe international de la Psychologyof Mathematics Education s'intéresse à l'étude de la proportionnalité[8].

La plupart de ces textes soulèvent la di�culté de l'apprentissage de cette notion, souvent liée à la multiplicité descatégories de problèmes. Notre angle de travail a fortement été inspiré par la lecture de travaux relatifs à la prégnance




du modèle linéaire (De Bock, Van Dooren, Janssens & Verschaffel [5]) dont voici un exemple : � Mama put3 towels on the clothesline. After 12 hours they were dry. Grandma put 6 towels on the clothesline. How long did ittake them to get dry ? 6 �.

La séquence élaborée vise ainsi principalement à ébranler les conceptions initiales des apprenants par rapport à larémanence du modèle linéaire.

Objectifs du travail et questions de recherche

Dès le début de ce travail, deux axes nous ont intéressés. D'un côté, l'insertion de situations de non-proportionnalitédans l'étude de la proportionnalité et de l'autre, l'intégration de manipulations pour les apprentissages liés à laproportionnalité. Cela nous a menés aux questions de recherches suivantes.

� Une séquence intégrant une situation de non-proportionnalité permettant la confrontation à une situation deproportionnalité est-elle un apport pour les apprentissages liés à la proportionnalité ?

� Une séquence intégrant des manipulations permettant la confrontation aux perceptions initiales des élèves est-elleun apport pour les apprentissages liés à la proportionnalité ?

� Une séquence intégrant situation de non-proportionnalité et manipulations permet-elle d'améliorer l'aptitude desélèves à choisir un modèle adéquat pour traiter les diverses situations rencontrées ? Et à long terme ?

Nous sommes e�ectivement convaincus que tant les manipulations que la confrontation de la proportionnalité à lanon-proportionnalité peuvent amener les élèves à un meilleur apprentissage de cette notion. C'est pourquoi la séquenced'apprentissage brièvement présentée à la section 4.6 a été mise au point

Cadre théorique

Notre ingénierie a été élaborée dans le cadre de la théorie des situations didactiques de Brousseau [3] et enutilisant la méthodologie décrite par Artigue [1] comme précisé dans la section ci-dessous. De plus, notre travails'appuie sur la conversion de registre de représentation au sens de Duval [7].

Méthodologie et description du dispositif expérimental

Pour construire l'ingénierie dont il est question dans ce papier, nous avons suivi le processus présenté par Artigue[1] pour lequel elle distingue quatre phases.

La première concerne les analyses préalables. Comme nous l'avons souligné dans la section 4.6, de nombreuxtravaux ont déjà été menés quant à l'apprentissage de la proportionnalité mais la lecture des travaux de De Bock,Van Dooren, Janssens & Verschaffel [5] sur l'illusion de linéarité ainsi que le regard sur le contenu de di�érentsmanuels belges nous ont amenés à préciser notre approche.

La deuxième phase reprend la conception et l'analyse a priori. La théorie des situations didactiques de Brousseau[3] a guidé l'écriture de la séquence d'apprentissage. Nous souhaitions notamment créer un milieu qui permette dedévoluer la situation aux élèves en étant confrontés à leurs préconceptions erronées. Quelques composantes de l'analysea priori sont exposées dans la section suivante.

Les troisième et quatrième phases sont respectivement celle de l'expérimentation et celle de l'analyse a posterioriet de l'évaluation (validation). Dans notre cas, ces deux phases se sont répétées car, a�n de mettre au point cetteséquence, nous l'avons testée de nombreuses fois en l'adaptant au fur et à mesure des expérimentations. Des élémentsde l'analyse a posteriori sont également présentés par la suite.

Comme présenté à la section 4.6, l'une des questions de recherche de la thèse porte sur l'apport d'une situa-tion de non-proportionnalité dans l'étude de la proportionnalité. Pour traiter cette question, nous avons notammentmis en place un protocole d'expérimentation basé sur l'introduction de la séquence dans certaines classes (classes-expérimentales) tandis que d'autres classes (classes-témoins) poursuivaient leur enseignement. Les enseignants dedeux écoles ont accepté de nous accueillir dans leurs locaux, ce qui a donné l'occasion de mener de nombreuses autresexpérimentations.

6. Maman a placé 3 serviettes sur la corde à linge. Après 12 heures elles étaient sèches. Grand-mère a placé 6 serviettes sur la corde àlinge. Combien de temps ont-elles pris pour sécher ?




L'activité proposée amène les élèves à étudier la variation du volume d'un cylindre en fonction de sa hauteur dansun premier temps et en fonction de son diamètre dans un second temps. Cela permet d'observer et de construire avecles élèves les caractéristiques d'un phénomène proportionnel par comparaison avec un phénomène qui ne l'est pas.Un enjeu important de l'activité est en e�et d'établir le lien entre phénomène linéaire, tableau de proportionnalité etgraphique en ligne droite d'une part et phénomènes non linéaires, tableaux de non-proportionnalité et graphiques defonctions non linéaires d'autre part.

Une situation d'introduction permet aux élèves de se familiariser avec le matériel et de mettre en évidence despoints essentiels d'une démarche scienti�que : placement des repères, précision, utilisation d'un matériel adéquat, etc.Ensuite, dans une première partie, on demande de remplir un cylindre jusqu'à certaines hauteurs après avoir estimé,pour chaque cas, le nombre de mesurettes nécessaires à cette opération. Les élèves écrivent leurs résultats dans untableau et il leur est demandé de repérer et d'écrire les di�érents liens qu'ils observent entre les valeurs du tableau, enles symbolisant par des �èches.

Dans une deuxième partie, on propose de remplir des cylindres de diamètres simple, double et triple jusqu'àune même hauteur après avoir demandé, dans chacun des cas, une estimation du nombre de mesurettes nécessaires.L'importance de la démarche qui consiste à ne faire varier qu'une seule grandeur à la fois est explicitée. De nombreuxélèves s'attendent à obtenir des rapports simple, double et triple comme lorsqu'ils font varier la hauteur.

À nouveau, il est demandé de placer les résultats obtenus pour la variation du diamètre dans un tableau. Les liensdécouverts entre ces di�érentes valeurs doivent être mis en évidence a�n d'en dégager les valeurs correspondant à desdiamètres par exemple quatre ou cinq fois plus grands de celui de départ.

Pour chacune des deux situations, les élèves placent les résultats dans un graphique. Sur base des tableaux etgraphiques construits au cours de l'activité, l'enseignant institutionnalise les savoirs visés, au sens de Brousseau [3], enréalisant une synthèse qui met en évidence les caractéristiques permettant de distinguer les phénomènes proportionnelsdes autres.

La description complète de cette séquence d'apprentissage se trouve dans le rapport de la recherche � Math &Manips � du CREM [9].

Analyse a priori

Une analyse approfondie de la séquence d'apprentissage a été menée avant de l'expérimenter dans les classes.Comme il n'est pas possible de faire état de l'entièreté de cette analyse ici, nous avons choisi d'en présenter deuxpoints importants.

Lors des deux phases adidactiques de la séquence, la dévolution de la situation est prévue grâce aux composantesspéci�ques du milieu : les �ches de travail, le matériel reçu et les di�érents registres utilisés doivent permettre auxélèves d'accepter le problème comme leur. L'intention d'enseigner du professeur disparaît ainsi, du moins en apparence.

Au terme de la séquence, l'enseignant institutionnalise les savoirs en considérant trois facteurs. Il doit mener lesélèves des manipulations à la conceptualisation, leur faire di�érencier les deux modèles issus des deux situations etamener le vocabulaire adéquat lors de la synthèse.

Analyse a posteriori

Les nombreuses expérimentations ont permis de dégager divers éléments lors de l'analyse a posteriori, trois enressortent.

Le premier concerne les �ches de travail. Leur clarté ainsi que le choix du vocabulaire ont dû être modi�és a�n depermettre une meilleure dévolution de la situation.

Le deuxième élément touche aux di�érents liens repérés par les élèves dans leurs tableaux de résultats. Dans celuiissu de la variation de la hauteur du cylindre, certains remarquent des liens de type multiplicatif et d'autres les écartsconstants. Lors des premières expérimentations, les élèves adoptaient implicitement ces écarts constants comme unepropriété de la proportionnalité. Pour éviter cette erreur, il a donc été nécessaire d'intégrer dans la séquence unediscussion au cours de laquelle les élèves conviennent avec l'enseignant que la multiplication est plus générale en cesens que les additions dépendent des valeurs initiales tandis que les liens multiplicatifs internes sont les mêmes pourtous les groupes, quelles que soient les valeurs de départ. De plus, les liens de type multiplicatif permettent de trouverla réponse pour n'importe quelle hauteur sans recourir aux étapes intermédiaires.




Le troisième élément se rapporte aux estimations lors de la variation du diamètre d'un cylindre. Au cours despremières expérimentations, il n'était pas explicitement demandé aux élèves d'écrire leurs estimations. Ils avaientalors tendance à occulter leurs idées premières, ce qui réduisait l'e�et positif du con�it cognitif induit par le milieu.L'introduction d'un tableau dans les �ches de travail permettant aux élèves d'écrire leurs estimations leur donnel'occasion de se confronter à la non-concordance des résultats avec leurs prévisions. Cela les incite à se poser davantagede questions et à chercher des justi�cations aux résultats obtenus.




[1] Artigue M. (1988). Ingénierie didactique. Recherches en didactique des mathématiques, 9 (3), 281-308.

[2] Boisnard D., Houdebine J., Julo J., Kerb÷uf M.-P., Merri M. (1994). La proportionnalité et ses prob-lèmes. Paris : Hachette Éducation.

[3] Brousseau G. (1998). Théorie des situations didactiques. Grenoble : La Pensée Sauvage.

[4] Comin E. (2002). L'enseignement de la proportionnalité à l'école et au collège. Recherches en Didactique desMathématiques, Vol.22/2.3, La Pensée Sauvage éditions, pp.135-182.

[5] De Bock D., Van Dooren W., Janssens D. & Verschaffel L. (2007). The illusion of linearity. FromAnalysis to improvement. New York : Springer.

[6] Dupuis C., Pluvinage F. (2003). La proportionnalité et son utilisation. Recherches en Didactique des Mathé-matiques, Vol.2/2, La Pensée Sauvage éditions, pp.165-212.

[7] Duval R. (1996). Quel cognitif retenir en didactique des mathématiques ? Recherches en Didactique des Mathé-matiques, Vol.16/3, La Pensée Sauvage éditions, pp.349-382.

[8] Géron C., Stegen P. & Daro S. (2007). L'enseignement de la proportionnalité : liaison primaire-secondaire.http://www.enseignement.be/download.php?do_id=2712&do_check= (HYPOThèse)

[9] Guissard M.-F., Henry V., Lambrecht P., Van Geet P., Vansimpsen S. & Wettendorff I. (2013).Math & Manips - Des manipulations pour favoriser la construction des apprentissages en mathématiques, rapportde recherche téléchargeable sur www.crem.be

[10] Hersant M. (2001). Interactions didactiques et pratiques d'enseignement, le cas de la proportionnalité au collège.Thèse, Paris : Université Paris 7 - Denis Diderot.

[11] Hoyles C., Noss R. & Pozzi S. (2001). Proportional reasoning in nursing practice. Journal for Research inMathematics Education, pp.4-27.

[12] Post T., Cramer K., Harel G., Kieren T. & Lesh R. (1998). Research on rational number, ratio andproportionality. Proceedings of the twentieth annual meeting of the north american chapter of the internationalgroup for the Psychology of Mathematics Education PME-NA XX, Vol.1, Raleigh NC, pp.89-93.

[13] Nowak M.-Th., Tran D., Zucchetta J.-F. (2001). La proportionnalité dans tous ces États. Lyon : IREM deLyon.


http://www.enseignement.be/download.php?do_id=2712&do_check=



4.7 An insight on children's ideas about the inverse relation betweenquantities

Ema Mamede*, Isabel Vasconcelos**

*Institute of Education - University of Minho, Portugal**Federal University of Rio Grande do Sul, Bairro Rio Branco, Porto Alegre, Brasil

Résumé : Cette étude visant à analyser comment la relation inverse entre la taille et le nombre de pièces dans des situations de division

est lié à la notion de fraction sur le sens quotient et le sens partie-tout. Nous avons analysé les réponses données par les enfants dans les

3ème et 4ème barres (8-10 ans) à une enquête par questionnaire, qui a été résolu individuellement. Les résultats suggèrent que les fractions

présentées avec le sens de quotient promouvoir la compréhension de la relation inverse entre la taille et le nombre de pièces plus que

ceux présentés avec le sens de partie-tout, ou même avec la division partitif ou situations quotitives. Contrairement à cela, des problèmes

fraction d'équivalence avec des signi�cations de partie à tout obstacle à cette compréhension. Les sens quotient et partie-tout permettent

le transfert de connaissances, la compréhension de la relation inverse entre la quantité est impliqué lors de la commande fraction. Dans les

situations de division, il existe une association signi�cative entre la division partitif et la division quotitive. Une analyse des justi�cations

des enfants pour les réponses aux problèmes étant donné nous permet de rassurer que les bonnes réponses n'ont pas été obtenus au hasard,

mais plutôt sont pris en charge par le raisonnement correct sur les quantités impliquées dans le problème.

Abstract : This study analyzes how the inverse relationship between size and number of parts in division situations is related to the

concept of fraction over the quotient and part-whole interpretations. A survey by questionnaire was carried out on 72 children at the 3rd

and 4th grades (aged 8-10). Results suggest that the fractions presented in the quotient interpretation promote more the understanding of

the inverse relationship between size and number of parts than when the part-whole interpretation is involved, or even when the partitive

and quotitive division situations are involved. Contrary to this, fraction equivalence problems with part-whole interpretation hinder this

understanding. Quotient and part-whole interpretations allow the transference of knowledge regarding the understanding of the inverse

relationship between quantities when fraction ordering is involved. In division situations, a signi�cant association is found between partitive

division and quotitive division. An analysis of the children's justi�cations for the answers given to the problems enables us to reassure

that the correct answers were not obtained randomly, but rather are supported by a correct reasoning about the quantities involved in the

problems.

Theoretical framework

This study investigates the understanding of the inverse relationship between size and number of parts in partitiveand quotitive division situations and with fraction in part-whole and quotient situations, with 3rd and 4th gradestudents. Considering the mathematical contexts approaching of the inverse relationship between quantities, fractionsand division situations are emphasized.

Several studies focused on the students' understanding of the inverse relationship between quantities. Some arefocused on the concept of division (see Correa, Nunes & Bryant, 1998 ; Squire, & Bryant, 2002 ; Correa, 2004 ; Spinillo& Lautert, 2011 ; Mamede & Silva, 2012), others focused on the concept of fraction (see Behr, Wachsmuth, Post &Lesh, 1984 ; Kornilaki & Nunes, 2005 ; Mamede, Nunes & Bryant, 2005 ; Nunes & Bryant, 2008 ; Magina, Bezerra &Spinillo, 2009 ; Mamede & Cardoso, 2010 ; Hallett, Nunes, Bryant & Thorpe, 2012).

In order to understand the aspects involved in division situations (see Correa, Nunes and Bryant, 1998), it isimportant to distinguish the di�erence between partitive division and quotitive division. In partitive division, thequantity is divided between the number of recipients, and the part received by each recipient is the unknown part(e.g., John has 8 sweets to be shared between 4 children. How many sweets each child will receive ?). In quotitivedivision, a quantity is divided and what each recipient will receive is already known ; what is left to know is thenumber of recipients (e.g., Mary has 6 sweets and will give 2 sweets to each child. How many children will receivesweets ?).

When the understanding of inverse relationship between divisor and quotient in division situations is investigated,it makes sense to consider two kinds of tasks when one of the dimensions is held constant (dividend or divisor). Forthe �rst situation, the divisor can be held constant and the dividend can be changed. In this case, children mustunderstand that the bigger the whole, the bigger the parts are, if the number of parts is held constant. For the secondsituation, the dividend is held constant and the divisor is changed. The divisor corresponds to the number of recipientsor the size of the quote. In either case, the inverse relationship is applied : the bigger the number of parts, the smallerthe size of the part, or vice-versa.




Correa, Nunes and Bryant (1998) investigated the development of the concept of division by 61 children aged 5-7,by analyzing how they understand the inverse relationship between divisor and quotient in partitive and quotitivedivision tasks. Authors investigated how children understood the relationship of quantity between the three terms ina division when the dividend was constant and the divisor changed. In partitive division, results showed that 30%of the children aged 5 were able to put into words in their justi�cations the inverse relationship between divisor andquotient, when the dividend is held ; the same was observed in about 55% of the 6-years-old children and 85% forthe group aged 7. Regarding the mistakes made, the authors did not �nd signi�cant di�erences in 5- and 6-years-oldchildren, and the 7-years-old children had the smaller percentage of mistakes made. In quotitive division, the authorsshowed results above expectations in di�erent age groups : 50% of the 5-years-old children, 38% of the 6-years-oldchildren and 40% of the 7-years-old children assessed correctly the inverse relationship between divisor and quotient.In the justi�cations given by the 5-years-old children, about 30% referred incorrectly to the direct relationship betweendivisor and quotient ; most of the 6-years-old children give logical-mathematical justi�cations, but about half of theminferred an incorrect direct relationship between divisor and quotient. The performance of 7-years-old children wassimilar to that of 6-years-old children.

The investigation carried out by Kornilaki and Nunes (2005) involved two studies : one with partitive divisiontasks and the other with quotitive division tasks, both with discrete and continuous quantities. The authors analyzedwhether children transfer their understanding of logical relationships of discrete quantities to continuous quantitieswith 96 children aged 5 to 7 years. In the problems proposed, the number of recipients varied producing two conditions :(1) in the condition of same divisors, the size of the divisor was the same ; and (2) in the condition of di�erent divisors,the number of recipients varied. The analysis detected that the condition of di�erent divisors was clearly more di�cultthan the condition of same divisors and that the inverse relationship between divisor and quotient is understood onlyafter the division equivalence principle. In partitive division tasks, 33% of the 5- and 6-years-old children's answerswere explained based on �the more recipients, the more they have�. Nevertheless, this answer decreased remarkablywith age, since only about 10% of the 7-years-old children used this incorrect reasoning. In quotitive division tasks,about 50% of the 5- and 6-years-old children and 25% of the 7-years-old children's answers were explained based on�the more recipients, the more they have�. This justi�cation was used by 75% of the children who answered incorrectly.Almost all children who answered correctly presented justi�cations based on the inverse relationship between divisorand quotient.

Kornilaki and Nunes (2005) argue that children understand more easily partitive division than quotitive division,because they use term-by-term correspondence as the procedure to solve this type of division, once it is more simplethinking about the inverse relationship than building each quote.

More recently, Mamede and Silva (2012) investigated children's understanding of partitive division with discretequantities with 30 children aged 4 and 5. In individual interviews, children were asked to make judgments in taskswith inverse relationship between divisor and quotient when the dividend is the same. Tasks involved division of 12and 24 discrete quantities by 2, 3 and 4 recipients. Results showed that children aged 4 and 5 have some idea aboutthe division, are able to estimate the quotient when the divisor changes and the dividend is constant, and are able tojustify their answers.

The inverse relationship between quantities is essential to understand the concept of fraction. However, this conceptis far from being easy for children. Research has been giving evidence that children struggles with the concept of rationalnumber (see Behr, Wachsmuth, Post & Lesh, 1984 ; Hart, 1981 ; Kerslake, 1986 ; Mack, 2001 ; Mamede & Cardoso,2010 ; Monteiro & Pinto, 2005). Besides, there is still a lack of research focused on children's understanding of theinverse relationship between quantities, which is essential for understanding the concept of fraction.

Information described by recent research (see Nunes, Bryant, Pretzlik, Evans, Wade Bell, 2004 ; Mamede, Nunes& Bryant, 2005 ; Magina, Bezerra & Spinillo, 2009) consider that the conceptual knowledge of fractions supposes :

1. the invariance principle, that is, dividing a whole in equal parts, while maintaining the initial quantity ;

2. the ability of representation, of being written as ab , where a and b are whole numbers (with b 6= 0) and the same

symbols can represent di�erent quantities (e.g., 12 of 8 and 1

2 of 12) ;

3. the understanding of equivalence logic ( 12 , 24 ,36 ) and ordering ( 12 > /frac13 > frac14 ) ; and

4. the area of more diverse and complex interpretations than the operations performed by children with naturalnumbers.

The literature presents di�erent classi�cations of interpretations or meanings for fractions. Kieren (1993, 1995)distinguishes four categories known as �sub-constructs� which are relevant for the knowledge about fractions : (1) quo-tient ; (2) measure ; (3) operator ; and (4) ratio. Berh, Lesh, Post and Silver (1984) rede�ned the previous classi�cation




establishing �ve �sub-constructs� understood to be su�cient to clarify the concept of rational number, which are :

1. part-whole ;

2. quotient ;

3. ratio ;

4. operator ; and

5. measure.

More recently, Nunes, Bryant, Pretzlik, Evans, Wade and Bell (2004) presented a classi�cation based on �situations�in which fractions are used, relying on the meaning of the magnitudes assumed in each case, considering :

1. part-whole ;

2. quotient ;

3. operator ; and

4. intensive quantities.

In the study reported here, this last classi�cation was adopted, concerning quotient and part-whole interpretations.Thus, for quotient interpretation or situation, a

b can represent the relationship between the number of recipients anditems to be distributed (e.g., 2

3 can represent 2 chocolate bars to be shared fairly by 3 children), but it also representsthe quantity of an item received by each recipient (e.g., 23 corresponds to the amount of chocolate received by eachchild). In the part-whole situation, a

b represents the relationship between the number of equal parts in which the wholeis divided and the number of these parts to be taken (e.g., 2

3 of a chocolate bar means that this was divided into 3equal parts and 2 of these parts were considered).

Studies focused on di�erent meanings of rational number have suggested that these a�ect di�erently how childrenunderstand fractions. Some authors argue that the quotient meaning favors the understanding of inverse relationshipbetween numerator and denominator of the fraction (see Stree�and, 1997 ; Mamede, Nunes & Bryant, 2005 ; Mamede,2007). Nunes et al. (2004) suggest that this understanding is facilitated for quotient meaning because numerator anddenominator are variables of di�erent nature.

Mamede, Nunes and Bryant (2005) investigated whether the fraction quotient and part-whole situations in�uencethe level of the children's performance in problem solving tasks. Eighty children participated in the study aged between6- and 7 year-olds, who have not had formal instruction on fractions, but some of them were already familiar withthe words �half� and �forth part� in social contexts. The authors analyzed how they understand fractions concerningpart-whole and quotient situations, in tasks related to equivalence, ordering, and labelling. Results indicated thecorrespondence (association established between one part and each recipient) and division of an item in equal partsas the most used procedures by children. Children presented success levels in ordering and fraction equivalence taskswith quotient situations, suggesting that they have some informal knowledge on the logic of fractions, developed intheir daily life, without school instruction.

As little children understand the inverse relationship between quantities in division situations and understand thelogical invariants (ordering and equivalence) of fractions it particular situations, it is important to know how theseaspects are related. In this study, it is considered the hypothesis that children who understand the inverse relationshipbetween divisor and quotient in division situations present better performance on understanding inverse relationshipbetween size and number of parts of a fraction in part-whole and quotient interpretations.

Thus, this study aims to analyze how the inverse relationship between size and number of parts in division situationsis related to the concept of fraction with quotient and part-whole situations. One tried to address three questions :

1. How do children understand the inverse relationship between size and number of parts in partitive and quotitivedivision situations ?

2. How do children understand this inverse relationship when fractions are involved in part-whole and quotientsituations ? And

3. is there a relationship between partitive and quotitive division situations and fractions in part-whole and quotientsituations ?

Methodology

Participants




To assess the children's understanding of the inverse relationship between quantities in division and fraction situ-ations, a survey by questionnaire was applied in the classroom to 72 children aged between 8 and 10 years, at the 3rd

and 4th grades from a public school in Braga, Portugal.TasksThe questionnaire included 22 tasks : 6 division problems (3 partitive division problems and 3 quotitive division

problems) ; 16 problems with fractions (8 problems in part�whole interpretation (4 of ordering and 4 of equivalence offractions) ; 8 problems in quotient interpretation (4 of ordering and 4 of equivalence of fractions)).

All fractions involved in the tasks were less than 1 and were the same for the problems presented in quotient andpart-whole interpretations. Tables 1 and 2 show an example of a task presented for each type of division and fractionsituation, respectively.

Division ProblemPartitive Mary and Louise have the same quantity of sweets. Mary will distribute her sweets by 3 children

and Louise will distribute hers by 4 children. Will the children at Mary's group receive more sweetsthan, less sweets than, or the same quantity of sweets as the children at Louise's group ?Explain your answer.

Quotitive John and Paul bought the same quantity de marbles. John will put 3 marbles in each bag andPaul will put 6 marbles in each bag. Will John need more bags than, less bags than, or the samequantity of bags as Paul ?Explain your answer.

Table 4.1 � Examples of tasks presented in division situations.

Problems Equivalence OrderingPart-whole situations Marco and Lara have each a pizza with the

same size. Marco divided his pizza into 5 equalparts and ate one part. Lara divided her pizzainto 10 equal parts and ate two parts. DidMarco eat more pizza than, less pizza than,or the same quantity of pizza as Lara ?Explain why.

Ana and Rita have each a chocolate with thesame size. Ana ate 1

2 of her chocolate andRita ate 1

3 of her chocolate. Did Ana eat morechocolate than, less chocolate than, or thesame quantity of chocolate as Rita ?Explain why.

Quotient situations Children share two same-sized cakes. Two girlsshare one cake fairly ; three boys share theother cake fairly. Does each girl eat more cakethan, less cake than, or the same quantity ofcake as each boy ?Explain why.

Two girls will share a chocolate bar and eachone will eat 1

2 of the chocolate. Three boys willshare a chocolate bar and each one will eat 1

3 ofthe chocolate. Does each girl eat more choco-late than, less chocolate than, or the samequantity of chocolate as each boy ?Explain why.

Table 4.2 � Examples of tasks presented in fractions situations

ProceduresThe questionnaire was solved individually and lasted for 40 minutes, being implemented and followed by the class

teacher. Each child received a booklet with one problem per sheet to be solved. In each problem, multiple-choicequestions were present, and the judgment for relative value of the quotients by using relations �more than/ less than/same quantity as� was favored. Questions were presented to the class and read by the researcher using PowerPointslides. Each child had to indicate the right answer on the booklet. The tasks used were adapted from the studies ofMamede, Nunes and Bryant (2005) and Spinillo and Lautert (2011).

Results

Results of the children's performances when solving the proposed tasks were analyzed, by assigning 1 to each rightanswer and 0 to each wrong answer. Table 3 presents the mean of the proportion of the correct answers and standarddeviations, according to the type of problem.




Quotient Part-whole DivisionOrdering Equivalence Ordering Equivalence Partitive Quotitive.80 (.27) .61 (.27) .59 (.35) .35 (.32) .43 (.40) .46 (.40)

Table 4.3 � Mean and (standard deviation) of the proportion of correct answers by type of problem.

Results suggest that, children seem to better understand the inverse relation between quantities when fractions inquotient situations are involved. Results also suggest that quotitive division seems to be easier than partitive division.The analysis by type of problem presented allows us to better understand the children's performance on the taskspresented. Graphs 1A-B present, respectively, the distribution of the ordering and equivalence of fractions problemscorrectly solved in quotient situation. Ordering problems seem to be more accessible to understand the inverse relationbetween the numerator and denominator. About 56.2% of the children answered correctly to all ordering problemsand 78.1% solved correctly at least 3 of the 4 problems ; 13.7% got all equivalence of fractions problems presented inquotient interpretation right, and 54.8% solved correctly 3 of the 4 problems of this type.

Figure 4.5 � Distribution of correct responses in ordering and equivalence problems of fractions presented in part-whole situations.

The fraction problems presented in part-whole interpretation seem to turn the understanding of inverse relationbetween the numerator and denominator even more di�cult to children. Graphs 2A-B present the number of correctanswers obtained in ordering and equivalence of fractions problems in part-whole interpretation, respectively.

Figure 4.6 � Distribution of correct responses in ordering and equivalence problems of fractions presented in part-whole situations.

In ordering problems presented with part-whole interpretation, only 23.3% of the children answered correctly to allproblems and about 56.2% answered correctly to 3 of the 4 problems presented. In equivalence problems, about 5.5%of the children answered correctly to all problems and 24.7% answered correctly 3 of the 4 problems. Concerning thedivision, children seem to struggle with partitive division situations on the inverse relation between quantities. Graphs3A-B present, respectively, the number of correct answers in partitive and quotitive division problems. In partitivedivision problems, the percentage of children who answered correctly all problems is about 24.7% ; 13.7% answeredcorrectly to 2 of the 3 problems, and 27.4% got only 1 problem correctly solved. In quotitive division problems, 26% ofthe children got all problems correctly solved ; 17.8% answered correctly to 2 of the 3 problems ; and 24.7% answeredcorrectly only to 1 problem.

These results suggest that the exploration of fractions in quotient and part-whole situations and the partitive andquotitive division situations contributes di�erently for the understanding of inverse relation between quantities.

A correlational analysis was conducted to identify potential associations between the inverse relation of quantities




Figure 4.7 � Distribution of correct responses in partitive and quotitive division situations.

Quotient Part-whole DivisionOrdering Equivalence Ordering Equivalence Partitive Quotitive

Ordering Quotient 1Equivalence Quotient .303** 1Ordering Part-whole .531** .196 1

Equivalence Part-whole .219 .206 .348** 1Partitive division .278* .088 .424** .337** 1Quotitive division .228 .099 .386** .356** .494** 1

*.p<.05 ; **.p<.001 ; Pearson's correlation coe�cient.

Table 4.4 � Correlations for each type of problem.

established in ordering and equivalence fractions problems and in partitive and quotitive division problems. Table 4summarizes the correlations registered for each type of problems.

It seems that there are di�erences on how the interpretations of fraction (quotient, part-whole) a�ect the under-standing of the inverse relationship between quantities. Children's performance in ordering and equivalence fractionsproblems presented in the quotient interpretation are related to each other ; ordering problems in quotient interpreta-tion are strongly related to the ordering ones presented in the part-whole interpretation. Possibly, this can be explainby the double representation of a fraction in a quotient situation referred previously (as in a quotient interpretationor situation, a

b can represent the relationship between the number of recipients and items to be distributed, but it alsorepresents the quantity of an item received by each recipient). The ordering problems in quotient interpretation arealso weakly related to partitive division problems. The equivalence problems presented in quotient interpretation onlyseem to be related to the ordering problems in quotient interpretation.

The ordering and equivalence problems in part-whole interpretation are related to each other, and are related topartitive and quotitive division problems. It is noteworthy that the success in partitive division problems is stronglyrelated to the success in quotitive division problems. Maybe this phenomenon occurs because, at this age, childrenalready have some consistent knowledge on multiplicative structures deriving from formal instruction.

The written justi�cations of the children's responses were analyzed to reach an insight on their reasoning. System-atizing these explanations, 5 categories of justi�cations were addressed : 1) inverse relationship �it attends to theinverse relation between the quantities involved in the problem, producing a valid justi�cation (e.g.,�[. . .] because hedivided his pizza into 2 equal parts and she divided hers into 4 equal parts and hers become smaller.�) ; 2) proportionalreasoning �it comprises a establishment of a proportional relation between the quantities of the problem, producing avalid argument (e.g., `They eat the same because there are 2 girls for 1 chocolate bar and the boys are the double ofgirls and they have the double of chocolate bars.�) ; 3 direct relationship �it sets a direct relation between the quantities(e.g., �He eats more because he has more cake, thus he eats more cake.�) ; 4) initial quantity - only corresponds tothe problem's initial quantity ignoring the relation between quantities (e.g., �Both eat the same because he has onepizza and she has one pizza�) ; and 5) inconclusive/invalid - corresponds to all inconclusive, inappropriate, or blankexplanations. Table 4.5 summarizes the percentages of each type of argument used by the children, according to thetype of problem.

Children performed better on ordering and equivalence of fractions problems presented in quotient interpretation.This interpretation seems to facilitate children's understanding of the inverse relation between quantities, allowingthem to solve correctly the problems and present a valid argument. Children's valid arguments were based on theinverse relation between the quantities involved in the problem and on proportional reasoning, conducting the children




Quotient (%) Part-whole (%) Division (%)Ordering Equivalence Ordering Equivalence Partitive Quotitive

Inverse relationship 91.8 56.2 49.3 17.8 38.4 42.5Proportional reasoning 1.4 15.1 2.7 6.8 0 9.6Direct relationship 2.7 13.7 9.6 42.5 32.9 30.1Initial quantity 1.4 6.8 26.0 8.2 15.1 6.8Inconclusive 2.7 8.2 12.3 24.7 13.7 11.0

Table 4.5 � Arguments used by the children according to the type of problems.

to correct responses. Figures 4.8 and 4.9 illustrate, respectively, examples of valid justi�cations presented when solvingfractions problems, in quotient and part-whole interpretations.

They eat the same because there are three girls for one cake and the boys and the cakes are in doubleBecause girls are two for one (chocolate bar) and the boys are the double with the double of chocolate bars.

Figure 4.8 � Valid arguments based on proportional reasoning when solving equivalence fraction problems in quotientinterpretation.

Marco divided his pizza into 2 equal parts and ate 1 part and Sara hers into 4 equal parts and also ate 1 part, butMarco ate less because he only ate 1

4Because Marco's pieces have double size of Rita's thus if Marco ate one and Rita ate 2, both of them had the same

amount.

Figure 4.9 � Valid arguments presented when solving ordering and equivalence fraction problems in part-wholeinterpretation.

Division problems also seem to help children to understand the inverse relation between quantities. However, inthese situations many children still establish a direct relation between quantities. These results suggest that the successlevels regarding the children's performances for the problems presented were not obtained randomly, since they seemto be followed by explanations supported by valid arguments.




Final remarks

Understanding the inverse relationship between quantities when fractions are involved is further facilitated whenquotient interpretation is involved rather than when the part-whole interpretation is involved. Consistent with previousstudies (see Mamede, Nunes & Bryant, 2005 ; Mamede, 2008), quotient interpretation still reveals to be important forthe understanding of inverse relation between quantities, regardless the age di�erences of the children. As Mamede,Nunes and Bryant (2005), also our results suggest that di�erent fraction interpretations involve di�erent levels ofunderstanding of the inverse relation between quantities for children.

The results of this study also suggest that children present higher comprehension levels about the inverse relationbetween quantities in quotitive division situations than the ones involved in partitive division. Probably, this can beexplained by the use of sharing procedures in quotitive division situations. It is an interesting result, as previousstudies reported in literature can o�er divergent results (see Correa, Nunes & Bryant, 1998 ; Kornilaki & Nunes,2005). Correa, Nunes and Bryant (1998) investigated the understanding of inverse relationship between quantitieswith 5-to 7-years- old children, and not with children who already had formal instruction on division and/or fractions.Their results highlight that little children understand this inverse relationship between divisor and quotient, whenpartitive division was involved. Also, the studies by Kornilaki and Nunes (2005) suggest that children have someideas on the inverse relation between divisor and quotient in partitive division tasks, when asked to judge the relativesize of the shared sets with 5- to 7-years-old children. The results of the present study suggest that 8- the to 10-years-old children also understand the inverse relation between quantities, but quotitive division seems to play a rolefacilitating this understanding. Perhaps, this might be due to the period of formal instruction to which these childrenwere exposed. Surprisingly, as fractions (that constitute another mathematical context where this inverse relationshipbetween quantities must be understood and applied) present success levels higher than the ones reported in divisionproblems.

The existence of associations between the kinds of problems where the inverse relation between quantities is involvedhighlights that fraction ordering and equivalence problems, presented with quotient interpretation, are related to eachother ; and the ordering problems with this interpretation are also related to the ordering ones presented with part-whole interpretation. It may suggest that children have some facility in applying transversely this knowledge on thesekinds of problems. This facility in understanding inverse relation seems to happen also between division problemsand fraction problems with part-whole interpretation. It is important to highlight that division problems are stronglyrelated to each other, demonstrating some di�culty in understanding inverse relation between quantities.

The e�ect existing between the relationships and the kinds of problems where the inverse relation between quantitiesis involved must be looked more closely, in view of the exploration of these relationships in the classroom. Furtherinvestigation is necessary about this topic, in order to stimulate this understanding in children at elementary schooleducation.

REFERENCES

Behr, M., Wachsmuth, I., Post, T. & Lesh, R. (1984). Order and Equivalence of Rational Numbers : A ClinicalTeaching Experiment. Journal for Research in Mathematics Education, 15 (5), 323-341.

Correa, J., Nunes, T., & Bryant, P. (1998). Young children's understanding of division : The relationship betweendivision terms in a noncomputational task. Journal of Educational Psychology, 90, 321-329.

Direcção Geral de Inovação e Desenvolvimento Curricular (2007). Programa de Matemática do ensino básico.Lisboa : Ministério da Educação.

Fonseca, H. I. (2000). Os processos matemáticos e o discurso em actividades de investigação. (Dissertação deMestrado, Universidade de Lisboa). Lisboa : APM.

Hallett, D., Nunes, T., Bryant, P., & Thorpe, C.M. (2012). Individual di�erences in conceptual and proceduralfraction understanding : The role of abilities and school experience. Journal of Experimental Child Psychology, 113,469-486.

Hart, K. (1981). Fractions. In K. Hart (Ed.), Children's Understanding of Mathematics :11-16, (pp. 66-81). London :John Murray Publishers.

Kerslake, D. (1986). Fractions : Children's Strategies and Errors - A Report of the Strategies and Errors in SecondaryMathematics Project. Berkshire : NFER-NELSON.

Kieran, T. (1993). Rational and Fractional Numbers : From Quotient Fields to Recursive Understanding. In T.Carpenter, E. Fennema and T. Romberg (Eds.),Rational Numbers - An Integration of Research (pp. 49-84). Hillsdale,




New Jersey : LEA.Kieren, T. (1995). Creating Spaces for Learning Fractions. In T. Sowder and B.P. Schapelle (Eds.), Providing a

Foundation for Teaching Mathematics in the Middle Grades (pp. 31-66). Albany, New York : SUNY Press.Kornilaki, E., & Nunes, T. (2005). Generalising Principles in spite of Procedural Di�erences : Children's Under-

standing of Division. Cognitive Development, 20, 388-406.Mack, N. (2001). Building on informal knowledge through instruction in a complex content domain : partitioning,

units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32, 267-295.Mamede, E. & Cardoso, P. (2010). Insights on students (mis)understanding of fractions. In : M. M. Pinto & T. F.

Kawasaki (Eds.), Proceedings of the 34th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol.3, pp. 257-264). Belo Horizonte, Brasil : PME.

Mamede, E., Nunes T. & Bryant, P. (2005). The equivalence and ordering of fractions in part-whole and quotientsituations. In : H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conf. of the Int. Group for the Psychologyof Mathematics Education (Vol. 3, pp. 281-288). Melbourne, Australia : PME.

Mamede, E. & Silva, A. (2012). Exploring partitive division with young children. Journal of the European TeacherEducation Network, (8), 35-43.

Monteiro, C. & Pinto, H. (2005). A Aprendizagem dos números racionais. Quadrante, 14(1), 89-104.National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Virginia :

NCTM.Nunes, T., Bryant, P., Pretzlik, U., Evans, D., Wade. J. & Bell, D. (2004). Vergnaud's de�nition of concepts as a

framework for research and teaching. Annual Meeting for the Association pour la Recherche sur le Développement desCompétences, 28-31. Paris.

Ponte, J. P., Boavida, A., Graça, M. & Abrantes, P. (1997). Didáctica da Matemática. Lisboa : Ministério daEducação �DES.

Spinillo, A.G. & Lautert, S. L. (2011). Representar operações de divisão e representar problemas de divisão : hádiferenças ?, International Journal for Studies in Mathematics Education, 4(1), 115 �134.

Stree�and, L. (1997). Charming fractions or fractions being charmed ?, In T. Nunes and P. Bryant (Eds.), Learningand Teaching Mathematics �An International Perspective (pp. 347-372). East Sussex : Psychology Press.




4.8 A priori analysis and its role

Hana Nováková, Jarmila Novotná

Charles University in Prague, Faculty of Education, Czech republic

Résumé : L'article est consacré à l'analyse a priori qui est un des concepts clés dans la Théorie des situations didactiques (Brousseau,

1997). Dans cette théorie l'analyse a priori représente un des moyens de base dont l'enseignant dispose pendant la plani�cation des cours

(Nováková, 2013). Dans cet article l'analyse a priori est présentée d'un point de vue di�érent �comme un outil qui aide à la préparation, à

la réalisation et à l'analyse des résultats dans la recherche en didactique des Mathématiques. Son importance dans la recherche est illustrée

par une recherche en cours qui est visée à l'amélioration de la culture scolaire des élèves dans le cas de la résolution de problèmes. L'étendue

de la contribution étant limitée, nous présentons un aspect de l'analyse a priori �les stratégies de la résolution du problème posé.

Abstract : The paper focuses on a priori analysis, one of the key concepts of the Theory of Didactical Situations (Brousseau, 1997).

According to this theory a priori analysis is one of the main tools a teacher uses when planning a teaching unit (Nováková, 2013). This

paper introduces a priori analysis in a di�erent perspective - as a tool helping in planning problems, using them in lessons and in analyzing

results in mathematics education research. Its importance for research is illustrated on an example from an ongoing research focusing

on improvement of pupils' culture of problem solving. Because of the limited scope of this paper, only one aspect of a priori analysis is

introduced, namely problem solving strategies.

What is a priori analysis

According to Brousseau (1997) and his Theory of Didactical Situations in Mathematics (TDSM) a priori analysisis one of the tools available to a teacher when planning a lesson. Its objective is to predict as accurately as possible thecourse of the relevant teaching unit, especially with respect to division of this unit into di�erent phases, to potentialpupils' reactions and attitudes and the teacher's reactions (obstacles, misconceptions and mistakes, correction of andfurther work with these mistakes), possible solving strategies (correct and incorrect), knowledge prerequisite for theuse of the di�erent solving strategies. Thus a priori analysis provides the teacher with a lot of valuable information.According to TDSM, a priori analysis is the condition for devolution and consequently for establishment of a-didacticalsituation. In a posteriori analysis, a priori analysis is compared with experience from realization in the classroom.Recommended changes are formulated.

A priori analysis as a tool in mathematics education research

Nováková (2013) shows how a priori analysis can be used by a teacher. This paper focuses on a priori analysis as atool for a researcher. We will show what role a priori analysis plays in an ongoing research project GA�R P407/12/1939Development of culture of problem solving in mathematics in Czech schools.

The goal of the research project is development of a theory of mathematics problem solving with focus on the roleheuristic strategies play in development of pupils' culture of solving mathematics problems KRP). KRP is understoodas a structure of internal factors that in�uence a pupil's performance and success in problem solving. It consists of fourcomponents : intelligence, creativity, ability to use existing knowledge, and reading comprehension skills (Eisenmann,Novotná, P°ibyl, 2014).

In a short-term (3 months) and long-term (13 month) experiment, pupils are introduced to heuristic strategies thatthey rarely or never come across in usual lesson but are very e�ective and useful in problem solving (e.g. systematicexperimenting, analogy, graphical representation, use of auxiliary element �see e.g. B°ehovský, Eisenmann, Ondru²ová,P°ibyl, Novotná, 2013). The pupils are lead systematically to use of a suitable heuristic strategy when they come acrossa problem they cannot solve using �school solving algorithm�.

For these ends the research team works on development of batteries of problems that can be solved e�ectively usingone of the above listed strategies. All these problems are carefully elaborated and commented upon and o�er moreways of solution. Selected problems are also subject to a priori analysis. The following text presents one example ofhow a priori analysis is used in the above described project.




A priori analysis of a problem

We usually conduct a priori analysis with respect to the following aspects of the problem : nature of the assignment(knowledge prerequisite to grasping the assignment, potential problems in comprehension of the assignment), thematicunit, goal of the problem (didactical goal for the teacher and mathematical background that pupils are expected tolearn or develop when solving the problem), time needed for solution of the problem, class management, aids, variables,pupils' reactions and attitudes, teacher's reaction, solving strategy (correct and incorrect), and prerequisite knowledge.

In our research it is necessary to know how the problem can be solved and whether there are some heuristicstrategies among the possible solving strategies. In the following text we discuss this aspect of a priori analysis.

1. Problem Area of a quadrilateral

Assignment : Triangle ABC in �g. 4.10 has a unit area. Points P , Q, R, S divide sides AC and BC into threeequal segments. What is the area of the coloured quadrilateral ? (Horenský et al., 2007, p. 29/6)

Figure 4.10 �

1.1 Analysis of di�erent solving strategies �correct strategies

I.Direct method

a Arithmetical solution : We use knowledge of similarity of triangles ABC and PRC, ABC and QSC (see Fig. 4.10).The ratio of similar triangles with coe�cient k equals k2.

b Graphical representation �solving drawingLet us move trapezium PRSQ to the line above the trapezium (see Fig. 4.11). We move parallelogram RSTB undertriangle UCV (see Fig. 4.12) ; thus we form three congruent trapeziums.Prerequisite knowledge : triangle and its properties, translation, composition of �gures.Possible obstacles : correct division of the triangle, correct composition of a trapezium.II.Introduction of auxiliary element

a Graphical representation �solution drawing II If we divide a triangle into nine congruent triangles as shown in Fig.4.13, we discover that the trapezium is covered by three triangles and so its area is SPQRST = frac39 = frac13.




Figure 4.11 �

Figure 4.12 �

Prerequisite knowledge : triangle and its properties, fractionsPossible obstacles : correct determination of the area of the trapezium (relation whole �part).

b Graphical representation �solution drawing IIILet us extend triangle ABC into parallelogram ABCD (see Fig. 4.14). Let us draw points E and F as intersectionsof half-lines PR and QS with line segment BD. Line segments PE and QF divide parallelogram ABCD into threecongruent parts. Triangle QSC is congruent with triangle ERB. As trapezium ABRP together with triangle ERBmake one strip, the area of the strip equals union of this trapezium and triangle QSC. The area of trapezium PRSQequals to one half of area of the whole strip, therefore area of ABRP in union with QSC is twice the area of PRSQ.Thus the area of the studied quadrilateral equals one third of triangle ABC.

Prerequisite knowledge : construction of a parallelogram on the basis of a triangle.Possible obstacles : correct determination of the area of the parallelograms and later of the required trapezium.

III.Speci�cation and generalization




Figure 4.13 �

Figure 4.14 �

Instead of the scalene triangle ABC let us select a right triangle. It is right-angled at C (see Fig. 4.15). Thefollowing holds :

SABC | =a · · · b2

= 1, SQSC =13a · · ·

13v

2, SPRC =

23a · · · 13v2

Thus :

SRSPQ =4

9· · · av

2− 1

9· · · av

2=

1

3· · · av

2

In this computation we worked with a right triangle but we can use the same procedure for a scalene triangle ABC.Also in this case triangles ABC, QSC and PRC are always equiangular (similarity coe�cients are the same as in thespecial case). Thus also altitudes of these triangles are in the same ratio.




Figure 4.15 �

Prerequisite knowledge : triangle and its properties, area of the right triangle, congruence of triangles.The answer : The area of the coloured quadrilateral is 1

3 of the area of the triangle.

1.2 Analysis of di�erent solving strategies �incorrect strategies

The following is an example of an incorrect strategy that can be come across in pupils' solutions. It is commonthat teachers cannot foresee all incorrect strategies their pupils will use. However, if the teacher/researcher considerthem before the lesson, they will �nd it easier to react to situations when incorrect solving strategies are proposed bypupils.

Extension into a parallelogramThis incorrect strategy comes out of the correct strategy b) Graphical representation �solving drawing III. The

pupil �nds out that the coloured part is one third of the area of the parallelogram. The triangle represents one half ofthe area of the parallelogram. The pupil works analogically to calculate the area of the coloured part of the triangleand divides 1

3 by 2. The result is then 16 .

2.Problem Kite

Assignment :Determine how much paper is needed for construction of a kite if its dimensions correspond to thosegiven in the picture (Fig. 4.16).

Figure 4.16 �

2.2 Analysis of di�erent solving strategies �correct strategies

I. Re-composition into a rectangle (Fig. 4.17)We draw both diagonals. We make use of the property of line symmetry of a trapezium and translate the left part

of the trapezium to the right so that a rectangle with dimensions 10 and 30 is created.




S = 10 · · · 30 = 300

The area of the kite is 300.

Figure 4.17 �

Prerequisite knowledge : deltoid and its properties, line symmetry, area of a rectangle, multiplication.Possible obstacles : multiplication if we do not have a calculator.

II. Introduction of an auxiliary element (Fig. 4.18)Let us circumscribe a rectangle to the deltoid. Its dimensions will be 20 to 30. Let us draw both diagonals of the

deltoid. We can see that its area is one half of the circumscribed rectangle. This is in fact the formula for calculationof the area of a deltoid.

S = frac20 · · · 302 = 300

Figure 4.18 �

Prerequisite knowledge : deltoid and its properties, area of a rectangle, multiplication.




Possible obstacles : realizing what part of the rectangle is the deltoid, correct multiplication if we do not have acalculator.

III. Division into two, three or four triangles (Fig. 4.19)We use the horizontal diagonal to divide the deltoid into two tringles whose area is easy to calculate using the

formula for the area of a triangle.

S1 =20 · · · 10

2= 100, S2 =

20 · · · 202

= 200

S = S1 + S2.

Figure 4.19 �

Another possibility is to draw both diagonals and to calculate the area of three (two congruent) triangles or four(two and two congruent) triangles. The process is the same as shown above.

Prerequisite knowledge : deltoid, triangles and its properties, the area of a triangle, the area of a right triangle,multiplication.

Possible obstacles : correct visualization of the situation, correct multiplication of dimensions and addition of theareas.

2.2 Analysis of di�erent solving strategies �incorrect strategies

Let us show here an example of a wrong strategy that can be come across in the pupils' solutions. The strategybuilds on strategy II. The pupils try to use the formula for calculation of the area of a deltoid but forget to divide theproduct of diagonals by two.

A posteriori analysis �solving strategies

Let us now compare this component of a priori analysis with results of pupils in one of the experimental classes.They were 3rd grade lower secondary grammar school pupils (aged 13). The problems were solved by 28 pupils, whowere working on their own. They recorded their solutions into work sheets.

1.Problem Area of a quadrilateral

The problem was solved correctly by 14 pupils, incorrectly by 13 pupils. One pupil did not know how to solve theproblem. There were three correct solving strategies. In most cases (15) pupils used a solving strategy we had notanticipated in our a priori analysis �calculation after measuring the sides. However, this strategy resulted in a correctanswer of only 7 pupils. Others in consequence to inaccurate measuring did not reach the correct result. The pupils




also used two correct solving strategies anticipated in a priori analysis : graphical representation - solving drawing II(6 pupils) and graphical representation �solving drawing III (1 case).

2. Problem Kite

The problem was solved correctly by 22 pupils. 13 pupils divided the kite into two triangles using the shorterdiagonal. 3 pupils divided the kite using the longer diagonal (strategy 3.1.3.), 6 pupils solved the problem using arectangle (strategy III). 6 pupils did not solve the problem correctly : 2 of were utterly helpless, 2 proceeded correctlybut made a numerical mistake in multiplication (incorrect number of zeros), 2 pupils calculated the area of one half ofthe �gure and forgot to multiply the result by two. The teacher was convinced the pupils had little di�culty solvingthe problem as it was the subject matter dealt with in mathematics lessons at that time.

Conclusion

A priori analysis is of great importance in the project both to the researchers and teachers : When selectingand posing problems for experiments, the researchers need to analyse whether the selected or posed problems meetthe before de�ned criteria. In the here reported case it is important to check that the problem can be solved usingdi�erent strategies and that one of the above described heuristic strategies is much more e�cient for its solution thanother strategies. A priori analysis also facilitates cooperation between researchers and teachers while planning theexperimental teaching unit.

As far as teachers are concerned, a priori analysis is an aid in planning the lesson in which the problem will be used.It facilitates preparation of teaching aids, suggests how to present the problem to their pupils, outlines how pupilsmight react to the problems, shows what correct and possibly incorrect strategies pupils might use when solving it.Thus a priori analysis helps the teacher be ready for contingencies.

REFERENCES

Brousseau, G. (1997). Theory of didactical situations in mathematics. Boston : Kluwer Academic Publishers. Frenchversion : 1998. Théorie des situations didactiques. Grenoble : La pensée sauvage.

B°ehovský, J., Eisenmann, P., Ondru²ová, J., P°ibyl, J., & Novotná, J. (2013). Heuristic strategies in problemsolving of 11-12-year-old pupils. In Novotná, J., Moraová, H. (Eds.), Symposium on Elementary Maths TeachingSEMT '13. Proceedings (pp. 75-82). Praha : UK-PedF.

Eisenmann, P., Novotná, J., & P°ibyl, J. (2014). �Culture of Solving Problems� �one approach to assessing pupils'culture of mathematics problem solving. In Ková£ová, M. (Ed.), 13th Conference on Applied Mathematics Aplimat2014 (pp. 115-122). Bratislava : STU.

Nováková, H. (2013). Analýza a priori jako sou£ást p°ípravy u£itele na výuku. Scientia in educatione, 4 (2), 20-51.Horenský, R., Molnár, J., Rys, P., & Zhouf, J. (2007). Po£ítejte s klokanem kategorie �Junior�. Olomouc : PRODOS.

Acknowledgement

The research was supported by the project GA�R P407/12/1939.




4.9 Developing hypothetical thinking through four cycles of informal stochas-tical modelling

Ana Serradó Bayés

La Salle-Buen Consejo, Puerto Real, Cádiz, Spain ;

Résumé : Nous présentons une tâche consistant en un processus de modélisation horizontal de la fréquence relative d'apparition

de chaque voyelle dans les chaînes de caractères avec quatre cycles : un modèle pseudo-concret élaboré par un processus statistique de

recherche, une modélisation statistique et validation par l'analyse de animations numériques et le transfert à une autre réalité. Chaque

cycle commence par une question où les étudiants ont à générer leur hypothèse. Un cadre bidimensionnelle a été construit pour analyser

l'hypothèse des élèves : le croissance de la pensée cognitive hypothétique et le cadre de la description de la distribution de la fréquence

relative stabilisée. Les résultats permettent d'identi�er les élèves d'identité maturation cognitive sur la pensée hypothétique avec un dessin

initial de leurs hypothèse percevais de la réalité, la description verbale de l'hypothèse de ce qui peut se produire lors de l'analyse des

animations numériques de données, le transfert de l'hypothèse de l'équivalence de certaines propriétés et processus, et la nécessité d'une

preuve de la Loi des Grands Nombres.

Abstract : We present a task consisting in an horizontal modelling process of the relative frequency of appearance of each vowel in

strings of characters with four cycles : a pseudo-concrete model developed through a statistical process of investigation, a statistical modelling

and validation through the analysis of digital animations and the transference to another reality. Each cycle begins with a question where

students have to generate their hypothesis. A bi-dimensional framework has been constructed to analyse students' hypothesis : the cognitive

grow of hypothetical thinking and the Stabilized Relative Frequency Distribution Description Framework. The �ndings identify students

cognitive maturation on hypothetical thinking with an initial drawing of hypothesis as perceptions of the reality, the verbal description of

the hypothesis of what can occur when analysing digital animations of data, the transference of hypothesis about the equivalence of some

properties and processes, and the need of a proof of the Law of Large Numbers.

Introduction

The Spanish Curriculum of Compulsory Secondary School indicates that students of grade 9 (ages 14 and 15) shouldformalize the notion of probability through the classical and frequentist approach of relative frequency estimations(Batanero et al., 2013). In the Spanish curricular desing and its textbooks, this estimation is taken as de�nition ofthe mathematical value, raising serious epistemological, ontogenic and didactic obstacles (Chaput et al, 2011 ; Serradóet al, 2005). Obstacles that could appear, in a �rst moment, when structuring the di�erent notions of probability ; orin a future, when understanding the random convergence with the aim of proving the Law of Large Numbers or theCentral Limit Theorem.

With the aim of trying to surpass these obstacles, other countries have opted for a progressive mathematizationof the notion of probability. In coherence with the Realistic Mathematics Education (RME), they propose beginningthrough a horizontal mathematization, which refers to modelling the problem situation into mathematics (Drijvers,2000). According to this modelling perspective the probability is de�ned as : the theoretical value of the degree ofcon�dence that one can give to a random outcome obtained by the observation of a stabilized relative frequency whenthe same random experiment is repeated a larger number of times under the same conditions (Chaput, Girard, &Henry, 2011).

This perspective concurs with the general process of contemporary statistical thinking, and contributes to the learn-ing of a modelling process through the development of statistical investigations. The concepts of sampling, variationand distribution are key when engaging students in this modelling process (Wild, 2006). When modelling, studentsare asked to reason about distributions, which imply establishing relations between the data (data distribution), thepopulation (distribution of probabilities) and the samples (sampling distribution). But, this kind of reasoning is intro-duced in the Spanish Curriculum three years later, in grade 11 (ages 17 and 18), in the context of Statistical Inference(Batanero et. al, 2013).

As a consequence, we are exploring here the possibility of introducing the Stabilized Relative Frequencies Dis-tribution (SRFD) in the context of Informal Inference Reasoning (IIR). Conscious of the importance that acquiresdrawing hypotheses in Inferential Reasoning, in this paper we are interested in understanding the process of hypothesisgeneration when involving students in a task consisting in an informal stochastical modelling process with the aim ofconstructing the SRFD. This task would be the �rst stage in a learning trajectory with the aim of proving the Lawof Large Numbers. Furthermore, we conjecture : a task that engages students in the informal statistical modelling




process of �growing samples� may help them to cognitively mature their hypothetical thinking about the Law of LargeNumbers.

Hypothetical thinking when de�ning stabilized relative frequency distribu-tion

We consider statistical models as oversimpli�cations of the reality, where all our statistical conceptions about theproblem to model are in�uenced by how we collect data, analyse and interpret it. In coherence, when learning aboutmodelling, students need to engage in progressive and simultaneous investigative and interrogative cycles.

Stabilized relative frequency distribution framework

The investigative cyclecycle concerns the way one acts and what one thinks about during the course of a statisticalinvestigation with an adoption of the PPDAC model (Problem, Plan, Data and Conclusions) (Wild and Pfannkuch,1999). Engaging on these progressive investigative cycles should provide students with the ability to learn about com-plex sequences of operations with data through action. And, in successive processes of interiorization, condensationand rei�cation what students initially conceive purely operationally can be conceived structurally at a higher level(Sfard, 1991). One example, of interest in this paper, about the intricate interplay between operational and structuralconceptions of an object is the analysis about the relation between data and distribution presented by Bakker andGravemeijer (2004). They examined aspects as centre, spread, density (relative frequency) and skewness to structurethe relationship between data (individual value to operate with) and distribution (conceptual entity). This structurecan be read upward, from data to distribution, which leads to a frequency distribution of a data set. And in the down-ward perspective, from distribution to data, we use probability distributions to model data (Baker and Gravemeijer,2004). Furthermore, the relationship between data and distribution provides a bridge between the relative frequencydistribution and the probability that can be conceived as equivalence in a system of increasingly sophisticated knowl-edge. Pegg and Tall (2005) de�ne equivalence when two-way relationships reveal the same general structure expressedin di�erent ways. For Pegg and Tall (2005), the equivalence stage is previous to a crystalline concept, where all theseequivalent ideas are seen as di�erent aspects of the same underlying conceptual entity. In our study, the stabilizedrelative frequency distribution should be seen di�erent from the probability, but equivalent ideas underlying the sameconceptual entity that is the crystalline concept of the Law of Large Numbers.

Bakker and Gravemeijer (2004) presented a framework that structure the statistical concepts that provide equiva-lences between frequencies, stabilized relative frequency and distribution and probability. And, based on this frameworkBen Zvi, Gil and Apel (2007) constructed a new framework that allows analysing the cognitive aspects of the dis-tribution in the context of the Informal Inferential Reasoning : reasoning about variability, distributional reasoning,reasoning about signal and noise, contextual reasoning and graph comprehension. These two frameworks, providepart of the picture underlying the conceptual structure of distribution, but are not adequate when thinking about,exploring and describing distributions. In order to solve this problem, Arnold and Pfannkuch (2012) propose the Dis-tribution Description Framework (DDF) organized by : (1) overarching statistical concepts that underpin distribution,(2) characteristics of distribution, and (3) the speci�c features that are used when describing distributions. But, stillthis framework does not provide any reference to concepts underlying sampling reasoning, crucial when structuringdata and distributions, as : sampling size, random process, distribution, intuitive con�dence interval and relationshipbetween sample and population (Dierdrop et al., 2012). With the aim of integrating those previous frameworks andsolving the problems observed, we propose a theoretical framework of the cognitive aspects related to sampling :contextual knowledge (samples, population, sample size. . .), distributional (error, reliability, law of large numbers),graph comprehension (smoothing), variability (tendencies, intuitive con�dence intervals).

The integration of the Distribution Description Framework (DDF) and the cognitive aspects of the sample ledus to propose the Stabilized Relative Frequency Distribution Description Framework (SRFD), in which we describetheoretically the overarching statistical concept and the characteristic of the distribution : contextual knowledge(population, sample, sample size, variable, interpretation and explanation), distributional (shape, skewness, error,reliability, individual cases, law of large numbers), graph comprehension (decoding visual shape, unusual features,smoothing, comparing samples), variability (spread, density, tendencies, intuitive con�dence intervals) and signal andnoise (centre, modal clumps).




Cognitive maturation of hypothetical thinking framework

However, we consider that engaging students in successive investigative cycles, it is not enough to crystalline theconcept of the Law of Large Numbers. We think that students should be conscious of the interplay that exists betweenthe investigative and interrogative cycle (Wild and Pfannuck, 1999). The engaging in successive interrogative cyclesde�ned as generic thinking process, is a distilling and encapsulating of both ideas and information. When involved inthe interrogative cycle students are engaged in actions as generate, seek, interpret, criticise and judge the di�erentcomponents of the Stabilized Relative Frequency Distribution. In this paper, we are interested in understanding theprocess of hypothesis generation when involving students in an informal stochastical modelling process for constructingthe SRFD with the aim of proving in a future the Law of Large Numbers. Furthermore, we are interested in analysing ifengaging students in successive cycles of informal stochastical modelling process should provoke on them the cognitivematuration of their hypothetical thinking. From the di�erent theories of cognitive growth that o�er di�erent aspectsof the development over the longer term, we adapt Tall et al (2011) broad maturation of proof structures to thecognitive maturation of hypothetical thinking. And in consequence, we present the framework for cognitive maturationof hypothetical thinking : (Level 1) drawing hypothesis as perceptions of the reality ; (Level 2) verbal description ofhypothesis of what can occur ; (Level 3) hypothesizing about the properties of the concept to be de�ned ; (Level4) hypothesizing about the equivalence of some properties and de�nitions of the concepts to be de�ned ; (Level 5)crystallization the hypothesis of the concepts constrained by the need of a proof ; and (Level 6) announcing thehypothesis of a deductive knowledge structure.

Methodology

To answer the main question of how students can develop the hypothetical thinking that should allow them toconstruct the notion of SRFD for in a future prove the Law of Large Numbers, we carried out a design-based researchstudy of a task (e.g. Bakker and Gravemeijer, 2004). The task was designed to generate students' activity with the aimof giving them the opportunity to encounter new statistical ideas and strategies, new modes of enquiry and developtheir hypothetical thinking. The main question of the task, or real problem, was : �Can I guess which language isspeaking my friend only counting the vowels ?� The problem generated three questions to be answered by the studentsthat guided their learning.

Problem 1 : �Which vowels are not used when writing a mobile message ?� The question was introduced to con-textualize the task in an environment where students have to think why certain vowels are not used in Spanish whenwriting a SMS or a Whatsapps. For example, the Spanish sentence �Mi casa es blanca� is written using sms languageas �Mi cas s blnc�, omitting those vowels that intuitively appear more often in our language. This intuition leads usto think that students in this case may have an everyday sense of each vowel density in the sentences. These �rstintuitions about the density are a key aspect for constructing the notion of distribution (Bakker & Gravemeijer, 2004).

Problem 2 : �Which problem can we plan to get to know what happens in Spanish ?�The question allowed to introduce students in an informal modelling process with three cycles : The cycle 1 consisted

in a pseudo-concrete model developed through a statistical process of investigation of the vowels that appear in chainsof characters. The learning objectives of this �rst cycle were : (1) formulate questions related to a context with theaim of converting those real problems in mathematical/statistical problems that could be mathematized through anhorizontal modelling, (2) formulate hypothesis about the answers of the mathematical/statistical problems posed, (3)devise a data collection collection plan coherent with the analysis and procedures planned, and (4) gain appreciationof the important role of sample size in statistical sampling, analyse and interpret the collected data using Geogebra.

The second cycle, is a statistical modelling process through the analysis of digital Java animations looking forpatterns when increases the sample size of characters ; The learning objectives of this second cycle were : (1) hypothesizeabout what happens when the sample size of characters increase, (2) enhance the understanding of the sample size onsample representativeness and variability, (3) increase recognition of the potential for bias due to poor sampling design,(4) enhance understanding of the relative frequency and stabilized relative frequency distribution, (5) reduce mistrustof single random sample, (6) develop facility with Java Animations for analysing data patterns, and (7) construct theLaw of Large Numbers through Java Animations.

The third cycle consisted in a validation of the model with other samples of characters. The learning objectiveswere : (1) drawing hypothesis about what could happen with other samples of characters, (2) compare distributionsthrough Java animations, enhance recognition of the centre and spread aspects of the relative frequency distribution,(3) validate and interpret the preliminary hypotheses using other samples, and (4) gain knowledge of bias error.




Problem 3 : (cycle 4) �Can we �gure out which language is speaking a friend only by counting the vowels ?� Theaim of this cycle is assessing if students are able to transfer the informal modelling process developed to solve thisproblem.

The teaching experiment lasted 11 sessions of 60 minutes in which there were developed four cycles of large classes,individual and small-group action. For each problem and cycle of modelling, the question was presented to the large-class in order to decide which statistical problem pose. Individually, students were asked to hypothesize about theanswer of the problem posed. Then, in small-group, students were asked to confront their hypothesis to promote adeliberative dialogue that guide the collection, analysis and interpretation of the data. Their interpretations werepresented to the large-class in order of drawing conclusions about the problem and structure the components of theSFRD. Those conclusions opened new problems, new cycles of modelling, and new methodological cycles of large-classdiscussion, individual re�ection, small-group action, large-class discussion.

On this study participated 49 students of grade 9 (ages 14, 15) composed by two groups of 25 (A) and 24 (B)students from a Spanish Compulsory Secondary School in a low socioeconomic coast city. The four individual re�ectionsof the students corresponding to their hypotheses about the answer of the problem posed were codi�ed twice. The�rst codi�cation consisted in discriminate in the students' individual answers the descriptions of the �ve componentsof the SRFD. The second one codi�ed for each student the level of cognitive maturation of the hypothetical thinkingin relation with the Stabilized Relative Frequency Distribution to prove the Law of Large Numbers.

Results

Students were asked to draw their �rst hypothesis when answering the problem 2. On one hand, the �rst generalperceptions of group A students was that the letter with more density of appearance was the �a�. And, in coherencewith their perceptions, students posed the problem : �Is the letter A the ones that more appears in Spanish ?� And, inconsequence, when asked to hypothesize which was the answer of the question, all answer the letter a. We interpretthat students are in a level of maturation 1, because they were able of drawing hypothesis about the density of thedata as perceptions of the reality about the structure of the Spanish Language.

On the other hand, the students of group B doubted about the density of appearance of the vowel "a" and "e",so they posed the question : "Which is the letter that is going to appear more ?" Students of group B were asked toindividually answer the question : "Which do you think is the answer to the statistical problem posed ?" We founddi�erences on the language of the answer. Some students used a dubitative language, as : "I do not know exactly whichis going to be the answer [. . .] But, although I haven`t done the calculus, I think that the vowel that it is going toappear more is 'a'" (Raquel, group B). We interpret the use of the dubitative language as a step forward the levelone because, when drawing her hypothesis as perceptions of the reality, she expresses the need of calculating beforeanswering. This answer should be interpreted as the need of the student of validating their hypothesis to answer theproblem, as logic of hypothesis, calculus, and solution. Other students' language was assertive. For example : "thevowel more used in Spanish due to its frequency of appearance is the 'a'" (Esteban, group B) or "the answer of theproblem is going to guarantee . . ." (Esther, group B). We consider that the use of this assertive language can be anexpression of their pragmatic view of the reality, and their deterministic conception of the nature of mathematics. Weconsider that this deterministic conception of the nature of mathematics it is going to constrain the possibilities of thecognitive maturation of hypothetical thinking from the level 1 to upper levels, due to the di�culties of hypothesizingabout the variability of the data and the contextual knowledge that allow students structure the SRFD.

In the second cycle of modelling there was a student that still used an assertive language when hypothesizing aboutwhat happens when increasing the sample size. He answered : "Increases the number of characters that we have in thetext. Increases the relative frequency" (Alex, group A). The other students verbalize their hypothesis, with more orless accuracy, using a dubitative language about how the distribution is going to vary in relation with the sample size.Comparing with their �rst hypothesis, they integrate the contextual knowledge looking for interpretations of the realityand trying to venture the shape of the relative frequency distribution. We interpret that we can observe a cognitivematuration on their hypothetical thinking. We interpret that except Alex, all the students have matured form thelevel 1 to the level 2, because they use a dubitative language, and, at least, they present a contextual description ofthe relative frequency consisting in the interpretation of the reality. We understand that some students have maturedfrom level 2 to 3, because they draw hypotheses about the shape of the distribution describing graphical properties ofthe SRFD.

At the beginning of the third cycle of modelling, students were asked if they could validate the conclusions obtainedin stage two, through comparing samples using digital Java animations. The direct question posed aimed to answer




yes or not. Almost half of the 49 students answer a�rmatively. For example : "Yes, I can. Although in some textswe are going to have a bigger relative frequency than in the others" (Alex, group A). We consider that Alex has stilldi�culties on deduce appropriate properties of the SRFD, that constrain the maturation of his hypothetical thinking.In contrast, an example of the negative answer is : "we are still not able to validate the conclusion obtained, becausewe have the same problems ; vary the sample size, the sample of texts, the frequency absolute and relative of vowels"(Gloria, group A). We interpret the quotation of Gloria as her need of integrating three components of the SRFD : thecontextual knowledge, the distributional knowledge and the variation of data. When integrating these components,the student might be involved in the maturation her hypothetical thinking from level 3 to 4 looking for the equivalenceof some properties of the SRFD.

In the last cycle students were asked to transfer their knowledge to solve the problem 3. Students were asked :"which do you think that could be the answer of the third problem ?" The least students answered negatively describingthe dependence of the solution to the sample. For example : "No, because depending of the sample that we chosewe are going to have a di�erent result" (Juan Manuel, group B). The student argues about how the solution isconstrained by the sample chosen in the modelling process. This constriction di�cult student cognitive maturationon his hypothetical thinking anchored in level 3. However we think that in the student description of "depending ofthe sample", he expresses the need of understanding the sampling distribution. The majority of the students thatanswered a�rmatively used their perceptions about the reality or reasoned about the complexity of the problem tosolve and described the four cycles of the informal stochastical modelling process. These answers can be consideredmaturation from the level 3 to 4 of their hypothetical thinking looking for the equivalence of properties of the SRFD.

Conclusions

Students were asked to solve a task related to the real problem (Can we �gure out which language is speaking afriend only by counting the vowels ?"). The students, when asked to solve this real problem, have been involved ina horizontal statistical modelling process in which they have to convert the real problem in a statistical one. In thisprocess they developed their �rst hypothesis about the distribution of the vowels in chains of characters.

When analysing students' individual hypothesis we have found maturation on their hypothetical thinking. Students'perceptions of the reality about the density of the vowels in the structure of the Spanish Language has allowedconsidering that most of the students are initially in the level 1. The maturation from the level 1 to 2, it is constrainedfor some students by their deterministic conception of the nature of mathematics and the use of assertive language. Wethink that the maturation from the level 2 to 3 occurs when the student describes the individual data related to realityas independent properties of the contextual, distributional or graphical knowledge. The need of understanding thecontextual knowledge of sampling constrains that some students evolve from level 3 to 4. Nevertheless, the expressionof the complexity of the problem and the need of integrating the contextual, distributional or graphical knowledgeprovoke in the students the maturation from the level 3 to 4.

These stages on maturation of the hypothetical thinking are coherent with the proposal of maturation of proofstructures introduced by Tall et al. (2011). And, it gives us information about the importance of introducing the hypo-thetical thinking on Compulsory Secondary School when developing structures of proof. However, involving studentsonly in this task is not su�cient in a learning trajectory that aims to prove the Law of Large Numbers. We conjecturethat this proof should be made by a series of successive tasks in which students engage in reasoning about sampling anddistribution in statistical problem solving processes, then being involved in informal modelling processes of "growingsamples" with the aim of crystalline the notions of probability and sampling distribution, and �nally prove the Lawof Large Numbers.

References

Bakker, A., & Gravemeijer, K. P. (2004). Learning to reason about distribution. In D. Ben-Zvi, & J. Gar�eld(Edits.), The Challenge of Developing Statistical Literacy, Reasoning and Thinking (págs. 147-168). Dordrecht : KluwerAcademic Publishers.

Batanero, C., Ortiz, J., Roa, R., & Serrano, L. (2013). La Statistique dans le Curriculum en Espagne. Statistiqueet Enseignement , 4 (1), 89-106.

Chaput, B., Girard, J.-C., & Henry, M. (2011). Frequentist Approach : Modelling and Simulation in Statistics and




Probability Teaching. In C. Batanero, G. Burrill, & C. Reading (Eds.), Teaching Statistics in School. Mathematics-Challenges for Teaching and Teacher Education : A Joint ICMI/IASE Study. (pp. 85-96). Dordrech : Springer.

Drijvers, P. (2000). Students encountering obstacles using a CAS. International Journal of Computers for Mathe-matical Learning, 189-209.

Serradó, A., Cardeñoso, J. M., & Azcárate, P. (2005). Obstacles in the learning of probabilistic knowledge : in�uencefrom the textbooks. Statistics Education Research Journal , 4 (2), 59-81.

Tall, D., Yevdokimov, O., Koichu, B., Whiteley, W., Kondratieva, M., & Cheng, Y.H. (2011). Cognitive developmentof Proof. In M. De Villiers, & G. Hanna (Edits.), Proof and proving in mathematics education (pp. 13-49). Dordrecht :Springer.

Wild, C. (2006). The Concept of Distribution. Statistics Education Research Journal, 5 (2), 1-26.Wild, C., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry (with discussion). International Sta-

tistical Review, 67 (3), 223-265.




4.10 Teaching and learning algebra in the transition between scholasticlevels : a preliminary study.

Daniela Sanna

Dipartimento di Matematica e Informatica, Italy

Abstract : This paper is focused on students' use of algebraic language to represent and solve "real-life" problems. The study is based

on recent developments in mathematics education, in particular related to the �eld of elementary algebra. We consider both studies on

school curricula and teachers' practices, and recent experimental studies on pre-algebra and the relationship between teaching and learning

of arithmetic and algebra. The research highlights some preliminary results of a two-year study aimed at analyzing spontaneous students'

use of algebraic symbolism and informal procedures for the solution of a particular kind of mathematical problems, namely generalization

problems. The study involves classes which, in the Italian scholastic system, correspond to the transition between primary and middle

school and between middle school and high school. These phases correspond also to a change of teachers who usually work with the same

class of students respectively for �ve (students between the ages of 6 and 10), three (students between the ages of 11 and 13) and �ve

years (students between the ages of 14 and 18). In these transitions a �rst approach to formalized language and mathematical axiomatic

structure occurs, together with the switch from arithmetic to elementary algebra.

Résumé : Dans ce texte on analyse l'usage de la parte d'étudiants du langage algébrique dans les représentations et la résolution de

problèmes concernant de contextes liés au réel. L'étude se fonde sur de résultats récents des recherches dans le domaine de la recherche en

didactique de l'algèbre et prend en considération tant les études concernant les programmes o�ciels et les pratiques de l'enseignant aussi

comme des études actuels sur la pré-algèbre et sur la relation entre l'enseignement e l'apprentissage de l'arithmétique et de l'algèbre. Cette

communication concerne certains premiers résultats de notre étude, de la durée de deux ans, ayant le but d'analyser l'usage spontanée du

symbolisme algébrique et des procèdes mois formels dans la réponse à question nécessitant un processus de généralisation. Les niveaux

scolaires choisis son ceux correspondants aux niveaux de la transition école primaire-collège et collège-secondaire. Pendant ces phases

de transition on retrouve la première approche avec le langage formel, la structure assiomatique des mathématiques, aussi bien que le

passage de l'arithmétique è l'algèbre élémentaire. Dans le système scolaire italien ces niveaux correspondent à un change d'enseignant, qui

d'habitude institutionnellement est chargé travailler avec les mêmes élèves pendant cinq (avec élèves de 6 à 10 ans), trois (avec élèves de

11 à 13 ans) et cinq ans avec étudiants de 14 à 18 ans).

Introduction

This study focuses on some aspects of the complex issue of algebra teaching and the development of students'algebraic thinking from the age of 9 to 16.

National systematic assessment devices (e.g. INVALSI test) testify that Italian students struggle with mathematicsin secondary school, when they face algebra for the �rst time.

The negative attitude toward this branch of mathematics is not peculiar to Italian students : previous studiescarried out in the Anthropological Theory of the Didactic framework (Chevallard 1985 ; Chevallard, 1994) reveal acultural worsening of society feeling of algebra.

The aim of the �rst part of the study, described in this article, is to investigate on the development of generalizationskills and on the spontaneous use of symbols to indicate unknowns or variables in generalization tasks. In this articlesome results of the activities carried out with students for this preliminary study are presented.

Literature review and theoretical framework

A description of the current situation about research on elementary algebra education is illustrated in Enseignementde l'algèbre élémentaire : bilan et perspectives (2012). Mercier (2012), analyzing the studies in the �rst part of thisbook, explains that the research follows two trends. In the �rst trend, the current situation in the teaching of algebra isanalyzed by studying and comparing o�cial curriculum, textbooks and teachers' practices. The other trend focuses ona discussion of new practices for algebra teaching in an attempt to characterize the main causes of students' di�cultiesin learning algebra.




Arithmetic teaching-learning as a possible cause of di�culties in algebra ?

The strong relationship between arithmetic and algebra, from the point of view of teaching and learning processes, isvery evident if we look at history of mathematics and algebraic thinking through commognition framework. Accordingto this learning theory, developed by Anna Sfard, �thinking is a form of communication and [. . .] learning mathematicsis tantamount to modifying and extending one's discourse� (Sfard 2007, p. 565). Sfard (1995, p. 16), contextualizingmathematics structure and development in this theoretical framework and applying the rei�cation 7 concept to algebra,claims that mathematical knowledge development is "a process in which the transitions from one level to another followsome constant course." So "each layer in such structure, except the �rst, would be a discourse about the discoursethat constitutes the preceding layer" (Caspi & Sfard, 2012, p. 46). According to this model elementary algebra is adiscourse about arithmetic. Also

this model does seem to indicate the trajectory that one should take in instruction in order to ensure a meaningful learning

of algebra. [. . .] if each layer in the hierarchy is a discourse about its predecessor, an introduction of a new layer before the

student mastered the preceding one carries the risk that the learner would simply not know what the new discourse is all about.

(Caspi & Sfard, 2012, p. 47)An analysis of the most common and frequent students' errors in algebraic tools manipulation, highlights some

widespread features of their mathematical thinking, that is : the one-way equal sign interpretation (Cusi & Malara,2012, p. 308 ; Navarra, 2009, p. 17.), the lack of closure of algebraic expressions (Cusi & Malara, 2012, p. 308 ; Caspi &Sfard, 2012, p. 51.) and the absence of �structure sense" (Hoch, 2003). Some researchers claim that some of the causesof these mistakes and lacks could be the sudden change of teaching methods and of mathematical symbols' meaningin the transition from arithmetic to algebra.

A solution proposal is based on the conjecture that 9-10 year-old children are already able to operate on and withunknowns. Researchers of Medford Tuft University have conducted experimental studies in this direction. Carraher,Schliemann and Brizuela (2000a, 2000b, 2001, 2012) e Carraher and Schliemann (2002) claim that 4th and 5th gradersare already able to use letters or unconventional symbols for unknown quantities spontaneously and to operate on andwith this symbols.

The stance of the American researchers is opposite to other researchers that a�rm �[. . .] if even after an introductionto algebra, students experience di�culties in performing operations with or on a letter representing an unknown or ageneralized number, one can hardly expect them to do so spontaneously without any instruction. Although the letterin an equation or an algebraic expression may have a numerical referent in the pupil's mind, this does not necessarilyrender it operational. Meaning for these operations with literal symbols still has to be constructed�(Herscovics &Linchevski, 1994, p. 63). During this study �at no time did we see any evidence of students directly performingoperations on or with the unknown. Thus we can conclude that students solve these equations by working aroundthe unknown at a purely numerical level�(Herscovics & Linchevski, 1994, p. 70). The mentioned study was aimedat investigating features of informal procedures of students without prior algebraic instruction, in solution of �rstdegree equations in one unknown. Whereas in this article "guess and check" procedures are considered as arithmeticprocedures, others assert that one can use the term algebraic thinking with respect to "any kind of mathematicalendeavor concerned with generalized computational processes, whatever the tools used to convey this generality" (Sfard,1995, p. 18) and that "algebraic thinking can be interpreted as an approach to quantitative situations that emphasizesthe general relational aspects with tools that are not necessarily letter symbolic, but which can ultimately be used ascognitive support for introducing and for sustaining the more traditional discourses of school algebra."(Kieran, 1996,p. 5 quoted in Johanning, 2004, p. 372). However Linchevski (1995) seems to agree with the �nal part of the quotedde�nition, so with the fact that the use of letters is not a necessary condition for the algebraic mode of thinking. Shealso stresses the importance of informal processes and preconcepts in constructing and understanding formal algebraicconcepts. In this sense she consider pre-algebra as a combination of activities aimed at developing �the more primitive,concrete preconcepts that are necessary for the development of the higher, more abstract concept"(Linchevski, 1995,p. 114). Pre-algebra is considered as a transition stage from arithmetic to formal algebra which can be used as apreparation course before the beginning of formal algebra or also intermittently at the beginning of a new chapter offormal algebra.

We will call into question these results in our study, also through experimentation in the classrooms.

7. The term rei�cation is used by Sfard (2008, p. 342) to indicate the �substitution of a discourse about processes with a discourse aboutobjects�




Description of the experimental study

On the basis of the mentioned theories and conjectures, we have planned an experimental study, now in the initialstage. The study attempts to understand whether the use of symbols could spontaneously arise in students. The word�spontaneously� is here used because students have been asked to solve word problems where symbolic-literal languageis not explicitly requested or strictly necessary. The second aim is to analyze the di�erent levels of generalization skillsattained : students have been asked to solve problems that result in numerical sequences, and the number of studentswho have given a general explanation of their answer even when this was not explicitly request has been taken intoaccount.

In this stage 8 of the study, activities have been carried out with classes of di�erent grades. In particular, the studyfocuses on the transition between 5th and 6th grade and 8th and 9th grade that, in the Italian school system, correspondto a change of scholastic institution and teachers as well as classmates. When necessary, the research project, which hasa planned duration of two years, will try to monitor evolution of competences related to algebraic thinking, followingstudents across the transition from primary to middle school and from middle to high school. So far we have carriedout the activities with 5th, 6tih and 8th graders (9th graders will be involved in a next planned activity). Therefore,our sample includes students who have not yet worked with literal expressions or with unknowns but also studentswho already know symbolism and tools of algebraic language. We have used the experiencial learning method. Eachsession was divided into phases. During the �rst phase students have worked individually ; for the second the classhas been divided into small groups. At the end of the group work, answers and procedures have been compared ina whole-class discussion. We have carried out the activities with whole classes and during a curricular lesson to besure that students worked in a usual scholastic environment. For the same reason, their teacher was in the classroomduring activities and, in some cases, she managed the discussion with us.

The problems used are adaptations of problems taken from INVALSI test, chosen on the basis of the activitiessuggested by Linchevski (1995) to improve skills that are considered peculiar features of algebraic thinking.

5th and 6th graders

The problems used to compare 5th and 6th graders di�er only for the starting �gure of the sequence and for thelast question, used only with 6th graders (Table 4.6).

We have worked with one 5th grade class and two 6th grade classes. The majority of the students have worked outthe answers to questions a) and b) only by using arithmetic and no one has written a general word explanation forhow to �nd the number of squares in a generic �gure.

To answer to question c), which clearly asks about a generic �gure, 7 students out of 19 in the �rst 6th grade classand 17 students out of 26 in the second one have written a general word explanation but no one of them has writtenany formula (Table 4.7).

Also for this problem the majority of the groups have used an arithmetic procedure to answer to the questionsabout a speci�c �gure (Table 4.8). Only one of the �ve groups (that means 4 students out of 26), and only in a class,has written an �unconventional formula�, that is a formula with words or abbreviations.

Figure 4.20 � The unconventional formula written by a group.

An example of formula written using whole word is circled in Figure 4.20. The translation is : the number of the�gure · · · 3 + 1�.

To answer to question c), in the �rst class only two of the �ve groups have written a word algorithm and no onehas written a formula. In the second class there were two word algorithm and three unconventional formulas (Table4.9).

8. Experimentation, started in 2013-14 school year and conducted according to the action research procedures, includes two previousteacher training moments.




Problem : The three following �gures are made up fromsmall squares

Figure 1

Figure 2

Figure 3

1. How many small squares will be there in Figure 10 ?

2. Marco says : �Figure 12 will be made up from 144small squares ! ! !�Marta claims : �Figure 25 will be made up from 225small squares ?�Do you agree with Marco ? And with Marta ? Explainyour answers.

Problem : The three following �gures are made up fromsmall squares.

Figure 1

Figure 2

Figure 3

1. Marco says : �Figure 12 will be made up from 144small squares ! ! !� Do you agree with Marco ? Explainyour answer.

2. Marta claims : �Figure 25 will be made up from 576small squares ?� Do you agree with Marta ? Explainyour answer.

3. Are you able to �nd the number of squares forwhichever �gure ? If you can, explain in which way.

Table 4.6 � The problems for 5th and 6th graders.

AnswersClass Number of

studentsDrawand/orcount

Give an ex-ample

Write aword algo-rithm

Write aformula

6th grade A 19 1 4 76th grade B 26 1 17

Table 4.7 � 6th graders' answers to question c) of �Squares problem�




Problem :Mara and Claudio are playing with some match-sticks. They make up a �gure sequence. The �rst three �g-ures of the sequence are sowed below.

Figure 1

Figure 2

Figure 3

1. Mara says : Figure 20 will be made up from 62 match-sticks ! ! ? Do you agree with Mara ? Explain your an-swer.

2. Claudio claims : Figure 32 will be made up from 128matchsticks ! ! ? Do you agree with Claudio ?

3. In your opinion, can one be able to �nd the numberof matchsticks of whichever �gure ?

AnswersDraw and/or count Use an arithmetic procedure Write a formula

Class Numberofgroups

without aword ex-planation

Add aword ex-planation





6th gradeA

5 3 1 1

6th gradeB

5 3 3 1

Table 4.8 � 6th graders' answers to questions a) and b) of �Matchsticks problem�.

AnswersClass Number of

groupsDrawand/orcount

Give an ex-ample

Write aword algo-rithm

Write aformula

6th grade A 5 1 2 26th grade B 5 2 3

Table 4.9 � 6th graders' answers to question c) of �Matchsticks problem�.




Figure 4.21 � An example of word algorithm.

Individual problem : Mara is playing with some match-sticks. She makes up a �gure sequence. The �rst three �g-ures of the sequence are sowed below.

1. How many matchsticks does Mara need to make upFigure 10 ? Show how you work out your answer.

2. Mara says that, to make up Figure 15, she needs 225matchsticks. Do you agree with Mara ? Explain youranswer.

3. Are you able to �nd the number of matchsticks whichMara needs for whichever �gure ? If you can do it,explain in which way.

Group problem : Claudio has a toy cars collection. Hewant to arrange them over a big table in his room in thefollowing way : in the �rst line he put 6 toy cars, in thesecond he put 4 more than in the �rst, in the third 4 morethan the second and so on. . .

1. How many toy cars will be there in the line 10 ?

2. How many in line 35 ? Show how you work out ouranswer.

3. Are you able to �nd the number of toy cars forwhichever line ? If you can, explain in which way.

4. Which line will have 94 toy cars ? And which line 200toy cars ? Show how you work out your answer.

Table 4.10 � The problems for 8th graders.

An example of word algorithm is shown in Figure 4.21 : the translation of the circled part is �We have found thealgorithm taking out a matchstick and multiplying the number of the �gure · · · 3. Then we have added the missingmatchstick�.

8th graders

The individual problem for 8th graders, in Table 6, was about �gure sequences. In contrast to the 6th graders, 8th

graders (5 out of 23) have written a word algorithm to determine the number of a generic �gure also to answer toquestions a) and b) which are about a speci�c �gure.

Figure 4.22 � An example of general word algorithm




AnswersDraw and/orcount

Give an exam-ple

Write a wordalgorithm

Write a for-mula

0 1 8 5

Table 4.11 � 8th graders' answers to question c) of �Matchsticks problem�.

AnswersUse an arithmetic procedure

Question Drawand/orcount

without aword expla-nation

Add a wordexplanation

Write aformula

a) 4 1b) 3 1 1

Table 4.12 � 8th graders' answers to questions a) and b) of �Toy cars�.

An example of general world algorithm written by an 8th grader student to answer to question b) is underlinedin Figure 4.22. He writes : �. . .because she has to multiply the number of the �gure (which is equal to the number ofmatchsticks in every side) times 4, that is the number of sides in a square.�

In question c), despite these students have already had an algebraic instruction, only 5 out of 23 have written aformula and 8 have written only a word algorithm. The written formulas are not written using conventional algebraicsymbolism but using words or abbreviations, an example is shown in Figure 4. The translation of circled part is :�Number of sides · · · Number of matchsticks on a side�.

Figure 4.23 � An example of general formula written by a student of 8 grade class.

For the group activity a no-geometrical problem has been chosen (Table 6). In this problem there is an additionalquestion which is equivalent to solving a linear equation.

Table 8 shows that, in questions a) and b), only one of the �ve groups (that means 5 students out of 22) has writtena formula for a generic �gure : we have to wait the explicit question c) to have a generalization by the other groups :three groups have written a word algorithm and two groups have written a formula with abbreviations.

The word algorithm written by a student to answer to question c) is �you have to multiply the number of the lineby 4 and to add 2.�(Figure 4.25).

No one has formed or solved any equation to answer to question d) : they clearly write �we have applied the inverseoperation in relation to the original formula� (Figure 6).

Conclusions

In the 6th grade classes, where only a group, that is 4 students out of 45, and only in the second problem haswritten a general procedure to answer to a question about a speci�c �gure, a spontaneous generalization is rare.




Figure 4.24 � An example of tables and formulas written by a group of 8th graders to solve the �Toy cars� problem.

Figure 4.25 � An example of word algorithm written by some students of 8th grade class.

For 8th graders, the percentage of students who have used generalization even when questions didn't ask it, is alittle higher. This fact con�rms the classic studies : development of generalization skills occurs in students from 10to 13 years of age. Anyhow, the not very high percentage of this students (about 22% in both 8th graders' problems)shows that these skills are not completely achieved, despite these activities were carried out at the end of the schoolyear when they have already studied polynomials and linear equations in one unknown .

After an explicit request, 10-11 year-olds have written formulas (with words or abbreviations) in simple general-ization processes : that leads us to say that they are able to deal with symbols.

In the next stages of this study we want to more deeply analyze didactical variables involved in the constructionof knowledge and skills in elementary algebra. In particular we are interested in didactical variables which foster : theunderstanding of the meaning of algebraic expressions and the algorithms linked to these ; the comprehension of themeaning of the solutions to an equation ; the use of an equation as a tool for solving problems in real-life contexts.

In particular, conditions will be analyzed which could foster or prevent the �rst approach to formalized languagein the transition from arithmetic to elementary algebra considering also the peculiarities of the Italian school systemwhere experiential learning is not very widespread (Lai & Polo, 2012) and where teachers usually work with the sameclass at least for three years.

We want to study the emergence of these skills also by monitoring some of the 5th graders already involved in thisactivity next year, when they will be at middle school and some of the 8th graders too, when they will be at high school.




REFERENCES

Carraher D., Schliemann A., Brizuela B. (2000a). From Quantities to Ratio, Functions and Algebraic Relations.Symposium paper. AERA Meeting. New Orleans, April.

Carraher D., Schliemann A., Brizuela B. (2000b). Bringing out the algebraic character of arithmetic : Instantiatingvariables in addition and subtraction. Proceedings of the XXIV Conference of the International group for the Psycologyof Mathematics Education, Hiroschima, Japan, Vol.2, pp. 145-152.

Carraher D., Schliemann A., Brizuela B. (2001). Can young students operate on unknowns ? Proceedings of theXXV Conference of the International Group for the Psychology of Mathematics Education, Utrecht, The Netherlands(invited research forum paper), Vol.1, pp. 130-140.

Carraher D., Schliemann A., Brizuela B. (2012). Algebra in elementary school. Dorier J. & Robert A. (Eds.)Enseignement del l'algèbre élémentaire, La Pensée sauvage, Numero speciale, pp 107-124.

Carraher D. & Schliemann A. (2002). Modeling reasoning. Gravemeijer K., Lehrer R., Oers B. & Verscha�elL. (Eds.). Symbolizing, Modelings and Tools Use in Mathematics Education. The Netherlands : Kluwer AcademicPublischers, pp. 295-304.

Caspi S. & Sfard A. (2012). Spontaneous meta-arithmetic as a �rst step toward school algebra. InternationalJournal of Educational Research, 51-52, pp. 45-65.

Chevallard Y. (1985). Le passage de l'arithmétique à l'algébrique dans l'enseignement des mathématiques au collège�Première partie : l'évolution de la transposition didactique. Petit x 5, pp 51-94.

Chevallard Y. (1994). Enseignement de l'algèbre et transposition didactique. Resoconti del seminario matematicoUniversità e Politecnico Torino 52(2), pp. 175-234.

Dorier J. & Robert A. (Eds.) (2012). Enseignement de l'algébre élémentaire : bilan et perspectives, La penséesauvage.

Filloy E. & Rojano T. (1984). From an arithmetical thought to an algebraic thought. Proceedings of PME-NA, VI,Madison, Wisconsin, pp. 51-56.

Herscovics N. & Linchevski L. (1994). A cognitive gap between arithmetic and algebra. Educational studes inMathematics, Vol 27, n°1, pp 59-78.

Hoch M. (2003). Structure Sense. Proceedings of the Third Conference of the European Society for Research inMathematics Education, Bellaria, Italia.

Johanning D. (2004). Supporting the development of algebraic thinking in middle school : a closer look at students'informal strategies. Journal of mathematical behavior, 23, pp. 371-388.

Katz V. J. (1997). Algebra and its teaching : an historical survey. Journal of mathematical behavior, Ablex Pub-lishing Corp., 16 (1), pp. 25-38.

Lai S. & Polo M. (2012). Construction d'une culture scienti�que pour tous : engagement de l'enseignant et del'élève dans la rupture de pratiques habituelle. Dorier J.L. & Coutat S. (Eds.) Enseignement des mathématiques etcontrat social : enjeux et dé�s pour le 21e siècle �Actes du colloque EMF2012 GT9, pp. 1213'1226.

Linchevski L. (1995). Algebra with numbers and arithmetic with letters : a de�nition of pre-algebra. Journal ofmathematical behavior, 14, pp. 113-120.

Mercier A. (2012). Vous avez dit 'Algèbre ?. Dorier J. & Robert A., (Eds) Enseignement de l'algèbre élémentaire.La Pensée sauvage, Numero speciale, pp 163-180.

Navarra G. (2009). �Il progetto ArAl�. Navarra G. (coord.), Mura F. & Sini S. (Eds) Attività in ambiente earlyalgebra, Collaborazione Progetto ArAl Quaderno n7, pp. 15-24. Self publishing.

Sfard A. (1995). The development of algebra : confronting historical and psychological perspectives. Journal ofmathematical behavior, 14, pp 15-39.

Sfard A. (2007). When the rules of discourse change, but nobody tells you : making sense of mathematics learn-ing from a commognitive standpoint. The journal of the learning sciences, 16(4), pp. 565-613. Lawrence ErlbaumAssociates, Inc.

Sfard A. (2008). Psicologia del pensiero matematico. Il ruolo della comunicazione nello sviluppo cognitivo. Erickson.




4.11 Reasons to Believe : Mathematics and the Reality of Consumerism

Hauke Straehler-Pohl, Uwe Gellert, Nina Bohlmann

Freie Universität Berlin, Germany

Résumé : Les mathématiques formatent certaines décisions que nous prenons et façonnent la manière de percevoir la réalité que nous

vivons. Comment l'idéologie in�uence-t'elle les mathématiques, en a�ectant nos valeurs ou, même, en pénétrant nos désirs ? Après �iºek,

l'idéologie est considérée comme la condition nécessaire pour comprendre le monde dans lequel nous vivons. Quand nous résolvons des

problèmes sociaux en utilisant la mathématisation, nous participons à la manifestation d'un fantasme idéologique : il y a une correspondance

entre la réalité des mathématiques et la réalité sociale. L'un des dilemmes éthiques de la réalité sociale du premier monde est celui des

conséquences de la consommation. Dans notre présentation, nous allons decomposer une video de publicité d'un producteur mondial de

limonade. Notre nous sommes concentrés sur le rôle idéologique des mathématiques : Comment les mathématiques sont utilisées a�n de

permettre au consommateur de se libérer de l'expérience inconscient de culpabilité lié à la consommation et à célébrer une fantaisie agréable

et libératrice ?

Abstract : Mathematics formats many of the decisions we make and it shapes the way we perceive the reality we live in. But what

function serves mathematics for a certain ideology, in in�uencing our values or even penetrating our desires ? Following �iºek, ideology is

seen as the necessary condition for making sense of the world we live in. When we solve problems that emerge from the social reality we

live in by means of using mathematisation, we partake in the manifestation of the ideological fantasy that there is a mapping between the

reality of mathematics and the social reality. One of the ethical dilemmas of the social reality of the �rst world is that of the consequences

of consumption. We will critically analyse a commercial of a global producer of lemonade. Our particular focus will be on the ideological

role of mathematics and how it is used in order to allow the consumer liberating herself from the unconscious experience of guilt related

to consumption and to solemnize a pleasant and liberating fantasy.

Introduction

If one is to answer the question �What is mathematics ¾` from an anthropological stance, one could look at whatmathematics does for those who make use of it. For some mathematicians, mathematics is a means for generatingproblems and solving them within the realm that is spanned by a small set of abstract principles. For other mathe-maticians, mathematics is a source for solving problems that are generated by the realms of nature and technology.For students mathematics is sometimes a means for achieving (or avoiding to be excluded from) a certain economicalstandard (Baldino & Cabral, 2013 ; Pais, 2013 ; Vinner, 1997). The question can also be asked within the realm ofpopular culture (Appelbaum, 1995). In our highly technologized societies, mathematics both explicitly and implicitlyformats many of the decisions we make and it shapes the way we perceive the reality we live in (Gellert & Jablonka,2007 ; Skovsmose, 1994). We could also generate an answer to the question �What is mathematics ¾` by asking forthe role that mathematics plays in the ways that we communicate, that we cook, make music, play sports and so on.We can conduct a further shift of perspectives and ask what function mathematics serves for a certain ideology ininterpellating us, in in�uencing our values or even penetrating our desires. With ideology, we do not refer to a coherent�plot�, with which a human agent, a state-run institution, or a company intentionally manipulates our perception.Instead, following Slavoj �iºek, we see ideology as the necessary condition for being able to make sense of a world,that otherwise would remain contradictory and chaotic to us (�iºek, 2008). Ideologies provide us with narratives thatallow us to compensate the �lack� that we experience, when being confronted with phenomena that contradict the waywe perceive the world. Ideology allows suppressing the dilemmas and cleavages that we face and it allows replacingthem with a coherent narrative. We ask for the function that mathematics serves when supporting an ideology inproviding us with a coherent narrative for an incoherent world. The aim of our paper is to explore ways, in whicha critique of ideology in popular media (viral commercials in our case) can be employed in order to foster students'critical awareness of the hidden functions of mathematics within modern society.

Mathematics and ideology

For any of the di�erent possible users of mathematics that we have hypothesized above, mathematics is a means forcreating an order where there has been disorder, of reducing a chaotic realm to a set of controllable variables. In thisway, mathematics can generate solutions for problems that appeared as unthinkable (Bernstein, 2000) from within thecontext in which the problems emerged. One might object that as long as the user of mathematics does not confuse her




mathematisation of a situation with the situation itself, s/he will not fall prey to an ideological fantasy. However, �iºekmakes us aware that fantasy is not a phenomenon on the level of consciousness, but has to be understood on the levelof our actions (�iºek, 2001). Ideologies do not exist because people believe in the narrative that replaces incoherencewith coherence, but because they act as if the narrative was true, no matter whether they believe in it or not. Hence,when we solve problems that emerge from the social reality we live in by means of mathematisation, we partake inthe manifestation of the ideological fantasy that there is a mapping between the reality of mathematics and the socialreality. Being aware of the gap between mathematics and the social reality and being aware that the mathematizedsituation needs interpretation in order to describe anything �real�, cannot immunise us against falling pray to anideological fantasy (Lundin, 2012). When learning mathematics, students are not seldom required to immerse in thatideology by means of fetishistic disavowal (�iºek, 2005) : They know very well that mathematics does not exactly mapthe world, but still they shall do as if it does, as mathematics provides very practical solutions.

However, as we have outlined above, being immersed in the ideological fantasy of mathematics is not an evil per se,as this immersion allows us to think solutions that have yet been unthinkable. The political question emerges, when weask for the broader ideological constellation that the mathematical fantasy Descartes' dream (Davis & Hersh, 1986)serves. In this paper, we will focus on the entanglement of mathematics with consumerism and re�ect on how thisentanglement could be deconstructed within the mathematics classroom.

The dilemma of consumerism

One of the ethical dilemmas that any person living in the �rst world inevitably faces is that of the consequencesof her/his consumption. It is almost impossible to completely refrain from consuming products that in some way con-tribute to the exploitation of people in a di�erent corner of the world or the exploitation of the planet. Simultaneously,social and ecological sustainability have become regulative ideals of our times. A climate-neutral life that does notnegatively in�uence the life of any other person is rather a desire for some of what ought-to-be than a realizable event.Almost any act of consumption of goods is related to at least a minimum amount of social or ecological �guilt� andconfronts humans with the disorder they create. Even though this guilt is one that the individual "lacking a real choice"can actually not be hold accountable for, this guilt is felt and it releases a desire for its extinction. In order to be ableto live with this supposed guilt (and hence keep on consuming) people have to develop an ideological fantasy-screenthat allows to suppress the traumatic truth, namely that despite their desire to make the world a better place, theyare undercutting this desire by means of their consumption. In this dilemma, the fantasy-screen of ideology providesa rationale that allows making sense of consumption or even allows integrating consumerist acts within the regulativeideal of sustainability (e.g., making the world a better place by consuming organic groceries or fair trade products). Inthis way, ideology allows people to keep on acting and consuming the way they do while keeping up the regulative idealof sustainability. In the following, we will critically analyse a commercial of a globally active producer of lemonade andexplicate the ideological junctions that allow the consumer to hail to a pleasant and liberating fantasy (liberating fromthe unconscious experience of guilt). Our particular focus will be on the role that mathematics plays in the enablingof this fantasy.

Reasons to believe in a better world - and the role of mathematics

The commercial that we are analysing starts by announcing to report on a �study� that has been �conducted�about the �real situation of the world� (Fig. 4.26).

Figure 4.26 � Beginning of the commercial




The mentioning of the real situation can be read as an indication that the following presentation is going todismantle false assumptions that circulate about the state of the world. Even though the small logo of the companyin the upper right corner signals the viewer that s/he is seeing a commercial, the claim for telling the "true story" islegitimated by an emblematic reference to the superiority of the sciences ("a study conducted in 2010"). Even thoughno "research question" is stated explicitly, the following scenes (Fig. 2 to 5) make quite clear that the question beinganswered is whether the world is becoming a worse or a better place.

Figure 4.27 � Capture 2a from commercial Capture 2b from commercial

Figure 4.28 � Capture 5a from commercial Capture 5b from commercial

Figure 4.29 � Capture 6b from commercial Capture 7 from commercial

A multi-ethnic children's choir adds music to the pictures : �I'm free to do whatever I - whatever I choose� . Veryquickly it is equally clear, that the answer to the question is positive : the world is indeed becoming a better place.In this way, the commercial indirectly addresses the viewer's unconscious guilt and releases her from it. The world isbecoming a better place and we are not ruining it. Every bad deed is compensated by a good deed. It is this pointat which numbers start playing their role. Just comparing the images would draw a quite depressive outlook on thesituation of the world. What can stu�ed animals do against a tank ? What can welcome mats do against barb-wiredwalls ? In order to still produce an optimistic outlook, the numbers have to e�ectively serve three functions : a)quantity matters : sure, a stu�ed animal cannot balance a tank, but 131.000 can, b) the choice of numbers suggeststo be actually derived from a �study conducted in 2010� and hence substantiate the scienti�c legitimacy of the claimsbeing made, c) numbers are stripped o� of time and space : it does not matter that the tanks kill people in a di�erentcorner of the world than where the stu�ed animals populate children's rooms ; it does not matter that the mats arenot meant to welcome Palestinians in an apartment of an Israeli family, while the barb-wired wall coops up millionsof Palestinians in one of the world's most crowded and simultaneously poorest areas.

What is striking is that the vast majority of arguments, which the commercial shows for proving that the �realsituation of the world� is improving, refer to consumption (see Fig. 4.28, 4.29 as examples). Summarizing, the com-mercial provides the viewer a fantasy-screen, which allows her to disavow her unconscious feelings of guilt : she canlay back and feel safe. Her consumption is not going to make the world a worse place. Even the opposite, by buying




a stu�ed animal, a welcome mat, a baking mixture for a chocolate cake or a monopoly game, she can even contributeto making the world a better place. As we have demonstrated, mathematics provides the fantastic material to createa coherent narrative for this ideology.

Disentangling mathematics and consumerism

Our analysis has shown, how the ideology of a supposedly value-free mathematics has not itself created a sociallyproblematic message, but how this ideology has been exploited in order to support a consumerist fantasy. Togetherwith their students, mathematics teachers could develop a similar critique in order to foster critical awareness of thepotential roles of mathematics in our modern society. However, teachers do not have to halt at the point of justcriticizing the role of mathematics, but can further broach the issue whether mathematics can also be exploited toundermine rather than to support consumerism.

Remaining within our theoretical frame, we seek support by what �iºek calls overidenti�cation. According to �iºek,�an ideological edi�ce can be undermined by a too-literal identi�cation, which is why its successful functioning requiresa minimal distance from its explicit rules� (2008, p. 29). This would mean to take the commercial more serious than ittakes itself and re�ect what the world would actually look like, if the numbers would really re�ect the �real situationof the world� and would really be results of a scienti�c study.

Students could research the �facts� displayed in the commercials and overidentify with the numerical relations thatare simultaneously supposed, e.g. : How many tanks are produced by the German arms industry ? In 2013, Germanyhas exported 164 tanks to Indonesia. What does that mean in terms of stu�ed animals ? Can Germany compensateby also sending 21,484,000 stu�ed animals ? These re�ections appear cynical en face the actual harm that is actuallydone to actual people. However, making use of overidenti�cation, students can reveal the cynicism underneath theideological fantasy produced by the commercial. Using mathematics in this way can help to e�ectively deconstruct theideological messages that we are confronted with in the media. It can help us resisting the temptation of the feel-goodpromise of consumerism and seek for ways to contribute to a better world on a di�erent terrain.

Figure 4.30 � Capture 12 from commercial

Consumerism is none of them.

REFERENCES

Appelbaum, P. (1995). Popular Culture, Educational Discourse, and Mathematics. Albany : SUNY Press.Baldino, R.R. & Cabral, T.C. (2013). The Productivity of Students' Schoolwork : An Exercise in Marxist Rigour.

Journal for Critical Education Policy Studies, 11(4), 71-84.Bernstein, B. (2000). Pedagogy, Symbolic Control and Identity : Theory, Research, Critique (Rev. ed.). Lanham :

Rowman & Little�eld.Davis, P.J. & Hersh, R. (1986). Descartes' Dream : The World According to Mathematics. San Diego : Harcourt

Brace Jovanovich.Gellert, U. & Jablonka, E. (2007).Mathematization and Demathematization : Social, Philosophical and Educational

Rami�cations. Rotterdam : Sense.Lundin, S. (2012). Hating School, Loving Mathematics : On the Ideological Function of Critique and Reform in

Mathematics Education. Educational Studies in Mathematics, 80(2), 73-85.




Pais, A. (2013). An Ideology Critique of the Use-Value of Mathematics. Educational Studies in Mathematics, 84(1),15-34.

Skovsmose, O. (1994). Towards a Philosophy of Critical Mathematics Education. Dordrecht : Kluwer.Vinner, S. (1997). From Intuition to Inhibition : Mathematics Education and Other Endangered Species. In E.

Pehkonen (Ed.), Proceedings of the 21th Conference of the International Group for Psychology of Mathematics Edu-cation, Vol. 1 (pp. 63-78). Helsinki : University of Helsinki.

�iºek, S. (2001). Enjoy Your Symptom ! : Jacques Lacan in Hollywood and out. London : Routledge.�iºek, S. (2005). The Metastases of Enjoyment : Six Essays on Women and Causality. London : Verso.�iºek, S. (2008). The Ticklish Subject. London : Verso.




4.12 Mathematics is more than the mathematics lesson

Audrey Cooke

Curtin University, Perth, Western Australia

Résumé : L'intégration des mathématiques tout au long curriculum se fait au béné�ce des élèves étudiant les mathématiques, parce

qu'elle permet de montrer comment les mathématiques sont utilisées dans le monde réel ce qui procure aux élèves une expérience qu'ils

apprécient. Cependant, la façon dont les enseignants intègrent les mathématiques varient et ces variations peuvent avoir des conséquences

sur l'expérience d'apprentissage mathématique. Les représentations que les enseignants se font de l'intégration des mathématiques peuvent

modi�er grandement cet apprentissage. De plus, les dispositions des professeurs vis-à-vis des mathématiques elles-mêmes, leurs croyances,

leurs craintes ou leur con�ance tout comme leurs conceptions des mathématiques peuvent in�uencer leur vision de l'intégration des maths

aussi bien que les activités d'apprentissage qu'ils créent. Ce papier, théorique, discute des possibles formes d'in�uence et propose des moyens

pour décrire leurs e�ets.

Abstract : Integrating mathematics across the curriculum is of bene�t for children learning mathematics as it can provide opportunities

that demonstrate how mathematics is used in the real world and create experiences that children enjoy. However, how teachers integrate

mathematics can vary and this, in turn, can impact on what happens in the mathematics learning experience. Teacher views of what it

means to integrate mathematics impact on how mathematics is integrated in learning experiences. In addition, teacher disposition towards

mathematics - their beliefs about, attitudes towards, anxiety and con�dence with, and conceptualisation of mathematics - can in�uence

both how teachers view mathematics integration and the mathematics learning experiences they create. This theoretical paper discusses

the potential forms of this in�uence and proposes ways to describe their e�ect.

Introduction

In her investigation into curriculum innovation in OECD countries, Kärkkäinen (2012) stated that the use ofintegrated studies across the curriculum is increasing. She found that literacy and numeracy were integrated withinother curriculum areas to enable children to develop these important life skills. However, mathematics integrationacross curriculum areas can be achieved in di�erent ways and it is teacher beliefs that can be the greatest determiningfactor on how integration is achieved. In their research with secondary teachers, de Araujo & al. (2013) created aframework to describe how teachers conceptualised mathematics integration. de Araujo & al. (2013) proposed thattheir framework addressed mathematics integration in terms of both how mathematics was integrated - between thestrands of mathematics, through topics, across disciplines, or through context - as well as how the integration wassituated temporally. Two of the types of integration were across mathematics rather than across the curriculum andtwo were across the curriculum. Of the two that were across the curriculum, one considered integration across thecurriculum as the insertion of mathematics into other disciplines. The other considered integration as based on context,where the focus was on creating a real world situation that may happen to utilise ideas from other disciplines (that is,across the curriculum as a by-product of the context). The creation of real world contexts is one of the key points theOrganisation for Economic Co-operation and Development [OECD] (2013) focuses on when discussing the developmentof mathematic literacy and, when the framework created by de Araujo & al. (2013) is considered, it would seem toindicate that context provides additional bene�ts of integration.

An integrated approach to mathematics necessitates changes in teaching and learning opportunities. Trammel(2001) described how integrated mathematics involved organising content and teaching di�erently to traditional math-ematics lessons. He outlined the class experience as starting with the presentation of a problem - within a realisticcontext - that needs to be solved and the change in the teacher role to that of facilitator - asking questions to getchildren to examine their ideas as they work on solutions. These changes will be dependent on the teacher as it isthe teacher who determines the classroom experiences (Katz & Raths, 1985). However, the importance of integrat-ing mathematics necessitates these changes, particularly the opportunities an integrated mathematics experience canprovide in terms of motivation for children and a recognition that mathematics can open up the world outside of theclassroom (Ellis 2005), as well as the development of numeracy or mathematical literacy (OECD, 2013).

More than mathematics

Teacher mathematical knowledge is required for teachers to be able to teach mathematics (Beswick, Watson, &Brown 2006). However, as Beswick, Callingham, and Watson (2012) proposed, it is not just mathematical knowledge




that is required - con�dence with and beliefs about mathematics are also involved. Ernest (1989) considered teacherconceptualisation of mathematics - their �mathematical philosophy� - as an additional factor that would impact on theteaching and learning of mathematics in their classroom, particularly in terms of the mathematical experiences theywould create. Teacher mathematics anxiety can also impact on teaching and learning experiences. Swars, Daane, andGiesen, (2006) demonstrated connections between mathematics anxiety and the willingness of the teacher to adapt theirteaching and learning experiences, with higher mathematics anxiety linked to decreased likelihood of changing. Theseelements - teacher beliefs about, attitudes towards, anxiety and con�dence with, and conceptualisation of mathematics- have been gathered as components of disposition towards mathematics (Cooke, 2014).

Disposition towards mathematics incorporates components that have been found to impact on how teachers ap-proach mathematics teaching and learning. As such, teacher disposition towards mathematics would also impact onmathematics integration across the curriculum. Ernest (1989) connects the teacher's view of mathematics with howthey believe mathematics should be taught and how they believe children learn mathematics, although these are ame-liorated by the social context of teaching (such as what happens in the classroom, the school, and the community).That is, teacher beliefs regarding mathematics could impact on what the teacher does when teaching (or preparingto teach) mathematics. Ernest (1989) proposed that it is the beliefs of the teacher, rather than the knowledge thatis held, that will di�erentiate what is done in the classroom. Remillard and Bryans' (2004) �ndings connect teacherbeliefs with how the curriculum is used, speci�cally, what mathematics is, how mathematics is learned, and the roleof the teacher were found to impact on how the curriculum was enacted in the classroom.

Beswick & al. (2006, p. 69) discussed the importance of teachers having su�cient con�dence and a level of joy withmathematics, together with su�cient knowledge, to enable them to �play with mathematical ideas. . . to see connectionsbetween ideas, and to imagine possible avenues for exploration�. The connection between knowledge and con�dence wasdeveloped further by Beswick & al. (2012). They proposed that teacher knowledge about mathematics incorporatedboth teacher beliefs about mathematics and con�dence with mathematics in terms of developing their students'understanding and critical numeracy, integration with other curriculum areas, and assessing student achievementagainst new state-based standards. Teachers with lower level of knowledge or con�dence may also be more likely to usetextbook and pre-prepared commercial worksheets for teaching and learning experiences in mathematics (Choppin,2011). Likewise, teachers with mathematics anxiety may focus more on procedural knowledge rather than problem-solving and reasoning (Swars & al., 2006).

Disposition towards mathematics and mathematics integration

Kemp and Hogan (2000) stated that mathematics can be imbedded into any curriculum area where students needto �use mathematics in order to do something - complete a task, make a model, understand a new concept, or solve aproblem� (p. 14). If integration of mathematics across the curriculum involves the application of mathematics, thenthere needs to be the desire and motivation to use mathematics - if not, there is the risk of what Kemp and Hogan(2000) described as students avoiding mathematics and using any other strategy that will provide �a good enoughresult� (p. 13). As the teacher is the linchpin regarding how mathematics experiences are created in the classroom(Ernest 1989), the teacher's disposition towards mathematics needs to be considered as this can impact on choicesmade regarding teaching, learning, and integration.

Disposition towards mathematics and integration of mathematics can be connected by the amalgamation of theframework proposed by de Araujo & al. (2013) with Ernest's (1989) conceptualisation of mathematics, mathematicsteaching, and mathematics learning, as shown in Figure 4.31. If a teacher's disposition towards mathematics incorpo-rates the view that mathematics is a set of rules to learn (Ernest, 1989), if they are not con�dent in using mathematics(Beswick & al., 2006), or if they rely on text books (Choppin, 2011), and they resist modi�cations to their teachingand learning experiences (Swars & al., 2006), then their disposition towards mathematics may be disparate with theintegration of mathematics across the curriculum. This would place them in the left-most vertical column and couldresult in situations where mathematics is not seen as essential to the learning experience (Kemp & Hogan 2000).

If the teacher conceives mathematics as �uid and connected (Beswick & al., 2006), was con�dent in its use (Beswick& al., 2012), had lower anxiety towards mathematics (Swars & al., 2006), saw the teacher's role as a facilitatorencouraging problem solving (Trammel, 2001), were more likely to engage in inquiry-based approaches (Wilkins,2008), and were more con�dent in their ability to teach mathematics (Swars & al., 2006), then their dispositiontowards mathematics may be more commensurate with integrating mathematics across the curriculum. This could bedue to the teacher being more likely to play with and explore mathematics (Beswick & al., 2012) and to adapt theirteaching and the learning experiences in their mathematics classroom (Swars & al., 2006) - elements of the right-most




Figure 4.31 � Connecting Ernest (1989) and de Araujo & al. (2013).

column of Figure 4.31.The right-most column of Figure 4.31 also re�ects the four dimensions Beane (1996) used to described curriculum

integration - the use of real world problems, the focus on the learning experience using knowledge (regardless of thecurriculum area the knowledge comes from), the learning experience as the outcome rather than a test on the knowledgeused (indeed, it could be argued that using the knowledge within the learning experience is su�cient assessment),and real and meaningful problem solving experiences that further develop that knowledge and problem solving. Thesedimensions re�ect the opportunities that are most likely to result in the bene�ts Czerniak, Weber, Sandman, & Ahern(1999) described - a more realistic re�ection of the world that generates student interest in the content.

Conclusion

The teacher needs to have a disposition towards mathematics commensurate with creating integrated learningexperiences where mathematics is used in a way that allows children to see that it is necessary and e�ective butalso generates a willingness to use it (Ellis, 2005 ; Kemp & Hogan, 2000 ; Trammel, 2001). However, integratingmathematics across the curriculum - as outlined in the right-most column of Figure 4.31 - requires teachers to havea level of knowledge about and understanding of mathematics and the other curriculum areas su�ciently to createrealistic, meaningful, and applicable experiences (Beane, 1996). They also need to have a willingness to explore andplay with mathematics (Beswick & al., 2006). As a result, more than knowledge is needed by e�ective teachers ofmathematics (Beswick, & al., 2012 ; Ernest, 1989) - teacher disposition towards mathematics - beliefs about, attitudestowards, anxiety and con�dence with, and conceptualisation of mathematics - can contribute to the teacher's creationand use of mathematical learning experiences (Beswick & al., 2006 ; Ernest 1989 ; Swars & al., 2006). If mathematicsis to be integrated across the curriculum in the ways outlined by Beane (1996) and evident in the right-most columnof Figure 4.31, then teachers' dispositions towards mathematics need to be considered together with mathematicalknowledge to ensure this occurs.

REFERENCES

Beane, J. (1996). On the Shoulders of Giants ! The Case for Curriculum Integration. Middle School Journal, 28(1),6-11. Retrieved from http ://www.jstor.org/stable/23024059

Beswick, K., Callingham, R., & Watson, J. (2012). The nature and development of middle school mathematicsteachers' knowledge. Journal of Mathematics Teacher Education, 15, 131-157. doi : 12.1007/s10857-011-9177-9

Beswick, K., Watson, J., & Brown, N. (2006). Teachers' con�dence and beliefs and their students' attitudes tomathematics. In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds.), Identities, cultures and learning spaces :Proceedings of the 29th annual conference of the Mathematics Education Research Group of Australasia, 1, 68-75).Retrieved from http ://www.merga.net.au/documents/RP42006.pdf




Choppin, J. (2011). The role of local theories : Teacher knowledge and its impact on engaging students withchallenging tasks. Mathematics Education research Journal, 23(5), 5-25. doi : 10.1007/s13394-011-0001-8

Cooke, A. (2014). Considering Pre-service Teacher Disposition Towards Mathematics. Manuscript submitted forpublication.

Czerniak, C. M., Weber, W. B., Sandmann, A., & Ahern, J. (1999). A literature review of science and mathematicsintegration. School Science and Mathematics, 99(8), 421-430. doi : 10.1111/j.1949-8594.1999.tb17504.x

de Araujo, Z., Jacobson, E., Singletary, L., Wilson, P., Lowe, L., & Marshall, A. M. (2013). Teachers' conceptionsof integrated mathematics curricula. School Science and Mathematics, 113(6), 285-296. doi : 10.1111/ssm.12028

Ellis, K (2005). Integrating integers across disciplines. Retrieved from http ://www.edutopia.org/math-coaching-integrated-curriculum

Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. In C. Keitel with P. Damerow, A. Bishop, &P. Gerdes, (Eds.).Mathematics, Education, and Society (pp. 99-101). Retrieved from http ://unesdoc.unesco.org/images/0008/000850/085082eo.pdf

Kärkkäinen, K. (2012). Bringing About Curriculum Innovations : Implicit Approaches in the OECD Area (OECDEducation Working Papers, No. 82). doi : 10.1787/5k95qw8xzl8s-en

Katz, L. G. & Raths, J. D. (1985). Dispositions as goals for teacher education. Teaching & Teacher Education,1(4), 301-307. doi : 10.1016/0742-051X(85)90018-6

Kemp, M. & Hogan, J. (2000). Planning for an emphasis on numeracy in the curriculum. Retrieved from www.aamt.edu.au/content/download/1251/25266/�le/kemp-hog.pdf

Organisation for Economic Co-operation and Development [OECD] (2013). PISA 2012 Assessment and analyticalframework : Mathematics, reading, science, problem solving and �nancial literacy. http ://dx.doi.org/10.1787/9789264190511-en

Remillard, J. T., & Bryans, M. B. (2004). Teachers' orientations toward mathematics curriculum materials : Im-plications for teacher learning. Journal for Research in Mathematics Education, 35(5), 352-388. doi :10.2307/30034820

Swars, S. L., Daane, C. J., & Giesen, J. (2006). Mathematics anxiety and mathematics teacher e�cacy : What is therelationship in elementary preservice teachers ? School Science and Mathematics, 106(7), 306-315. doi : 10.1111/j.1949-8594.2006.tb17921.x

Trammel, B. (2001). Integrated mathematics ? Yes, but teachers need support ! Retrieved from http ://www.nctm.org/resources/content.aspx ?id=1712Wilkins, J. L. M. (2008). The relationship among elementary teachers' content knowledge, attitudes, beliefs, and

practices. Journal of Mathematics Teacher Education, 11, 139-164. doi : 10.1007/s10857-007-9068-2




4.13 Didactical Desiderata from a Sociological Approach to Logic

David Kollosche

Universität Potsdam, Potsdam, Germany

Résumé : En tant que phénomène culturel, le raisonnement logique n'est pas exempt de dimensions culturelles. Bien que la recherche

sur l'impact de la logique sur l'enseignement des mathématiques se concentre souvent sur les questions épistémologiques, psychologiques

et didactiques, cette contribution o�re une vue sociologique sur les dimensions culturelles de la logique dans la société et la classe de

mathématiques. Il esquisse un cadre sociologique sur la base duquel l'impact social de la logique peut être analysé, et puise dans une

étude antérieure sur la généalogie de la logique, et propose quelques conclusions didactiques pour l'apprentissage de la logique et des

mathématiques.

Abstract : As a cultural phenomenon, logical reasoning is not free from cultural dimensions. While research on the impact of logic

on mathematics education often concentrates on epistemological, psychological and didactical issues, this contribution o�ers a sociological

view on cultural dimensions of logic in society and the mathematics classroom. It sketches a sociological frame on the basis of which the

social impact of logic can be analysed, draws from an earlier study on the genealogy of logic, and proposes some didactical conclusions for

the learning of logic and mathematics.

Towards a Sociology of Logic

Logic is a central theme in any philosophy of mathematics education, not only because it is consid-ered an essentialfeature of mathematics itself, but also because logical reasoning is believed to hold potential for individual emancipationand social progress. The Commission for the Study and Im-provement of Mathematics Teaching (CIEAEM) dedicatesone of four sub-themes of its annual con-ference in 2014 to the discussion of the nature and learning of logicalreasoning. This sub-theme focuses logic from the perspectives of epistemology, cognitive psychology and didactics,whereas sociological perspectives on the phenomenon of logic are widely neglected. A sociological approach towards theuse of logic faces two main challenges. Firstly, it needs to provide theoretical tools for the analysis of social dimensionsof logic both in the macro-domain of society and in the micro-domain of the mathematics classroom. While in the�rst domain we face questions of power and control executed by groups of people, the second domain makes us thinkabout issues of emancipa-tion and subjection of individuals. The usefulness of a sociological approach will dependon how well it can provide a combined view on both domains. Secondly, the epistemological range of a sociologicalapproach will depend on its ability to critically handle convictions connected to the issue in focus. Concerning logic,it has often been argued that logical thinking is the only 'reasona-ble' or 'legitimate' form of thinking and that,therefore, education in logical thinking necessarily bene�ts both the learner and society in general. However, as thereare numerous other productive forms of thinking and as the existence of a 'natural' development towards a logicalform of thinking cannot be proved, these claims cannot be solidi�ed in any academic sense, but belong to the coreconvictions of the philosophy of Ancient Greece and the enlightened modernity. In order to avoid any ideologicalrestriction of the sociological perspective, these very convictions themselves shall be handled as objects of analysis ina sociological approach towards logic.

The work of the French philosopher, psychologist and sociologist Michel Foucault provides a theory of the socialwhich allows considering the before-mentioned challenges. Foucault (1991 ; 2011) rejects the idea that power is a goodwhich a person or a group can possess. Instead, he considers power as the control of techniques for the governmentof others or the self. For example, an em-ployer may demand punctuality from his employees. Threatening with wagecuts or dismissal may be his techniques for the government of his employees. However, the employee is not told howto achieve punctuality. He in turn has to cultivate techniques for the government of the self ' a process Foucault callsascesis ', in this case techniques that allow punctuality, e.g. by buying a watch or adjusting personal attention. Thisexample does not only illustrate techniques for the government of others or the self, it also points to a very special formof techniques which Foucault (1977) calls disciplinary techniques. These techniques allow the government of othersthrough their government of the self. The e�ectiveness of disciplinary techniques lies within the subjecti�cation of theindi-vidual who develops techniques of the self and thereby internalises the originally external demands. Eventually,the disciplinary demand becomes a part of the personality of the individual who, from then on, may even defend thisdemand or claim it from others. Apart from that, knowledge is noth-ing isolated from power, but connected to it ina symbiotic sense. On the one hand, techniques of power can allow or prevent knowledge to develop, to spread andto gain legitimisation. On the other hand, knowledge itself can develop, improve and legitimise techniques of power.




As an example, Foucault (1970) argues that in modernity, the humanities develop hand in hand with disciplinarytechniques throughout society.

Based on the theories of Foucault, Valerie Walkerdine (1988) discussed how 'reason' is not the re-sult of a 'natural'development of individual thinking, but a social construction. But while her work focuses on the construction of 'reason'in early childhood, cultural dimensions of 'reason' are not systematically analysed. Di�erent to Walkerdine's approach,my analysis will address the following Foucaultian questions around social dimensions of logic : Can logical reasoningbe considered a technology for the government of others, of the self, or even as a disciplinary technology ? Whichascesis does it require' Wither the social need for such a technology' Who and which knowledge legitimises logic ? Whoand which knowledge is legitimised by it ? And how are mathematics and mathematics education connected to thesesocial dimensions of logic ?

Findings from a Genealogy of Logic

An earlier study (Kollosche, 2013) following a genealogic approach aimed at identifying social di-mensions of logic.Genealogy is a method introduced by Nietzsche und Foucault (1984 ; comp. Lightbody, 2010). It aims at �nding socialdimensions of phenomena that have become familiar and natural but once had been original and controversial. Placinga taken-for-granted phenomenon in history provokes its alienation and allows its return as a concrete object of study.By looking at the genesis of a phenomenon, at its struggles, the alternatives it competed with and the interests itserved, genealogy highlights social connections, possibilities and restrictions of the phenomenon.

The idea of logic has a wide range of meanings in mathematics, philosophy, psychology and public discourses.In order to focus on a well-determined object of analysis, the following thoughts will address four principles whichScholasticism has identi�ed within the work of Aristotle. These prin-ciples are a sensible reduction as they constitutethe basis of Aristotle's work on logic which in�u-enced both the discourses on logic throughout the Middle Ages andthe structure of mathematics as it was coined by Euclid's Elements :

1. Law of identity. Everything stays the same, nothing changes. This law postulates the existence of never-chancingand ever-reliable objects or concepts for which the term 'truth' was introduced by the Pre-Socratic philosopherParmenides (2009).

2. Law of excluded middle. Everything is or is not ; there is no other way. This law restricts our judgements to twocategories, e.g. truth and the false, and leaves no other option.

3. Law of excluded contradiction. Nothing is and is not. This law demands a decision between the two categories.Combined with the law of excluded middle, it forces our judgements into an an-tagonism of true and false, ofbeing and not being, and leaves no room beyond the extremes.

4. Law of su�cient reason. Everything but one thing has a reason and is de�ned by it. Distancing himself frommythological thought, the Pre-Socratic philosopher Anaximander introduced the science-founding claim thatnearly everything has a reason which is its destiny.

The genealogical analysis of the four principles of Aristotelian logic reveals that logic has religious, epistemologicaland political dimensions which can be understood in a dialectics of possibilities and restraints. On a religious dimension,the belief in an imperishable truth has the potential to appease people who feel threatened by changes or decay, whereasthe inalterability of truth has the potential to frighten people who have an actively formative attitude towards ourworld. On an epistemologi-cal dimension, logical thinking introduces a productive order of thought by abstracting thefamiliar patriarchal order of society, whereas it narrows down the intellectual focus to those aspects of a phenomenonwhich can be expressed in the terms of logic. On a political dimension, logical reason-ing allows public decision-making on the basis of discussions instead of physical violence, whereas the rhetorical power it provides is not equallydistributed among society and can be used to subju-gate 'the uneducated'.

It shows that logic can be understood as a disciplinary technology. Firstly, logical thinking is a tech-nology for thegovernment of the self which is attractive for the religious, epistemological and polit-ical bene�ts it promises to theindividual. Secondly, logical reasoning is a technology for the gov-ernment of others. Either it operates through thelogical thinking of the individual on the basis of which it can legitimise its academic, ethical or political claims, orit excludes the individual from the discourse. Such exclusion becomes apparent in the writings of Parmenides (2009).But also Aristotle attests a 'lack of education' to those unwilling to think logically (1933, p. 1006). Thus, logic can beused to legitimise academic, ethical or political discourses. At the same time, these very discourses, e.g. the philosophyof Aristotle, help to legitimise logic as a technology of power.




As mathematics is no empirical science, its objects can be abstracted as far as necessary from reality to perfectly �tinto the patterns of logical argumentation. This is why rationalism regards mathemat-ics as a role model 'in our searchfor the direct road towards truth ? (Descartes, 1990, pp. 225). Ob-viously, a large part of the authority of mathematicsis legitimised by its dedication to logic.

Logic in School Mathematics

The school mathematics discourse is widely in�uenced by logic. Although the extend of logical rea-soning byteachers and students in mathematics classrooms varies among countries and teachers, the contents of mathematicseducation are essentially formed by logic. Most curriculums only include contents which demonstrate the power of logicwhile contents that could threaten its prestige are excluded. E.g. they exclude non-Euclidean geometry, paradoxes of settheory or alternative logics, whereas they add calculus and probability theory to the Euclidean core ' contents whichdem-onstrate how even the in�nite and chance can be mastered by logic. The logical form of school mathematicsimplies that its understanding depends on and/or cultivates logical thinking. An exam-ple from a common schoolbook for the 7th grade of German high school might illustrate this claim. The school book presents the following textwithout any further explanation, but with a sketch showing a circle and a line of each type (Brückner, 2008, p. 142 ;my translation) :

Lines and circle can have di�erent locational relations.Secant : a line that cuts a curve (g1)Tangent : a line that touches a curve (g2)Passant : a line that avoids a curve ? the passing line (g3)A student not yet knowing these terms could rightly ask whether a tangent can be a passant as it does not intersect

but passes the circle, or whether a secant can be a tangent as you cannot cut with-out touching. Only a student familiarwith the idea of classi�cation will know that it is forbidden to place a line in more than one or in none of the threecategories and that the de�nition is 'meant like that'.

Classi�cation, however, is a logical concept resting on the laws of excluded middle and ex-cluded contradiction.Thus, this is an example of mathematical contents whose logical formation is not stated explicitly. Instead, theunderstanding of the student depends on the understanding of the logical formation of mathematics which can onlybe learnt implicitly from such examples. Thus, mathematics education does not only cultivate a culturally unre�ectedform of thinking, it also pro-vides this power unequally. Those who understand the latent order of mathematics gainthe possibil-ity to perform well and become con�dent in the use of mathematics while those who do not under-standits latent order are excluded from its power. As Bernstein-based socio-linguistic research in mathematics education(Cooper & Dunne, 2000 ; Gellert & Jablonka, 2007) has shown, the access to this understanding largely depends onthe socio-economic background of the students.

In Foucault's terms, mathematics education can then be understood as a disciplinary institution. The exposureof students to situations whose mastery depends on logical thinking is a disciplinary tech-nology which requires thestudents to develop techniques for the government of the self. Such a technique either allows success by participatingin the logical discourse or results in exclusion by rejecting the logical form of reasoning. As the latter form of ascesis issanctioned by unfavourable grading, students' achievements in school mathematics may be considered an indicator forthe abil-ity of and will to logical thinking. Thus, mathematics education may serve as an institutionalised mechanismfor the (re-)production of a logically thinking 'elite' and the exclusion of potential threats to the logical order of socialdiscourses which has become constitutive for modern societies.

Conclusion

Logic is not the only legitimate form of reasoning, but a very speci�c one that promises bene�ts and demandssacri�ces. While the techniques for the government of others and the self provided by logic can explain its social impactand the importance of an education in logical thinking, the short-comings of logical thinking show why it is neitherwise to overrate the potential of logical thinking, nor to regard students who might have good reasons not to thinklogically as intellectually de�cient. Eventually, I argue that both the students' emancipation and the understanding oflogic and mathe-matics would bene�t from a more critical approach towards logic. Such an approach would includean explicit exposure of the logical imprint on mathematics, a discussion of the possibilities and re-strictions of logic as




well as the approval of an individual ' sometimes even critical ' estimation of logical thinking. Only such an approacho�ers the opportunity to learn both how to apply logical thinking correctly and where to trust or not to trust in it.

REFERENCES

Aristotle. (1933). Metaphysics. In H. Tredennick (Ed.), Aristotle in 23 Volumes. London : Heinemann.Brückner, A. (Ed.). (2008). Mathematik 7. Gymnasium Brandenburg. Berlin : Duden.Cooper, B., & Dunne, M. (2000). Assessing Children's Mathematical Knowledge : Social Class, Sex, and Problem-

solving. Buckingham : Open University.Descartes, R. (1990). Rules for the Direction of the Mind. In M. J. Adler (Ed.), Great Books of the Western World

(pp. 223-262). Chicago : Encyclopædia Britannica.Foucault, M. (1970). The Order of Things : An Archaeology of the Human Sciences. New York : Pantheon (Original

work published 1966).Foucault, M. (1977). Discipline and Punish : The Birth of the Prison. New York : Pantheon Books (Original work

published 1975).Foucault, M. (1984). Nietzsche, Genealogy, History. In P. Rabinow (Ed.), The Foucault Reader. 1st ed., pp. 76-100.

New York : Pantheon.Foucault, M. (1991). Governmentality. In G. Burchell, C. Gordon, & P. Miller (Eds.), The Foucault E�ect (pp.

87-104). Chicago : University of Chicago Press.Foucault, M. (2011). The Government of Self and Others : Lectures at the Collège de France, 1982-1983. New

York : Picador.Gellert, U., & Jablonka, E. (Eds.). (2007). Mathematisation and Demathematisation : Social, Philosophical and

Educational Rami�cations. Rotterdam : Sense.Kollosche, D. (2013). Logic, Society and School Mathematics. In B. Ubuz & al. (Eds.), Proceedings of the Eighth

Congress of the European Society for Research in Mathematics Education (pp. 1754-1763). Ankara : Middle EastTechnical University.

Lightbody, B. (2010). Philosophical Genealogy : An Epistemological Resonstruction of Nietzsche and Foucault'sGenealogical Method. Volume I. New York : Peter Lang.

Parmenides. (2009). Fragmente. In M. L. Gemelli Marciano (Ed.), Die Vorsokratiker. Band 2 (pp. 6'41). Düsseldorf :Artemis & Winkler.

Walkerdine, V. (1988). The Mastery of Reason : Cognitive Development and the Production of Rationality. London :Routledge.




Table des �gures

4.1 Répartition des tâches de chaque communauté et interactions . . . . . . . . . . . . . . . . . . . . . . . 1344.2 Rapprochement des problèmes et des thèmes mathématiques . . . . . . . . . . . . . . . . . . . . . . . 1354.3 Problèmes de recherche et n÷uds de savoir, en lien avec les programmes de la classe de quatrième en

France . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.4 Liens entre un problème de recherche et les savoirs en jeu . . . . . . . . . . . . . . . . . . . . . . . . . 1374.5 Distribution of correct responses in ordering and equivalence problems of fractions presented in part-

whole situations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604.6 Distribution of correct responses in ordering and equivalence problems of fractions presented in part-

whole situations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604.7 Distribution of correct responses in partitive and quotitive division situations. . . . . . . . . . . . . . . 1614.8 Valid arguments based on proportional reasoning when solving equivalence fraction problems in quotient

interpretation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1624.9 Valid arguments presented when solving ordering and equivalence fraction problems in part-whole in-

terpretation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1624.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1674.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1674.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1694.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1694.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1704.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1704.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.20 The unconventional formula written by a group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814.21 An example of word algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1844.22 An example of general word algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1844.23 An example of general formula written by a student of 8 grade class. . . . . . . . . . . . . . . . . . . . 1854.24 An example of tables and formulas written by a group of 8th graders to solve the �Toy cars� problem. 1864.25 An example of word algorithm written by some students of 8th grade class. . . . . . . . . . . . . . . . 1864.26 Beginning of the commercial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894.27 Capture 2a from commercial Capture 2b from commercial . . . . . . . . . . . . . . . . . . . . . . . . . 1904.28 Capture 5a from commercial Capture 5b from commercial . . . . . . . . . . . . . . . . . . . . . . . . . 1904.29 Capture 6b from commercial Capture 7 from commercial . . . . . . . . . . . . . . . . . . . . . . . . . . 1904.30 Capture 12 from commercial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1914.31 Connecting Ernest (1989) and de Araujo & al. (2013). . . . . . . . . . . . . . . . . . . . . . . . . . . . 195


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