L'ÉCRITURE DE L'EXIL DANS LIBRO DE NAVÍOS Y BORRASCAS DE DANIEL MOYANO

L'écrivain argentin Daniel Moyano, né en 1930 à Buenos Aires et mort à Madrid en 1992, élabore une mythologie personnelle qui porte le sceau de l'exil en se présentant, dans un entretien avec Virginia Gil Amate et dans sa lettre à Claude Cymerman du 22 août 1989, comme un homme constamment déplacé et déraciné. Marqué dans son enfance par ce qu'il nomme « l'exil argentin »1 de son grand-père maternel, l'Italien Giuseppe Bellini, Moyano interprète comme un exil intérieur ses déplacements d'une province argentine à l'autre2 , avant le début, en 1976, de ce qu'il désigne comme son « dur exil externe » 3 en Espagne. Comme l'indique Sara Bonnardel dans un article intitulé « Libro de navíos y borrascas, los aprendizajes del exilio », si le thème de l'exil est très présent dans l'oeuvre narrative de Daniel Moyano, le cinquième roman de cet auteur « amplifie et enrichit la réflexion sur l'exil ». 4 Dans Libro de navíos y borrascas, roman d'inspiration autobiographique écrit par Moyano lors de son exil définitif en Espagne, la fiction trouve sa source d'inspiration dans le référent historique de la dictature des généraux en Argentine entre 1976 et 1983, date de publication de l'oeuvre. Libro de navíos y borrascas narre le voyage en exil de 700 « conosurenses » 5 victimes de la répression militaire, qui font la traversée de Buenos Aires à Barcelone à bord d'un bateau italien symboliquement baptisé Cristóforo Colombo. Alfred Melon souligne à quel point le nom et la nationalité du bateau se chargent d'une portée symbolique, en « [rebroussant] à la fois le temps et l'espace du périple colombien, mais aussi le temps et l'espace qu'ont parcourus les immigrés du peuplement argentin ». 6 Libro de navíos y borrascas s'impose comme une oeuvre d'une grande variété thématique, relevant d'une complexe élaboration littéraire tant par sa structure romanesque et sa polyphonie narrative que par la réflexion métatextuelle qui l'habite, sans oublier le recours à l'intertextualité musicale. L'écriture de l'exil de Daniel Moyano est d'ailleurs placée sous le sceau de la musique, qui lui prête son caractère singulier.

Le titre Libro de navíos y borrascas joue sur le double registre de l'Histoire et de la fiction à travers la polysémie du mot « libro », tout à la fois annonciateur d'une chroniquejournal de bord d'une traversée -et d'une oeuvre littéraire. Les substantifs « navíos » et « borrascas », étroitement liés par la conjonction de coordination, acquièrent une forte valeur symbolique en convoquant le motif de la mer, indissociable, tout au long du roman, du thème de l'exil. L'alliance des deux pluriels indéfinis éveille, comme le remarque Sylvie Koller, des réminiscences culturelles et mythiques, « rappel de l'Odyssée, de l'Arche de Noé, des récits de naufrage, du vaisseau fantôme »,7 qui élargissent l'expérience de l'exil à une expérience humaine universelle. L'association de termes « navíos y borrascas » se teinte par ailleurs d'une connotation politique par allusion au naufrage historique qu'ont subi les victimes des dictatures militaires du cône sud. Naufrage historique explicitement évoqué dans les premières pages du roman :

« El juego consiste ahora en mover un barco italiano real llamado Cristóforo Colombo, a punto de zarpar del puerto de Buenos Aires con setecientos no deseables a bordo, sobrevivientes de un naufragio cuidadosamente buscado por eso que llaman la Historia, la aburrida suma de los acontecimientos menudos de todos los días, entre los que la gente vive y muere casi sin saberlo ». 8 Le substantif « juego », qu'il convient de relier au projet littéraire de transposition fictionnelle de l'Histoire, entre en tension avec l'adjectif « réel » qui qualifie le référent spatial du bateau, attirant l'attention sur la constante interaction entre réel et imaginaire dans ce récit de départ en exil.

Le roman commence par l'arrivée en fourgon cellulaire de Rolando, principal narrateur du roman, au port de Buenos Aires, et s'achève sitôt après le débarquement des passagers du Cristóforo Colombo au port de Barcelone. Moyano fait ainsi le choix de ne pas raconter les années d'exil mais de se centrer sur le temps de la traversée, tout en se tournant vers le passé et le futur grâce à de nombreux analepses et prolepses.

La structure romanesque est organisée en quinze chapitres qui présentent une certaine indépendance les uns par rapport aux autres tout en suivant le fil chronologique du voyage transatlantique. Les multiples jeux de mise en abyme, digressions, interpolations de récits secondaires et de spectacles improvisés sont autant de développements imaginaires qui contribuent au foisonnement de la trame narrative. Libro de navíos y borrascas est un roman polyphonique qui revêt la « forme musicale d'un choeur de voix qui se répondent et reprennent tour à tour le thème principal de l'exil ». 9 Les protagonistes masculins et fémininscomme Rolando, Sandra, Contardi, Bidoglio, el Gordito, Paredesforment un personnage collectif, identifiable à l'exilé du cône sud. Comme l'affirme Ana Vásquez : « […] el comienzo del exilio se vive en un modo colectivo, amarrado por un pasado, por ideales comunes, por una voluntad de retorno. » 10 Le roman commence comme un conte qui s'inscrit dans la tradition orale grâce à la mise en scène élaborée par le personnage-narrateur, qui conduit à un jeu de mise en abyme : « Hágamos de cuenta que estamos en un viejo caserón de piedra, antiguo refugio de pescadores rodeado por jardines sombríos, en una noche de invierno europeo. […] Nos hemos reunido aquí para oír la historia de un viaje. […] El viajero que acaba de llegar y va a contarnos una historia se saca las botas junto al fuego como en un cuento nórdico, les quita el barro del camino y acariciándose una barba de largas travesías se queda mirando fijamente el fuego, oyendo su chisporroteo milenario. Viene de los mares del sur, donde están las ballenas y los albatros, los naufragios y los grandes cementerios marinos.

[…]

Permítanme entonces ocupar durante algunas horas el lugar de ese viajero nórdico para contar mi propio viaje. Como él, también vengo de los mares del sur ». 11 La perspective narrative focalise l'histoire de ce voyage depuis le lieu de l'exil, sous forme de rétrospection. Tout au long de la traversée, les voix de l'exil se succèdent, se croisent et se répondent, multipliant les histoires racontées qui se greffent sur la trame narrative principale, celle du récit de voyage, euphémisme de l'exil. Les changements de voix narrative orchestrés par Rolando se prêtent à des jeux métafictionnels qui mettent en évidence la construction polyphonique du roman : « Para llevar adelante la pequeña historia de la sobrina del cocinero preferiría pasarme a otra voz, como si la contase otro. Y voy a elegir la voz de Bidoglio, que me parece más adecuada que la mía para este asunto […]. » 12 La multiplicité des voix narratives se conjugue à la pluralité des langues qui résonnent dans le roman. Se mêlent à la langue parlée argentine des phrases en italien, historiquement justifiées par l'importance de l'immigration italienne en Argentine et personnellement motivées par le lien affectif qui unit Daniel Moyano à la langue de son grand-père maternel 13 , mais aussi du portugais, de l'anglais et du français, reflet des diverses nationalités cohabitant sur le bateau. Le mélange de différents niveaux de langue contribue également à la variété linguistique du roman.

La relation privilégiée que Daniel Moyano entretient avec la musique 14 exerce une influence capitale sur l'écriture de l'exil, empreinte de métaphores musicales. Le départ en exil est raconté par le prisme de la sensibilité de Rolando, narrateur-musicien plus à l'aise avec les notes de musique qu'avec les mots : « Lo mío es la música, antes que las palabras. » 15 Le chemin de l'exil commence par la douloureuse séparation de Rolando et de son cher Gryga, que le violoniste est contraint d'abandonner lors de son arrestation. Le violon est alors condamné à la destruction, destruction dont Rolando reconstitue les probables étapes sur le mode de la conjecture, sans pouvoir se déprendre de son souvenir pour avancer dans le récit. Le motif du violon concentre ainsi l'expression de la déchirure du départ : « Difícil mover del puerto un barco real si seguimos con este asunto del violín. Pero sucede que ese violín está entre las palabras, hacia donde mire está su brillo, y francamente no encuentro otra manera de llegar al barco. Allá lo habrán mojado las lluvias y secado el viento tantas veces. Las avispas y los pájaros que picotean las uvas lo habrán llenado de estiércol y zumbidos. Las clavijas, saltadas de su sitio, se aferran todavía a las cuerdas, como corcheas cuelgan tan negritas las clavijas de ébano legítimas, y el puente delicado abonará la tierra, lo comerán pacientes los gusanos, despanzurrado mi violín como una araña que aplastaron ». 16 Le voyage s'achève par la chute d'un autre instrument de musique, celle de la guitare façonnée dans un arbre argentin et qui, tombée à l'eau à l'entrée du port de Barcelone, se transforme en bateau pour prendre le large alors même que son propriétaire arrive en Espagne. Cette scène à la fois visuelle et sonore fait se dérouler une longue métaphore filée où le champ lexical de la musique se mêle à celui de la navigation pour assimiler la guitare à une embarcation musicale, grâce au pouvoir poétique de l'analogie qui métamorphose également en voile un mouchoir emporté par le vent : « El impulso de las olas alejó rápidamente la guitarra a unos diez metros del muelle. […] El pañuelo dio unas volteretas arremolinándose, y cayendo en vaivenes fue a posarse sobre el pedazo de caña encajado entre las cuerdas a manera de mástil. Allí quedó como una vela bien braceada y recibió las brisas. El instrumento, al garete, vacilaba por diferentes rumbos. […] Cabeceaba frente a nosotros, y el mango, convertido en bauprés, daba golpes contra el agua. La vela, enrollándose en el mástil, con sus puntas daba zurriagazos sobre las cuerdas del pequeño navío musical arrancándole sonidos. En una seguidilla de golpes a manera de rasguidos sobre las tres primeras cuerdas, se oía un acorde enriquecido por el sonido de dos armónicos producidos por el roce de la caña y la bordona al mismo tiempo sonaba por simpatía, todo mezclándose al cloqueo del agua. Un golpe de viento desenrrolló el pañuelo, y con la vela desplegada la guitarra orzó hacia barloventoto. Un extremo del pañuelo fue a clavarse en las puntas sobrantes de las cuerdas, en el soporte. El paño blanco gualdrapeó contra el mástil y se infló lo mismo que una vela. Unas brisas encontradas hacía dar bandazos al navío de seis cuerdas, pero cuando consiguió alejarse del muelle, en aguas más tranquilas, recibió un viento de popa y empezó a alejarse de nuestra vista mar adentro ». 17 Le sort auquel est vouée cette guitare représente l'appel des origines, l'attrait des racines culturelles argentines. Si la remarque de l'Espagnol -« […] no tienes por qué afligirte tanto : estás en el país de las guitarras » -18 invite l'exilé à s'adapter au patrimoine culturel du pays d'accueil, le tiraillement entre avant et après, ici et là-bas, est néanmoins inévitable.

La notation musicale parsème les pages du roman créant un authentique langage musical, par exemple lorsqu'elle sert à inventer un prénom inédit en composant des accords imaginaires indéchiffrables. La musique renferme alors un secret, elle sert de rempart contre la violence de la Dictature : « Necesito un nombre difícil de nombrar, especialmente por los que tienen el maldito oficio de ir a sacar a la gente dormida de su casa. Imposible pensar entonces en los nombres convencionales.

Escondérselo en un sonido que ninguna voz humana pueda cantar y sólo algunos instrumentos ejecutar […] o complicarlo más todavía, algo que exija abrir bien la mano, o mejor un acorde teórico, de modo que aun con una guitarra a cuestas fuera difícil llamarlo […] » 19

Le langage de la musique n'envahit pas seulement le chapitre III, intitulé « Rasguidos » 20 , d'où est extrait ce passage, mais aussi le roman tout entier. Les nombreuses références à des paroles de tangos et de milongas créent un complexe réseau intertextuel, pièce maîtresse de l'écriture de l'exil de Daniel Moyano dans Libro de navíos y borrascas. Comme l'indique Nilda Díaz21 , le premier chapitre du roman, « Y chau Buenos Aires »22 , trouve son titre par relation d'intertextualité avec le dernier vers du tango « Anclao en París » (1931), écrit par Enrique Cadícamo sur une musique de Guillermo Barbieri :

« ¡No sabés las ganas que tengo de verte! Aquí estoy parado sin plata y sin fé; ¡Quién sabe una noche me encane la muerte y… chau, Buenos Aires, no te vuelva a ver! »23 À la lumière de cet intertexte, qui se clôt sur l'éventualité d'une séparation irrémédiable, le titre du chapitre inaugural de Libro de navíos y borrascas laisse augurer un départ sans retour, entre dérision et nostalgie. Le jeu intertextuel avec le tango « Anclao en París », qui se poursuit quelques pages plus loin par une parodie du vers final, creuse ironiquement le sentiment de perte lorsque le Cristóforo Colombo prend la route de l'exil en laissant à quai la valise du Flaco, porteuse d'un passé révolu : « Y chau valijita, no te vuelvo a ver. » [START_REF] Moyano | Libro de navíos y borrascas[END_REF] Daniel Moyano reprend d'autres paroles de tango dans le dernier chapitre du roman, intitulé « ¿Fin? », pour représenter la figure de Sandra lors de l'escale du Zampanó 25 au port de Lisbonne. Le personnage féminin est évoqué à plusieurs reprises « con su sombrerito pobre y su tapado marrón » [START_REF] Moyano | Libro de navíos y borrascas[END_REF] , telle la figure de « María » dans le tango éponyme de Cátulo Castillo et Aníbal Troilo (1945) :

« El otoño te trajo mojando de agonía tu sombrerito pobre y el tapado marrón… Eras como la calle de la melancolía que llovía, llovía sobre mi corazón ». [START_REF] Castillo | María[END_REF] Cette insistante référence intertextuelle nimbe Sandra d'un halo de mélancolie, faisant de la jeune femme marquée dans sa chair par la répression une figure emblématique des adieux. La relation amoureuse qui s'esquisse entre Sandra et Rolando avant que le chemin de l'exil ne les sépare à jamais s'inscrit ainsi sous le signe de l'échec.

L'intertextualité musicale avec « Volver » d'Alfredo Le Pera et Carlos Gardel (1935) [START_REF] Pera | Volver[END_REF] abolit tout espoir de retour à la terre natale grâce au jeu de réécriture qui transforme certains vers de ce célèbre tango en phrases négatives et interrogatives: « […] Pero mire qué preciosura de ciudad, mirelá bien porque lo más probable es que no volvamos nunca.

¿Así que nunca ? ¿Ni siquiera con la frente marchita dentro de veinte años? ¿Ni siquiera sintiendo que la vida es ffffuu, un soplo? ¿Ni siquiera con miedo al encuentro? ¿Ni con esperanza humilde? Mire, yo quiero volver y apenas estamos saliendo. Pero hay que sacárselo de la cabeza, no volveremos nunca […] » [START_REF] Moyano | Libro de navíos y borrascas[END_REF] La substantivation de l'adverbe « nunca », animalisé sous la forme d'une menaçante nuée d'oiseaux nocturnes, anticipe la perte de l'objet de la nostalgie. L'antanaclase zarpar-zarpas accentue la blessure du départ et de la perte :

« Más bien palabra de bicho. Gallinácea gris alechuzada. La nunca, ave de hábitos nocturnos, casi seguro que carnívora. En cuanto quiere caer la noche empiezan a revolotear las nuncas. Rondan los puertos y lechucean mástiles de barcos. Vaya palabra para empezar un viaje tan largo; revoloteando alrededor del buque listo para zarpar. Las zarpas de las nuncas, rapiñando ». 30 La créativité linguistique de Moyano dans le traitement littéraire de l'exil transparaît aussi à travers l'attribution du prénom « Volver » 31 au fils irréel né de l'union amoureuse imaginaire de Rolando et Nieves. Cette nouvelle onomastique prolonge le phénomène d'intertextualité avec le tango de Gardel tout en traduisant l'obsession du retour qui hante Rolando et ses compagnons d'exil. Comme le remarque Marina Gálvez Acero, l'imagination et la parole deviennent un « palliatif de l'exil » [START_REF] Acero | Daniel Moyano: la imaginación y la palabra como paliativo al exilio[END_REF] dans Libro de navíos y borrascas. D'où la présence dans le roman de constructions allégoriques, euphémismes, périphrases et tournures figurées, comme autant de stratégies de distanciation qui équivalent, selon Beatriz Sarlo, à des modalités discursives obliques. [START_REF] Sarlo | Literatura y política[END_REF] L'intertextualité musicale acquiert une importance toute particulière dans les chapitres XII et XIV, respectivement intitulés « El faro » et « Como una ilusión marina », où les passagers du Cristóforo Colombo construisent un récit allégorique sur le sort des disparus à partir de l'intertexte de deux valses des années vingt et trente : « Ilusión marina » de Antonio et Gerónimo Sureda, et « La pulpera de Santa Lucía » 34 de Héctor P. Blomberg et Enrique Maciel. Les réminiscences de la culture musicale populaire argentine servent de point de départ à l'élaboration d'une histoire de disparus qui s'appuie sur le motif maritime du phare et sur le symbolisme de l'ombre et de la lumière pour dénoncer de façon métaphorique la répression militaire en Argentine et suggérer le vide absolu laissé par les disparus. Le titre de ce récit collectif s'inspire du premier vers de « Ilusión marina » -« Era la hija del viejito guardafaros » -35 pour s'en éloigner progressivement au rythme des variations insufflées par les personnages moyaniens : « Historia del guardafaro », « La hija del guardafarro », « Historia del viejito guardafaro », « El faro de los desaparecidos », « El faro », « El farol », « El farolito », « El fósforo », « El apagón ». [START_REF] Moyano | Libro de navíos y borrascas[END_REF] Le titre définitif se réduit à un souffle, celui de l'onomatopée « Fsss » qui figure le son produit par les disparus lorsqu'ils sombrent à jamais dans l'absence :

Pero ninguna cosa que se apaga lo hace en silencio. Siempre hay un sonido, una protesta, algo así como una escupida de desprecio contra lo que no se puede sostener más. No tengo idea del sonido final de los desaparecidos. Pero sé que todas las cosas, salvo las plantas inocentes, cuando desaparecen lo hacen con un ruido […]. Y en ese caso, para esta historia de nuestro humilde farero yo preferiría, para acabar de una vez con las ilusiones de ese viejo, algo tan drástico y definitivo como un Fsss 37 La dénonciation des exactions de la dictature passe par le filtre de la subjectivité des personnages de fiction en route vers l'exil. L'expérience personnelle de l'emprisonnement, des interrogatoires et de la torture, l'évocation de la disparition et des assassinats politiques émaillent de façon plus ou moins elliptique, avec une plus ou moins grande distance humoristique, la narration de Rolando, les dialogues des autres protagonistes et leurs monologues intérieurs. Les voix narratives sont partagées entre la volonté de porter témoignage et le désir d'oublier, voire de nier la violence et le traumatisme subis. Sandra répugne à raconter son histoire de viol et de torture dont elle souhaiterait effacer l'existence 38 pour préserver la continuité de son être : « Lo mejor es olvidarlos, al menos eso me da la ilusión de que no existen y me permite seguir siendo Sandra. Y es muy feo andar ventilando esas intimidades, sacar los trapos al sol como quien dice. » 39 De même, Rolando passe sous silence l'épreuve traumatisante de son séjour en prison, importante lacune dans le récit de son départ en exil. La métaphore du petit serpent coupé en deux et dont les moitiés se ressoudent est une image de régénération qui symbolise le désir de réunifier une identité brisée : « Entrar en el camarote, por fin, como volver a casa, a una casa, y ver que lo sucedido entre el dejar el violín bajo la parra y entrar en el camarote quedaba entre paréntesis. […] Las dos partes de la viborita cortada con la pala se pegaron y ahora anda reptando tan campante. Y a lo sucedido entre la parra y el camarote taparlo con un trapo negro, como las viejas del campo tapan los espejos cuando hay trances de muerte. Mirarme al espejo sin trapos negros, yo, vivo ». 40

Moyano D., « D. Moyano : lettre à C. Cymerman », América, Cahiers du CRICCAL, n° 7 (1er trimestre 1990), p. 191 : « Por otra parte, Argentina es un país de exiliados europeos. Yo me crié en el exilio argentino de mi abuelo materno, que era italiano, sin saber que eso era el exilio, que la casa donde yo vivía y descubría el mundo era el exilio ».

Gil Amate V., « Lleno de ardor, con las manos tendidas. Entrevista con Daniel Moyano », Quimera, n° 86 (1988), p.

33: « Hay un exilio político forzoso, el de 1976, pero el exilio comenzó hace mucho en mi vida[…]. Mi padre era técnico en construcciones y se fue a trabajar a Buenos Aires en la época de Hipólito Yrigoyen. Después del golpe militar lo echaron, por eso tuvo que volverse a su tierra. Esto significó para mí el primer exilio. Es decir, yo había nacido en Buenos Aires y se supone que tenía que haber vivido allí. Me crié en La Falda y desde los catorce años viví en Córdoba, hasta que decidí, con ventitantos, irme a La Rioja. Este fue otro exilio. […] Otras variaciones del exilio fueron los distintos hogares de diferentes tíos con los que viví la infancia. No es un exilio como el que nos tocó vivir al venirnos a España, no, no, hablo de esos exilios que tienen todos los seres humanos y que consisten en ir dejando cosas ».3 Moyano D., art. cit., p. 192 : « En 1976 me echan del país los militares, y empieza el duro exilio externo, donde realmente soy 'de afuera', hasta hoy, 23 de agosto del 89, pese a haber obtenido la nacionalidad española hace como nueve años ».4 Bonnardel S., « Libro de navíos y borrascas, los aprendizajes del exilio », Novela y exilio, Olver G. de León (coord.), Uruguay, Editorial Signos, 1989 [en ligne : http://www.cervantesvirtual.com/obra-visor/libro-de-naviosy-borrascas--los-aprendizajes-del-exilio/html/7a5ff597-dfa2-47ff-9c98-8041be92734b_4.html ; consulté le 8 février 2019)].5 Moyano D., Libro de navíos y borrascas, Buenos Aires, Legasa, 1983, p. 27. 6 Melon A., « Le roman de l'exil : entre le témoignage et la sublimation », América, Cahiers du CRICCAL, op. cit., p. 64-65.

Koller S., « Un exil peut en cacher un autre », América, Cahiers du CRICCAL, op. cit., p. 31.

Moyano D., Libro de navíos y borrascas, op. cit., p. 11.

Koller S., art. cit., p. 31.

Vásquez A., « La leyenda de Ulises, mito del exilio », América, Cahiers du CRICCAL, op. cit., p. 23.

Moyano D., Libro de navíos y borrascas, op. cit., p. 9-10.

Ibid., p. 62.

Après la mort de sa mère, Moyano fut élevé par ses grands-parents paternels, qui parlaient un mélange d'italien et de portugais. Il apprit l'espagnol au collège.

Moyano reçut une formation musicale classique et enseigna au Conservatoire Provincial de Musique de La Rioja dont il fut aussi violoniste du Quatuor à Cordes et de l'Orchestre de Chambre. Il affirme dans son entretien avec Virginia Gil Amate : « Escribo como quien compone […] y procuro que mis palabras se sostengan en verdades auditivas o sonoras iguales a las que soportan la música. » Gil Amate V., art. cit., p. 32.

Moyano D., Libro de navíos y borrascas, op. cit., p. 11.

Ibid., p. 12.

Ibid., p. 312-313.

Moyano D., loc. cit.

Ibid., p. 74.

Ibid., p. 67-88.

Díaz N., « Daniel Moyano: entre la música y el oficio de escribir », América, Cahiers duCRICCAL, op. cit.,

Moyano D., p. 7. Le titre du chapitre est repris sans virgule p.


Macarena Flores González
email: macarena.flores-gonzalez@cyu.fr

Fabrice Vandebrouck

Semiotic control in calculus: the case of limits of functions at the beginning of the university

Keywords: Analysis, limits, semiotic control, localisation perspectives, secondary-tertiary transition

This work is situated in the context of the secondary-tertiary transition in analysis. From the results of a test in the first year of university, we find low success rates in the calculation of limits of functions, which do not increase considerably when the graph of the function is given. To understand the lack of success in these tasks, we interview students and analyse the mathematical activities they develop (or not) when answering the test. For this, we use Activity Theory in Didactics of Mathematics, focusing on the notion of semiotic control as well as that of deconstructions with localisation perspectives.

Introduction

For several years now, we have been setting up tests (MCQ) for students at the start of their university course. In these tests, we ask questions about the calculation of limits of functions to identify the knowledge acquired at the end of their secondary school education. In an earlier paper [START_REF] Vandebrouck | Students' visualisation of functions from secondary to tertiary level[END_REF], we identified that students did not necessarily use the expected algebraic methods to calculate the requested limits. They were however able to operate what we called decompositions of algebraic formulas according to the local perspective [START_REF] Montoya Delgadillo | Deconstruction with Localization Perspective in the Learning of Analysis[END_REF]. This helped to explain why some calculations were more successful than others, even if the complexity was higher and it also helped to explain some significant student errors. For this paper, we try to go further and find out about the control that students can exercise with the help of graphical decompositions. Indeed, mathematical control has been targeted as a key issue to study and explain the difficulties of students in Analysis in the secondary-tertiary transition [START_REF] Flores González | Activité et travail mathématique à la transition lycée-université en Analyse : le cas de suites un+1 = f(un)[END_REF].

We have asked some of the students to compute the limits of some functions by also providing the graph of the functions in play. It has been observed that success rates are not radically higher when the graph of functions are available. Interviews were conducted with eight students to understand when and how graphical representations helped them to calculate the limits of functions.

Theoretical framework

We situate our work in Activity Theory in Didactics of Mathematics (ATDM), Leontiev activity theory developed within French didactic [START_REF] Rogalski | Theory of Activity and Developmental Frameworks for an Analysis of Teachers' Practices and Students' Learning[END_REF][START_REF] Vandebrouck | Activity Theory in French Didactic research[END_REF]. It considers activity as multidimensional and focusing on both cognitive and mediative dimensions, related to mathematical knowledge and interactions between the actors in the classroom respectively.

In this paper, we focus only on the cognitive dimension. We use mathematical activity to refer to students' activity on a specific mathematical task in a given context. We use what we call students' mathematical activities (and sub-activities) which are operationalisation (or not) of mathematical knowledge [START_REF] Robert | A didactical framework for studying students' and teachers' activities when learning and teaching mathematics[END_REF]. Mathematical activities are algebraic or graphical according to the register of semiotic representations to which they are related [START_REF] Duval | Registers of Semiotic Representations and Analysis of the Cognitive Functioning of Mathematical Thinking[END_REF]. Duval identifies treatments and conversions as crucial mathematical activities with semiotic representations. He also develops the fundamental role of visualisation, especially visualisation with dimensional deconstructions (Ibid.). In [START_REF] Vandebrouck | Students' visualisation of functions from secondary to tertiary level[END_REF], we introduced DWP (deconstructions with perspectives) as students' ability to visualise representations of functions (graphical or algebraic) according to the three perspectives of localisation: local, global and punctual perspective [START_REF] Montoya Delgadillo | Deconstruction with Localization Perspective in the Learning of Analysis[END_REF]. In this paper, we rely on treatments and visualisations with DWP to characterize students' mathematical activities on functions given with or without graphs. We also adopt concepts coming from [START_REF] Galperine | Essai sur la formation par étapes des actions et des concepts [Essay on the staged formation of actions and concepts[END_REF]: orientation, execution and control. As we analyse answers to a MCQ (closed questions), we do not focus on orientation, only on execution of the task and control. During execution, students develop either treatment (mostly with applications of knowledge, organisation, with or without recognising sub-activities) or visualisations with DWP (with recognising sub-activities of pieces of knowledge about representations). Control permits the subject to ensure that the answer produced corresponds to the objective aimed at by the activity [START_REF] Flores González | A classic recursive sequence calculus task at the secondary-tertiary level in France[END_REF]. During control, students mostly develop visualisation, especially with DWP. Then the control is mostly a semiotic control [START_REF] Flores González | Activité et travail mathématique à la transition lycée-université en Analyse : le cas de suites un+1 = f(un)[END_REF]. For instance, in the case of the limits of functions, the visualisation of an asymptote on a graph of a function f (decomposition of the graph with local perspective) must be consistent with the calculation of a limit obtained with treatments on the algebraic expression of the function. Finally, we refer to the didactic contract defined as a set of rules (often implicit) that refer to the mathematical practices of the teacher and the student [START_REF] Pepin | Using the construct of the didactic contract to understand student transition into university mathematics education[END_REF].

Research questions

Calculus, especially computation of limits of functions supposes treatment activities on algebraic representations. In [START_REF] Vandebrouck | Students' visualisation of functions from secondary to tertiary level[END_REF], we had identified that some students use DWP instead of rules of computations. Increasingly, students are doing algebraic calculations with the help of calculators and the graphical representations. We wonder to what extent students are able to control their algebraic treatments when graphical representations of the functions are given. What kind of mathematical activities do students develop during computations of limits? Do the graphical representations change the execution of the computation? With which kind of visualisation?

Methods

A compulsory MCQ was given to 690 students at the beginning of their Bachelor's degree (September 2022) in science 1 at the Université Paris Cité. These students will be following quite similar courses in mathematics in the first year. This test included several questions about the limits of functions, some questions were provided with the graph and others without graph, randomly. Among the students who answered the test, eight volunteers had an interview on the computation of three of the limits. Of these volunteers, six were from French high schools, two from Tunisian high schools, and all of them obtained a scientific baccalaureate in 2022. This means that they should be able to recognise indeterminate forms for the calculation of limits. They all stated that they had not done any mathematics since the end of the last school year. All eight students gave their consent for the recordings of the interviews and their written output to be used for the research. Some students had already encountered these limits in the test, others not, some with graph, others not. For the interview, the three questions provided the graph of the functions. In order to answer our research questions, the interview with the students was designed to find out their mathematical activities for the calculation of the limit, and also to find out to what extent the graph of the function influenced (or not) the mathematical activities. We recorded and transcribed the entire interview, and collected the students' written work. We also collected their answers to the test that took place just before the interview. This allowed us to make an initial comparison between the test responses and the interview responses. To analyse the transcripts and the students' written work, we used task analysis, which is described in the next section. The three questions we are focusing on concern the calculation of limits in -∞, in +∞ and in one point (see Table 1). We make a task analysis assuming that the students do not have the graph at their disposal. In this context, students must first recognise that Q1 and Q2 are algebraic indeterminate forms (recognising algebraic sub-activities). In both cases, the expected activities are treatments in the algebraic register, especially application of the "dominant term rule" which is supposed to be available in students repertory of knowledge: to factorise by exp(-x) for Q1 and by x for Q2 in numerator and denominator in order to remove the indeterminate form. However, there exists a specific adaptation for Q1 because x goes to -∞ and therefore exp(-x) goes to +∞. That is to say that there is a composition of limits for Q1. Moreover, for Q2, students can only apply a simpler rule: to factorise by the term of highest degree because it is a rational fraction. By factoring in this way and then simplifying the algebraic expressions, it remains a direct application of the algebraic rules about operation on limits to find 1 in both cases (answers a and b). With this task analysis, we understand that Q1 is more complex than Q2 because of the adaptations: composition of limits and so the more complex form in Q1 than the rational fractions in Q2. For question Q3, students must also recognise an indeterminate form 0/0, which students do not easily recognise. Often for them, 0/0 gives 0 (answer c). To remove the indeterminate form, there is no generic rule. This increases the complexity of the task compared to Q1 and Q2. Students must recognise the remarkable identity x²-1=(x-1)(x+1), then simplify to remove the indeterminate form and find 2 (answer a). With this analysis, we understand that Q3 is more complex than Q1 and Q2 because of the two adaptations which call for recognising activities: the specific indeterminate form 0/0 and the remarkable identity.

Tasks analysis

From the graph, students can use a local perspective (graphical visualisation) with recognition of the horizontal asymptote y=1, at the neighbourhoods of -∞ for Q1 and +∞ for Q2. For Q3, students can use a punctual perspective by obtaining the image of x=1 as the function is extendable by continuity. These procedures can be carried out as execution or control activities.

Results on students' activities

Table 2 gives the success percentages of the students with and without graphs. In the last line, we give the number of students (among 690) who had this question in their test. The success percentages without graph are 40,45% (Q1), 68,38% (Q2), 18,99% (Q3) and this is coherent with our tasks' analyses. However, when the graphs are available it should be expected that the success percentages are almost 100%, especially because it is a multiple-choice test. The success rates for the limits at +∞ and -∞ should be closed because the graphical visualisation with local perspective at infinity (DWP) seems to be of the same nature.

lim 𝑥→ -∞ exp(-𝑥)-1 exp(-𝑥)+1 = 1 (Q1) lim 𝑥→ + ∞ 𝑥-1 𝑥+1 = 1 (Q2) lim 𝑥→1 + 𝑥 2 -1 𝑥-1 = 2 (Q3) Without G With G Without G With G Without G With G 40,45% 65
The table 3 shows the 8 students' activities during the interviews for the calculation of each three limits when the graph is available. During execution or control, activities can be algebraic or graphical and activities can be treatments or visualisations with DWP. In this table, (1) means not spontaneous -(2) local perspective in +∞ instead of -∞ for Q1 -(3) the control is done with external factors (didactic contract for instance) -(4) algebraic control using symmetry of the algebraic form of Q1.

This synthetic table shows that most of the students develop graphical activities to find the limits at +∞ and -∞ (first column). However, execution with visualisation is not spontaneous for E1, and furthermore, E4 and E8 do not do it. In addition, E5 and E6 make errors in the graphical execution when they adopt the local perspective in Q1: they are looking toward +∞ instead of -∞. Then E1 and E5 develop algebraic activities (execution) to find limit of Q2 (third column). Concerning control, it is possible to develop graphical activities when the execution has been done with algebraic activities (E4 and E8). This is not spontaneous for E4, and E8 makes a mistake by adopting the local perspective on the graph Q1 (fourth column). Only three students are able to control their graphical executions with activities in the algebraic register (E1, E2, E3). This control is semiotic and uses visualisation with local perspective for two of them (E1, E2). Three students are not able to control at all (E5, E6, E7). Regarding the students' activities for solving Q3, 3 students are not able to develop either algebraic or graphical activities (E6, E7, E8). Most of the other students solve the task using graphical activities with visualisation, and only one student (E3) is able to control with algebraic rules.

Some examples of students' mathematical activities during the interview

Graphical execution then algebraic control

Student E3 starts with graphical executions and visualisation with a local perspective. He controls his/her answers with algebraic treatments for Q1, Q2 and Q3 (see Figure 1).

M (Macarena):

How do you find the limit? (Q1) Student E3: Graphically, when x tends to -∞, the asymptote has the equation y=1. M:

And without the graph how do you do it? Student E3: I will add 1. This will give 2 on +∞, and it tends towards +∞, so towards 0. The result is 1. M: How did you find the limit? (Q3) Student E3: Graphically, as x tends to 1+ it is 1 but greater than 1, the orthogonal projection cuts the line in 2. But it can be done algebraically.

His controls are based on putting into operation the rules of algebraic calculation to prove the values of the limits. E3 is the only student who develops the activities expected in these situations. Moreover, in the test, even when the graph was not given, the student could still successfully complete the task.

Table 2 : success percentage of students with and without graphs (G)

 2

Table 3 : Students' activities during the interviews

 3

				Limits at infinite (Q1-Q2)			Limit at a point (Q3)
	Student	Execution Graphic	Execution Algebraic	Execution Algebraic	Control Graphic	Control Algebraic	Control Algebraic	Execution Graphic	Control Algebraic
		visualisat	treatem.	visualisat.	visualisat.	treatem.	visualisat.	visualisat.	treatem.
	E1	x (1)		x			x	x (3)
	E2	x					x (4)	x (3)
	E3	x				x		x	x
	E4		x		x (1)			x
	E5	x (2)		x (1)				x
	E6	x (2)
	E7	x
	E8		x		x (2)

Degrees in Mathematics, Computer Science, Chemistry, etc. (and no Prep School).

Answer to Q2

Answer to Q3

Figure 1: Student E3's written answers to the three questions

Algebraic then graphical execution with algebraic visualisation control only on infinite limits

Student E1 develops mathematical activities in the graphical register for executions (infinite or finite limit) even if for Q1 the visualisation was spontaneously in the algebraic register. Then he does a control with algebraic treatments, but this activity includes visualisations once more (deconstruction of the algebraic formula with local perspective: it is an operationalisation of the notion of equivalence of the numerator and the denominator).

M: How do you find the limit? (Q1) Student E1: Here I put 1, but I don't know exactly why, it's more instinctive. However, this visualisation with local perspective is not operational for the finite limit (Q3), and the student proceeds by an execution with graphic visualisation using the punctual perspective. Finally, he cannot control his execution other than with the didactic contract (MCQ).

M: How do you find the limit? (Q3) Student E1: We are looking for the limit of this (the function) when x tends to 1, and since the function is defined for x=1, the limit will be the value of f(1), here f(1)=2 graphically. And then in the answers we are proposed 2, so it joins what we read graphically.

Execution with algebraic treatments but then graphical control only at infinite limit

Two students (E4 and E8) are in this category. We show the example of E4 which allows execution when there is no basis for algebraic orientation.

M: How do you find the limit? (Q1) Student E4: To calculate the limit, we will have to put it in another form because it is an indeterminate form. So, we will factor by the dominant term. So here we know that the limit at -∞ of the exponential is 0. So, we're going to have the limit which is equal to 1, because it (-𝑒 𝑥 ) tends towards 0, and it (𝑒 𝑥 ) also tends towards -∞.

These students orient and execute their calculations by algebraic treatments, as if the graphs were not provided. They put into operation the rules of calculation for the infinite limit (Q1 and Q2). Nevertheless, the calculation for the finite limit (Q3) being more complex, these two students encounter difficulties: they apply the rule of action of infinite limit with the same calculation rules (factoring the dominant terms), which is not operational in this case:

M: How do you find the limit? (Q3) Student E4: Here you have to factor by the dominant term, the one with the highest power. Here, we factor at the bottom too... (she writes). We study... ah... bah no!... I've made a mess of things, I think. But, in this case, um... I need to use the graph for this. But normally we find the same thing...

Only E4 is able to reorient and execute his task with visualisation in the graphical register. For E8, the visualisation in the neighbourhood of x=2 is not possible and the graph does not help at all (for execution or control). As soon as the algebraic knowledge is insufficient to find the limit, the student cannot catch up.

Execution based on fragile visualisations, without control

Student E5 performs visualisations in the algebraic or graphical register. Nevertheless, these visualisations with the local perspective are fragile because the student spontaneously looks to the right for the in -∞ (Q1). For the limit in +∞ (Q2), the student favours visualisation on the algebraic formula:

M: How do you find the limit? (Q1) Student E5: I just look at the graph, and when I see that x tends to -∞, I see that it tends to -1. M:

Did you have a good look? Student E5: Yes (pause)... I was wrong, x tends to -∞. So, it's 1 (...).

On the other hand, two students (E6 and E7) do a graphical execution (visualisation with local perspective) only for Q1 and Q2. However, for them, there is no possibility of control using the algebraic register in Q3.

M:

How do you find the limit? (Q3) Student E7: I would have said 0 positive (0+). We look at the function when it approaches 1, I said something wrong. So, we look at where it is in 1, in 1 we see that it's between 1 and 2. So we can't use the graph. But I wouldn't know how to answer you...

Conclusions

From the results of the MCQ, even though success rates are increasing (by 25% for Q1, 5% for Q2 and 27% for Q3), the students' answers do not improve significantly when the graph of the function is given. The small sample of eight students surveyed may explain some of the students' difficulties and allows us to understand this limited increase: not all students are able to execute spontaneously the limit at +∞ and -∞ with graphical activities (visualisation with DWP), they may even make mistakes (wrong perspective when calculating the limit at -∞ for example), and few of them are able to use semiotic control. Some students find limits more easily with algebraic treatments (E4, E8) or algebraic visualisation with DWP (E1, E5) and in these cases the semiotic control with the graphic representation is not so immediate or with mistakes. Moreover, for the finite limit (Q3) only 5 out of 8 students managed to read the limit with the graph, but they couldn't control their execution with algebraic treatments (except E3). We can conclude that the semiotic control and the execution with the graph, when the graph is given, are not spontaneous, and many students make mistakes in the visualisation activity with DWP. This may explain a default of coordination between semiotic registers [START_REF] Duval | Registers of Semiotic Representations and Analysis of the Cognitive Functioning of Mathematical Thinking[END_REF], and reinforces our arguments about the lack of mathematical control at the secondary-tertiary transition. Furthermore, even if the graph is provided, some students still reason in a purely algebraic manner: knowing that they are in a MCQ, is this a phenomenon of the didactic contract?


Thomas Hausberger
email: thomas.hausberger@umontpellier.fr

Reinhard Hochmuth

From real to abstract analysis: the development of structuralist praxeologies in the analysis path

Keywords: Real analysis, abstract analysis, university mathematics education, Anthropological Theory of the Didactic, structuralist praxeologies

This article is a continuation of Winsløw's work on the concrete to abstract transition in analysis and Hausberger's work on structuralist praxeologies, developed in the paradigmatic context of abstract algebra. It aims at giving a foundation to the notion of structuralist praxeology in analysis. After a brief epistemological study of abstract analysis, we discuss the praxeological analyses of excerpts from a real and abstract analysis textbook used in the transition between Bachelor and Master degree programs in mathematics.

Introduction

The starting point of this research is the model proposed by [START_REF] Winsløw | Transformer la théorie en tâches : la transition du concret à l'abstrait en analyse réelle [Turning theory into tasks: the transition from concrete to abstract in real analysis[END_REF], based on the Anthropological Theory of the Diactic (ATD), that aims to describe the "transition from concrete to abstract" in real analysis. Winsløw focuses primarily on the institutional transition from high school to university, which is accompanied by ruptures in students' relationships to mathematics. He takes the analysis path as an example, so that this transition coincides, in epistemological terms, with that from calculus to real analysis. After defining real analysis as the study of real functions and the limiting processes (continuity, derivability, integration, series,...) associated with them, Winsløw distinguishes two components: a concrete analysis centred on the study of particular classes of functions, and an abstract analysis that deals with axiomatic systems (metric spaces, operators,...). While high school calculus deals mostly with the practical-technical 1 blocks Π₁ of concrete analysis, more complete praxeologies are developed at university, in the sense that they are endowed with a formal technological-theoretical 1 block Λ1, mobilizing abstract analysis concepts (convergence, continuity, neighbourhood, compactness, etc.) that provide a more rigorous justification of the praxis and organize the logos in a coherent whole. The result, in a second stage, is the appearance of abstract tasks dealing with these concepts (praxis Π2): this is the beginning of abstract analysis. As an example taken from the end of the analysis path, Winsløw gives a task of the type "show that a given linear application between two normed spaces is bounded and determine its norm", the solution of which uses the Cauchy-Schwartz inequality and the interpretation of the sum of a certain series as a scalar product in ℓ 2 . In other words, solving the task requires the reformulation in the language of Hilbert spaces of elementary results of real analysis.

In later research, [START_REF] Winsløw | An institutional approach to university mathematics education: from dual vector spaces to questioning the world[END_REF] highlight the generality of this model (Figure 1) to describe ruptures in the development of praxeologies taught in university mathematics curricula in both analysis and linear algebra, mainly during the first 2 years. The first transition is associated with rigor and the learning of proof, while the second highlights the didactic need to anchor concepts in previously constructed praxeologies before engaging them as objects in new tasks of a higher level of abstraction. The construction of institutional praxeological models (relative to universities that serve as case studies) of the knowledge taught, on the basis of the syllabi and course documents, supports the relevance of the model as a descriptive model of didactic transposition choices made by its actors at university and as an explanatory model of certain learning difficulties observed in students. In doing so, the horizon of abstract analysis (in terms of structures, as illustrated by Winsløw's first example) is lost sight of and the dialectic between concrete and abstract that impacts praxeologies is not further clarified from an epistemological point of view. The aim of this article is to clarify these aspects, in line with the research program initiated by [START_REF] Hausberger | Structuralist praxeologies as a research program on the teaching and learning of abstract algebra[END_REF] around the notion of structuralist praxeology. Starting from an epistemological analysis of mathematical structuralism, Hausberger proposed a second model (see theoretical framework) where the type 1 transition is no longer conceived as a phenomenon of praxeology completion, but of the construction of structuralist praxeologies according to the structuralist methodology (as described notably by Bourbaki). These evolve and generate tasks dealing only with concepts and structures, hence the second type of transition. In the emblematic case of abstract algebra, Hausberger relies on the analysis of course documents and the analysis of forums to show the relevance of the model in a teaching-learning context. Is Hausberger's model suitable for describing the development of praxeologies in a university analysis path, from the Bachelor to the Master of Mathematics, or are adaptations required with respect to the paradigmatic case of abstract algebra? What didactic phenomena does the lens of structuralist praxeologies reveal? These questions guide the exploratory study presented in this paper. We begin by a brief account of the historical study that we performed to elucidate the development of structuralist thinking in the scholarly practice in analysis (first section). We then present our theoretical framework and apply our model to extracts from a textbook entitled "Foundations of Real and Abstract Analysis" [START_REF] Bridges | Foundations of real and abstract analysis[END_REF]. Used at the transition between Bachelor and Master degree programs, this textbook was chosen by virtue of the didactic project that underlies it: to make visible how the concepts and theorems of abstract analysis enlighten real analysis, which is first recapitulated in view of its generalization.

Abstract analysis: a brief epistemological account

First of all, modern analysis is rooted in the 19th-century development of the foundations of calculus, which has several striking epistemological features. The 18th century concept of quantity in analysis has been replaced by the concepts of set and real numbers, presented axiomatically (Jahnke, 2003, chap. 10). The rigourisation and arithmetisation process of analysis (led by Cauchy and Weirestrass, among others) is thus also a movement of generalisation and axiomatisation due to its foundational aspects. Problems with the proofs of basic theorems concerning limiting processes such as the convergence of sequences of functions or the representation of functions by means of series (in particular Fourier series) led to foundational questions such as: how could the domain of the functions be characterized in a technically satisfactory way? This required the clarification of the notion of real number (Cantor), continuity, differentiability, integrability. Moreover, mathematicians actively sought pathological functions to delineate the limits of those concepts and challenge their associated intuitive backgrounds. The result is a new style in analysis where theorems now take the form : "Let A be a … subset of R which is … (closed, open … everywhere dense, measurable, simply connected, etc.) and let f be a function defined on A (or 𝐴 or…) which is C n or … in A and …. on 𝐴 and let x0 be a point of A such that …. , then …" (Jahnke, 2003, p. 188).

The study of sets of other objects than points of the real line or R n was necessary for the birth of point set topology as a branch of the emerging general/abstract/modern analysis. The calculus of variations by the Italian school (Ascoli, Volterra, Arzela) who considered sets of curves and functions (an epistemological breakthrough) played a major role and led to the early developments of yet another branch of abstract analysis: functional analysis (Jahnke, 2003, chap. 13). Both concrete applied problems (e.g. Dirichlet's problem of potential originating in physics or biology phenomena pointed out by Voleterra) and the search for a unifying generalising point of view stimulated its development.

Frechet was the first to define functionals on abstract metric spaces (called classes E) in his 1906 thesis and he called the emerging subject "analyse générale dans les espaces abstraits" (general analysis in abstract spaces). In 1914, Hausdorff began the development of topology on the basis of neighborhood axioms (and introduced the terminology metric space), which marked the development of topology as a separate discipline.

The Göttingen school led by Hilbert gave a foundation to the theory of linear integral equations (originating from the problem of potential). Hilbert used the analogy with the eigenvalue theory of linear algebra to deal with operators with symmetric kernels. The transition from the finite to the infinite relied on set theory and opened up the new realm of what Schoenflies called Hilbert spaces in 1908, and which were later axiomatised in their modern form in the 1920s by von Neumann in connection to the development of an adequate formalism for quantum mechanics. Riesz, building upon Frechet's concept of distance, established the connections between the French school of real function theory (Lebesgue) and the Göttingen school (in modern terms, he established an isomorphism between the spaces ℓ 2 and L 2 , the Riesz-Fischer theorem). Riesz also introduced the new, linear, normed function space L p which is not a Hilbert space for p ≠ 2 and pointed the way to the Hahn-Banach theorem (on the extension of linear functionals). The abstract axiomatic presentation of complete linear normed spaces was achieved by Banach (thus the Polish school) in his 1920 thesis. Topological tools (based on Baire's category theory) were used frequently in the later development of functional analysis, for instance the open mapping theorem proved by Banach. Three monographs based on axiomatic presentations by Banach, von Neumann, and the American Stone in 1932 marked the establishment of functional analysis as an important and solid mathematical subdiscipline.

To summarize, the generalisation of concepts (boundedness, distance, convergence, completeness, scalar product, compactness, continuity,...) from the n-dimensional Euclidean space R n and functions defined on it to infinite-dimensional function spaces and their operators was a main mathematical motivation to the development of functional analysis, which needed to pass from the finite to the infinite. The axiomatic definitions of the spaces, with R n as a reference providing geometrical insights, allowed to elucidate the logical relations among a diversity of new general concepts (e.g. convergence which became diversified). Properties such as boundedness and compactness separate from each other and new concepts appear with no classical counterpart in R n (separability, Hahn-Banach extension). Conceptual analogies with linear algebra and real functions, together with generalised geometric intuition and approximation principles, allow mathematicians to think about and process the passage from the finite to the infinite (Jahnke, 2003, chap. 13).

The French group Bourbaki, inspired by van der Waerden's textbook Moderne Algebra (published in 1930), set out to apply the structuralist methodology developed by the German algebraists to all fields of mathematics. Bringing to the forth the theme of unity of mathematics, they systematised the perspective of structures in a treatise Eléments de Mathématique whose first part, in several volumes, bore the subtitle "The fundamental structures of Analysis". The first volume dedicated to General Topology appeared in 1940 and the first volume in functional analysis, entitled Topological vector spaces in 1953.

Theoretical framework

The structuralist methodology, described by Bourbaki in its Manifesto (the Architecture of Mathematics), is the epistemological anchor-point of Hausberger's model, which provides an interpretation in the praxeological language of ATD. According to ATD [START_REF] Chevallard | Anthropological Theory of the Didactic (ATD)[END_REF], every human activity consists in the coordination of a praxis and a logos, hence the key notion of a praxeology, represented by a quadruple [T/τ/θ/Ө]. Its practical-technical block (or know-how) consists of a type of tasks T together with a corresponding technique τ (useful to carry out the tasks 𝑡 ∈ 𝑇). The technological-theoretical block (or know-why) comprises the technology θ, a discourse on the technique, and the theory Ө, the ultimate level of justification.

In the praxeological terms of ATD, the structuralist method consists in the passage from a praxeology P = [T/?/?/Өparticular] where it is unclear which technique to apply to perform a type of tasks T concerning particular objects, to a structuralist praxeology Ps = [T g /τ/θ/Өstructure] where, modulo generalization of the type of tasks (T g ), the theory of a given type of structure guides the mathematician in solving the problem. The first structuralist transition thus consists in the passage from a praxis П = [T/?] to a structuralist praxis Пs = [T g /τ] endowed with a structuralist logos Λ (figure 2). In this transition, [START_REF] Hausberger | Structuralist praxeologies as a research program on the teaching and learning of abstract algebra[END_REF] distinguishes two structuralist levels of praxeologies: at level 1, structures act as a vocabulary and appear mainly through definitions (e.g. a task of type T is solved by showing, by hand, that the definition is satisfied); at level 2, the technique mobilises general results about structures and structuralist technologies come into play. These consist in structure theorems (how structures canonically decompose into sub-substructures), structural stability of conceptual properties under operations on structures, isomorphism theorems, expressed abstractly in the language of Set Theory or conceptualized on a meta-level in terms of Category Theory. Although the structuralist principles may be described transversally to all domains of mathematics, structuralist technologies still need to be specified in the context of analysis to account for the structuralist dimensions of the latter field. In the application of the structuralist methodology, the stage is reached when structuralist statements are no more contextualized to specific objects or domains of objects (e.g. functions in one real variable) but concern abstractly defined classes of objects (e.g. generic functions between generic metric spaces). Although generic objects such as real functions or sequences/series already appear in early analysis courses in the context of abstract tasks of what we called pre-structuralist praxeologies [START_REF] Laukert | Calculus at the school to university transition: early stages of a structuralist perspective in real analysis[END_REF], the properties of functions and their domain (R or R n ) that play a role are now fully elucidated in terms of topological concepts and concepts of functional analysis (theories of Hilbert, Banach and metric spaces). This process is achieved through the second type of transition, situated at the institutional transition of Bachelor and Master degree programs. Figure 2 highlights that the new purely abstract praxis П' is anchored on reasoning with concepts that take their origin and rationale in the logos of previously developed structuralist praxeologies. The latter are fundamental to reduce the level of abstraction and integrate the objects of П' (highly abstract concepts) as a form of concrete knowledge.

Praxeological analyses of excerpts of the real and abstract analysis textbook

What makes the presentation in the textbook special, and in this respect different from usual real analysis textbooks, is that the arguments in real analysis are already oriented towards corresponding structuralist arguments: the logos block of core real analysis praxeologies is already shaped by corresponding structuralist concepts and practices. It would not actually be necessary in terms of mathematical content, but it is intended to make the following step towards abstract analysis easier for the students. This central didactic gesture is pointed out in the preface by the author as follows: I have tried, wherever possible, to present proofs so that they translate mutatis mutandis into their counterparts in a more abstract setting, such as that of a metric space (for results in Chapter 1) or a topological space (for results in Chapter 3). On the other hand, some results first appear as exercises in one context before reappearing as theorems in another [...] I hope that this procedure of double exposure will enable students to grasp the material more firmly. (Bridges, 1998, p. xi) We will thus base our observations on chapters 1 (Analysis on the Real Line) and 3 (Analysis in Metric Spaces). Instead of metric spaces, other topics such as Hilbert spaces, which generalises the inner product in R n , would also be possible. Our goal is to illustrate the application of our model and provide preliminary results. Therefore, we are not going to detail the analysis methodology (choice of extracts, modelling difficulties,...) because we are not aiming at systematicity. In Chap. 1, concepts such as distance, sequences, convergence, compactness and continuity of functions are presented in the context of real numbers. Regarding our model (Fig. 2), we consider the praxeologies established in Chap. 1 as an intermediate stage in the transition step of type 1. Indeed, compared to school analysis, the practical blocks are already based on an abstract axiomatic foundation (of the real numbers) and conceptual definitions are formulated on this basis. The logos block already appears fully developed. Only against the background of the treatment of metric spaces (see below) does it then become clear that they can be identified mostly as concretisations in the context of real numbers of more abstract praxeologies involving the metric space structure. This is a reason why we refer to the corresponding praxeologies of real analysis as pre-structuralist (see also [START_REF] Laukert | Calculus at the school to university transition: early stages of a structuralist perspective in real analysis[END_REF] and we see them as precursors of the structuralist praxeologies that compose the middle stage of our model.

The metric space structure is introduced in the beginning of Chap. 3. Concepts such as openness, closeness, convergence of sequences and their limit value or continuity are then defined as verbatim generalisations of the corresponding notions from Chap. 1 where they have been introduced in the context of real numbers using the absolute value: Accordingly, there are several tasks where students have to generalise proofs from the real analysis chapter: "prove that any constant mapping between metric spaces is continuous", "prove that a contractive mapping is continuous", or the proof that the sum, difference, max, min, product of realvalued functions defined on a metric space X and continuous at a point a are again continuous at a (p. 136). The last task, phrased in the same terms except the omission of X, was assigned in Chap. 1 as an application of properties of limits. The pre-structuralitst local praxeology Pps around the type of tasks T "prove that a real function is continuous" and the unifying logos of which contains definitions and properties of limits and continuity thus gives birth to a full structuralist praxeology Ps in which R is replaced by X. Structuralist technologies are now explicit in the form of theorems established through the assigned abstract tasks. This achieves the type 1 transition (with respect to the targeted praxeology), which is facilitated for the learner by the close relationship between Pps and Ps. It also prepares the type 2 transition by an emphasis on abstract tasks.

For reasons of space, we will limit our account of further relationships between the contents of chapters 1 and 3 with a focus on compactness. In Chap. 1, we can read: "A bounded closed interval in R is also called a compact interval". Later in the chapter, the concept of open covering of a subset S of R is defined and the Heine-Borel-Lebesgue Theorem is stated and proved: "Every open cover of a compact interval I in R contains a finite subcover of I". It is noteworthy that this result is introduced by the following meta-discourse (p. 47): "deeper results about continuity depend on two fundamental properties of the real line, described in our next two theorems. […] Although there exists shorter proofs, the one we present is adapted to prove a more general result in Chap. 3".

Open coverings are the essential ingredient to define compactness in the general context of metric spaces: "a metric space X is called compact if every open cover of X contains a finite subcover" (p. 147). The following task revisits the original real analysis definition: "prove that a subset of the Euclidean space R n is compact if and only if it is bounded and closed" (p. 148). This example shows very well how the dialectic between the particular and the general is addressed in the textbook. Furthermore, three types of structuralist assertions (including proofs) can be identified in the context of the concept of compactness. A first type is about generalising theorems that were already formulated and proved in the real analysis context: "a compact subset of a metric space is separable and bounded", "a compact set in a metric space is closed" (p. 147). A second type concerns statements that highlight the specificities of the real analysis context: "a normed space is finite-dimensional if and only if its unit ball is totally bounded, in which case that ball is compact " (p. 191). Finally, a third type of assertions are proven which are not only new, but also simplify (in the sense of the structuralist methodology) proofs of statements that are easy to formulate in the context of real analysis but are cumbersome to prove: "if f is a continuous mapping of a compact metric space X into a metric space Y, then f(X) is a compact set" (p. 148). Indeed, this result provides an immediate argument for the compactness of the circle line in R 2 , since it can easily be identified as the image of a continuous map from [0,2π] to R 2 . If all three types of structuralist assertions we have highlighted have in common that they are facets of the relationship between the particular and the general, i.e. between contents of real and abstract analysis, only the first facet is fully explicit in the textbook.

In the preface, the author classifies the 750 exercises contained in the textbook in 4 categories: "applications and extensions of the main propositions and theorems; results that fill in gaps in proofs or that prepare for proofs later in the book; pointers towards new branches of the subject; deep and difficult challenges for the very best students". This typology is centred on the theoretical development of abstract analysis and generalisation processes. Indeed, the following assigned task illustrates the transition step of type 2: "prove that if f is a continuous one-one mapping of a compact metric space X onto a metric space Y, then the inverse mapping is continuous". The type of tasks is still T, but the technique uses the topological criteria for continuity, so that knowledge on compact subsets is required to fulfil the task. By contrast, the typology lacks a category of tasks related to the investigation of new classes of concrete problems: applications of results are mainly conceived of in relation to theoretical extension. In our view, this hinders the type 2 transition due to a feeble concreteabstract dialectic within Chap. 3. Referring again to the middle stage of our model, we didn't encounter the contextualised level 2 structuralist praxeologies Ps that we expected to see and which could have contributed to the rationale of the assigned abstract tasks.

Conclusion and outlook

We note that the praxeologies of real analysis in Chap 1 already have an axiomatic foundation and take into account conceptual definitions that build on them. The same applies to elementary arithmetic, founded on the Peano axioms, without the perspective of Ring Theory. Nevertheless, a difference with abstract algebra resides in the quite different possibilities for structuralist generalisations of real analysis concepts besides metric (or normed, etc.) spaces, such as distributions or Hilbert space theory. Beyond the elaboration of structuralist aspects of real analysis, their justification is to make problems solvable that can actually be formulated in terms of real analysis but cannot be fully developed in its context, such as Dirichlet problems for elliptic partial differential operations and much more. Considering only metric spaces, such a strong justification appears to be missing. Since Hilbert spaces and also Sobolev spaces turn out to be metric spaces, the results regarding metric spaces can be transferred to those examples. The didactic issue arises to make coherent choices between theoretical exposition of concepts and structures and classes of problems (such as partial differential equations) to motivate and apply the structuralist insights.

Our model has unveiled a realm of structures that generalises real analysis and whose abstract face at the master's level largely rivals abstract algebra, posing the same didactic problems of developing a concrete-abstract dialectic conducive to learning. Reciprocally, the advent of structures explains the setting up of pre-structuralist praxeologies in real analysis, which serve as precursors to structuralist praxeologies in abstract analysis. Praxeological analyses of other textbooks is necessary in order to consolidate our model and deepen the study of the transpositive choices around the two types of structuralist transitions we have highlighted.

Figure 1 :

 1 Figure 1: Winsløw's model of transitions

Figure

  Figure 2: Structuralist transitions

Figure 4 :

 4 Figure 4: generalisation in terms of metric spaces (Bridges, 2018, p. 130)


Elizabeth Kimber
email: elizabeth.kimber@open.ac.uk

Cathy Smith

Identifying static-and emergent-shape thinking in students' language about graphs of functions

Keywords: Graphs, functions, covariation, language, systemic functional linguistics (SFL)

Perspectives on graphs of functions can involve static-shape thinking, which focuses on perceptual shapes, or emergent-shape thinking, which views a graph as a trace of covarying quantities. These, and more formal covariation perspectives, are used flexibly and together to solve graphical problems. Researchers have grappled with establishing empirical-theoretical relationships between the language students use and the meanings made available. Systemic Functional Linguistics (SFL) offers tools to analyse how linguistic choices contribute to the creation of mathematical texts and can thus be used to investigate how features of students' language communicate forms of shape-thinking or covariation. In this paper we present an SFL analysis of students' short written descriptions of a graph. We uncover similarities to the language used in their lessons and how shape thinking or covariation may be constructed by linguistic features of students' discourse.

Theoretical background

Functions are a core area of the secondary mathematics curriculum and are important for progression to calculus and for participation in a wider discourse on change. Research has suggested different interpretations of graphs, e.g., as picture or icons resembling objects [START_REF] Leinhardt | Functions, graphs, and graphing: Tasks, learning, and teaching[END_REF]; as an inherent shape that is independent of scales and orientation of axes (static-shape thinking); or as a trace of covarying quantities (emergent-shape thinking) [START_REF] Moore | Shape thinking and students' graphing activity[END_REF]. These are not mutually exclusive types of reasoning. We have inherited the names 'static-and emergent-shape thinking' but see them as forms of reasoning constructed in discourse, not as states of mind.

Understanding covariation involves coordinating how the values of two varying quantities change in relation to each other. It is fundamental to a conceptual development of functions and may occur at different levels of sophistication [START_REF] Thompson | Variation, covariation, and functions: Foundational ways of thinking mathematically[END_REF]. Teachers are necessarily constrained by the curriculum and we are interested in whether the richness of covariation is made available in lesson topics that focus on graphs and functions, for example, here, lessons on transforming graphs.

Students' learning and discourse on graphs

Learning mathematics can be seen as developing and participating in mathematical discourse. This includes learning how to use technical terms and grammatical structures that differ from everyday language, with teachers' language playing a crucial role in this development [START_REF] Schleppegrell | The linguistic challenges of mathematics teaching and learning: A research review[END_REF][START_REF] Sfard | When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint[END_REF]. Researchers have called for more analysis of topic-specific discourse [START_REF] Erath | Designing and enacting instruction that enhances language for mathematics learning: A review of the state of development and research[END_REF][START_REF] Prediger | School academic language demands for understanding functional relationships: A design research project on the role of language in reading and learning[END_REF] and [START_REF] Thompson | Variation, covariation, and functions: Foundational ways of thinking mathematically[END_REF] have asked researchers to articulate in more detail the basis of claims that someone is reasoning covariationally. Temporality and fictive motion (where movement verbs are used for objects that do not move) are important for developing concepts of smooth, continuous covariation [START_REF] Thompson | Variation, covariation, and functions: Foundational ways of thinking mathematically[END_REF]. Antonioni et al. (2020) further found that, after working with functions in dynamic environments designed to highlight motion, students' written texts involved dynamic language expressing variation around points. While not formal, this 'transitional discourse' can be related to formal discourse, and its expression of variation could be beneficial for learning. In linguistic terms, dynamism and temporality would be evident in choices that foreground actions (in SFL terms, material processes with explicit Actors), but these are not typical of formal mathematics discourse, which tends to concern timeless relations between objects rather than material processes [START_REF] Antonini | From experiences in a dynamic environment to written narratives on functions[END_REF][START_REF] Morgan | Writing mathematically: The discourse of investigation[END_REF][START_REF] Sfard | When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint[END_REF].

In this paper, we use tools from Systemic Functional Linguistics (SFL) to analyse students' written descriptions of graphs for linguistic features that may correspond to static-or emergent-shape thinking or covariation. This analysis is part of a larger study that focuses on teachers' classroom discourse about functions and graphs and contributes to answering our research question: 'What features of language are evident in classroom discourse on graphs?' And in particular for this paper, 'How do linguistic features of students' written descriptions of graphs construct or interweave shape thinking or covariation perspectives?'

Systemic Functional Linguistics

SFL is used to analyse written or spoken texts, and offers a theoretical basis for distinguishing the characteristics of formal, scientific language-in-use [START_REF] Halliday | An introduction to functional grammar[END_REF]. Within SFL, texts are seen as the product of ongoing conscious or unconscious choices. These affect how a text functions for the audience via its interrelated ideational, interpersonal and textual metafunctions. SFL analysis of mathematics texts has identified characteristic features of the mathematics register, such as technical vocabulary, nominalisation (through which processes become objects) and the prevalence of relational processes (such as being or having) which identify objects or assign attributes [START_REF] Schleppegrell | The linguistic challenges of mathematics teaching and learning: A research review[END_REF]. Through the ideational function, choices of process types and participants contribute to the construal of mathematical activity in a text. In particular, in the mathematics register relational processes are typically used to set up static, timeless relationships between mathematical objects [START_REF] Morgan | Writing mathematically: The discourse of investigation[END_REF]. In contrast, material processes of doing and happening are common in everyday use; they may have mathematical objects or humans as Actors and provide an actionoriented narrative that unfolds over time [START_REF] Halliday | An introduction to functional grammar[END_REF]. For example, the x-intercept is at (1,0) is a timeless, identifying description, whereas the graph crosses the x-axis at (1,0) allows an interpretation of unfolding -emerging -movement with the graph as Actor.

Related to choice of process, the choice of active or passive voice contributes to the ideational and interpersonal functions [START_REF] Halliday | An introduction to functional grammar[END_REF]. The interpersonal function constructs relationships between author, audience and mathematics. For example, the active clause I translated the graph has a human Actor, whereas a passive form the graph has been translated omits the Actor. Mathematicians and mathematics teachers commonly use passive voice to indicate generality but, by obscuring human agency, this diminishes the perception of mathematics as a classroom activity [START_REF] Morgan | Studying the role of human agency in school mathematics[END_REF]. Finally, the textual function organises the text to facilitate the other two functions.

It is these different portrayals of mathematics, mathematical participants and mathematical activity that can be brought into focus using SFL and may have the potential to suggest different forms of shape thinking or covariation.

Method and data

We are investigating classroom discourse in lessons on functions, graphs and change in secondary mathematics classrooms in England. Our main focus is on teachers' classroom discourse, but we also analyse students' language, to compare it with their teacher's language and language-focused practices in the lesson. The schools, teachers and student groups are an opportunity sample, and the data presented here was selected because its analysis allowed comparisons between two student groups taught the same topic by the same teacher. The topic of graph transformations is relevant as it offers opportunities to discuss the behaviour of individual graphs and the relationships between them.

The students were from two groups at a state school in England, taught by the same experienced teacher. The lessons involved similar activities using software to plot and compare graphs. Lesson 1 was for 30 students aged 15-16 who were due to take national examinations within eight months. Lesson 2 was for 28 students aged 14-15, much earlier in the same course. Ten lesson 1 students and 23 lesson 2 students gave consent for their responses to be analysed.

The data analysed includes lesson videos, teaching materials, a teacher interview and, centrally, students' written responses to an exit pass, i.e., a short task used at the end of a lesson to inform future lessons [START_REF] Wiliam | Embedded formative assessment[END_REF]. We wrote the task (Figure 1) for the lessons in consultation with the teacher.

Language-focused teaching in the lesson

We now describe features of the teacher's language, to illustrate our use of SFL tools and to allow later comparison with students' language. In lesson 1, the teacher emphasised that examinations required students to use words such as translation and vector (verbal and symbolic nominalisations) rather than everyday words such as move. To demonstrate this, he displayed the following model text, which students were asked to copy and complete. He used similar text three times in the lesson.

Given the graph of 𝑦𝑦 = 𝑓𝑓(𝑥𝑥), the graph of 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) + 𝑐𝑐 represents a translation.

The translation is given by the vector � 0 𝑐𝑐 �.

This model text aligns with several features of the mathematics register: it uses a relational process represents to describe a relationship between graphs; translation occurs as a noun, i.e. the material process translate has been nominalised; is given by is a passive construction which, textually, achieves a shift in attention between mathematical objects, from translation towards vector.

The teacher also discussed two other forms of words with students:

Teacher: you can say it is being translated that's ok or it is a translation either is ok Describe the blue graph.

(Please write 1-2 full sentences.) The second of these proposes the simpler relational process is instead of represents. The first introduces the material process translated, but in passive voice with an unspecified participant it. The teacher appears to be endorsing this material process+passive form as a more familiar but acceptable wording and the same form was used in a worksheet The graph … is transformed by the vector … .

Lesson 2 did not involve model text. Instead, the teacher encouraged students to develop their own articulations. He displayed descriptions of transformations in note form rather than sentences; most used the material process+active form but lacked Actors, e.g. translates the graph up by c units.

In the interview the teacher explained that he included the model text in lesson 1 "to make sure all their books had accurate notes that were easy to read". He also explained that students are expected to support each other to use technical language in discussions.

Analysis

In the first stage of analysis we considered each exit pass response holistically in terms of mathematical content, informed by the literature on shape thinking and covariation [START_REF] Moore | Shape thinking and students' graphing activity[END_REF][START_REF] Thompson | Variation, covariation, and functions: Foundational ways of thinking mathematically[END_REF].

The main SFL analysis considered the form of words in individual clauses based on the definitions and examples in [START_REF] Halliday | An introduction to functional grammar[END_REF] and using the local mathematical context. We focused on features of text that contribute to the portrayal of mathematics and thus we identified relational and material processes and their participants. Finding few human Actors, we also identified whether material processes were used in active or passive voice.

The final stages were to make detailed comparisons of responses from each lesson and between the lessons, and to consider the alignment of students' language with their teacher's language. By combining linguistic analysis with our initial holistic analysis, we considered whether certain linguistic features of students' responses suggest static-or emergent-shape thinking or covariation.

Findings

Holistically, most exit pass responses fall into two clear categories. Some describe the graph in terms of transformations (9 responses), echoing the lesson language. Some describe the shape or graphical features such as intercepts (22 responses). Finally, four responses, all from lesson 1, could not be categorised as the meanings are unclear: three refer to a transition and one may refer to intercepts.

Describing transformations

All six categorised lesson 1 responses refer to translations and offer no description of the graph's covariational behaviour. For example Response 1A: the curve has been translated by the vector � 2 3.5 �

These responses from lesson 1 show a uniformity and clear alignment with the lesson language. They use translation or translated and give the vector symbolically. Two use the teacher's nominalisation, but five use the teacher's alternative form of material process+passive, as in 1A. Whether they use a relational process or material process+passive, all the responses start with a mathematical object as participant, either the curve or it. In most cases it is likely to refer to the graph students were asked to describe, but in It is a translation, the It is more ambiguous and could equally refer to the relationship between the given curve and a perceived 'parent' curve. We note that none of the transformation responses explicitly refer to the idea of two curves from the model text.

In contrast with lesson 1, only three responses in lesson 2 describe transformations at all and do so without using translation or translated. Use of processes is similar to the lesson 1 responses as they include a mix of relational processes and material processes+passive. An example is Response 2B: It is flipped in the x-axis and placed in a weird angle. It's moved up in the y-axis and left in the X axis and it is a blue line.

As with the lesson 1 responses, 2B does not describe the shape of the curve, though the two other responses do include this using a relational process, e.g. It is a blue parabola but it has been stretched.

Overall, most of the responses (7/9) that describe transformations use material processes (translated, moved, flipped, placed, stretched), and all in passive voice, so these are processes that students considered are being done 'to' a graph rather than 'by' it.

Describing the graph's shape, features or activity

The 22 responses in this category are all from lesson 2. They are highly varied, as the examples below show, but refer to a named graph (12), points on the graph (15) and/or the graph's activity (13).

Response 2C: It is a parabola that goes through the point (2,3.5). The y-intercept is (0,2.1).

Response 2D: the blue graph is a parabola that peaks at (2,3.2) and intercepts the y-axis at 2.1

Response 2E: The graph heads as a linear line in a positive x-axis direction. Continuing along the xaxis, it begins to curve at a plateau, and gradually curves in a negative direction.

Responses 2C and 2D are typical of responses in this category: they talk about mathematical objects (parabola, y-intercept) and use a relational process is to name the graph. But they also use material processes (goes through, peaks) that construct the vertex as part of the graph's activity. They use different process types to describe intercepts: 2D continues the description of activity by using the material process intercepts, while 2C locates the object y-intercept using a relational process. The responses that described the graph via points on an unnamed curve were similarly split between using material or relational processes, or both. Response 2E shares only some features with other responses in this category: it refers to a linear line but does so using a material process heads; it describes the graph's activity, but in relation to the x-axis rather than locating particular points.

Unlike the material clauses in descriptions of transformations, these responses all use active voice.

There are also patterns in the types of material processes chosen: as responses 2C-2E show, students chose material processes that invoke visual features of the graph (peaks, curves) or movement and direction (goes through, intercepts, heads, crosses). These portray the graph as participating in processes through which its shape emerges or which relate to motion.

Discussion

We only have a small number of responses, especially from lesson 1, so we can only make tentative claims about students' language.

Uniformity and alignment with the lesson language

Lesson 1 responses align closely with the lesson language including the use of technical vocabulary and stating the vector, which were emphasised by the teacher as examination requirements. A difference is that the model text used the nominalisation translation in a relational process, whereas most students adopted the teacher's alternative of using the material process+passive.

One reason for common features of responses in the same lesson is that students were allowed to discuss their responses to the exit pass. From the seating plans we know that only some of the lesson 1 students who described translations in a similar way were seated together, so this does not account for all the similarity. We suggest that a stronger reason could be the language-focused practices in lesson 1, providing model text and emphasising the use of technical language.

In contrast, only three of the 23 responses from lesson 2 describe the graph as a transformation and the variety between these responses corresponds to the variety in lesson language. Lesson 2 had lower emphasis on formal language, which may have contributed to the overall variety of responses, as the students did not have immediate access to an authorised way to describe the graph. As with lesson 1, students' responses to the task will have been filtered by their perceptions of what was expected.

Passive voice is a distinctive feature of all the transformation responses. In English, the start of a clause is usually what the clause is about [START_REF] Halliday | An introduction to functional grammar[END_REF]. Since most students started with the curve (sometimes referred to as it), we suggest the curve was their focus of attention in this task. However, from the transformations work they were also concerned with describing this curve as the result of activity, thus used a material process form familiar from the everyday register. The material process+passive construction, which was made available in the lesson, achieves a description of activity but with the curve as the starting point. Previous research compares using passive rather than active voice and concludes that the passive retains a focus on activity but obscures human agency [START_REF] Morgan | Writing mathematically: The discourse of investigation[END_REF][START_REF] Morgan | Studying the role of human agency in school mathematics[END_REF]. However, here we consider students' adoption of material process+passive, e.g. the curve has been translated, instead of a more formal relational process with nominalisation, e.g. the curve is/represents a translation. In this context we agree that the material process+passive construction introduces activity and temporality, which are less apparent in the relational form. However, rather than obscuring human agency, we see it as allowing for human activity, as it portrays the given graph as a product of activity of an unspecified but potentially human actor.

Shape thinking and covariation in students' responses

We now examine how students' linguistic choices may communicate shape thinking or covariational reasoning. Our analysis is based on a small dataset generated in the context of graph transformations, so these are tentative ideas. Furthermore, we do not claim that features of students' language mean they are or are not thinking in particular ways.

In the transformations responses, the material process+passive form, combined with choices of material process that involve deformation or repositioning (translated, moved, flipped, placed, stretched), suggest that the given graph results from physically moving a material object around the plane. This invokes the material 'graph as wire' metaphor within static-shape thinking rather than more sophisticated emergent-shape thinking [START_REF] Moore | Shape thinking and students' graphing activity[END_REF]. We note that this portrayal involves only one curve, rather than a relationship between two curves afforded by the model text.

Within the graphical responses, we see many material processes in active voice with the curve as Actor (as in 2C-2E). When material processes are connected with shape, such as peaks and curves, this may allow for static-and emergent-shape perspectives to be intertwined or co-exist. These word choices describe perceptual features, suggesting or allowing static-shape thinking. But the students describe features of the graph using a material process, which portrays the visual features as arising through unfolding change, and therefore could simultaneously afford emergent-shape thinking.

Other material processes relate to movement as travel (goes through, intercepts, heads, continuing along, crosses). This is fictive motion, as the graph does not actually move. Notions of continuous covariation may require this conception of travel, but also require coordination of x-and y-values [START_REF] Thompson | Variation, covariation, and functions: Foundational ways of thinking mathematically[END_REF]. Language that combines movement processes with coordination of variables, such as 2C a parabola that goes through the point (2,3.5) or 2D the blue graph … peaks at (2,3.2), could therefore express covariational reasoning. In contrast, the use of material processes in 2E suggests a different form of covariational reasoning. This relates the behavior of the graph to movement along the x-axis (continuing along the x-axis it begins to curve) but does not state values of variables, which could be interpreted as emergent-shape thinking with gross coordination of values, in which "the person envisions a loose, nonmultiplicative link between the overall changes in two quantities' values." (Thompson & Carlson, 2017, p. 441).

We also see relational processes that classify and identify. Many responses use relational clauses to name the curve e.g. it is a parabola. Given students' probable prior work with functions and that the curve is actually part of a cubic, this is likely to have been based on the graph's perceived shape. A simple interpretation of this could be that the students are foregrounding static-shape thinking.

However, many expand their responses to include material process descriptions, offering richer interpretations. For example, the relational form in 2C The y-intercept is (0,2.1), treats intercepts as objects and expresses a timeless relationship, suggestive of static-shape thinking. In contrast, the material form in 2D the blue graph … intercepts the y-axis at 2.1, does not require the x-or y-intercept to be objects: instead they are part of the graph's unfolding behaviour.

These contrasts between relational and material processes are largely due to the detemporalising effect of the relational form, in which activity becomes static [START_REF] Morgan | Writing mathematically: The discourse of investigation[END_REF]. Relational processes and passive voice are features of formal mathematics texts. However, in this context, we note that our argument above suggests that they may both indicate less-sophisticated static-shape thinking, or at least do not challenge that, while certain material processes in active voice have potential to communicate the richness of emergent-shape thinking.

Conclusion

Within the limitations described, and although it is not surprising, we have provided tentative evidence that teachers' language and choices about language-focused practice in lessons affects students' language. We have also demonstrated how tools from SFL can help to identify how students' choices of words or grammatical structures indicate static-or emergent-shape thinking or covariation. Encouraging certain choices of words or phrases may help students both express and understand covariation, but further work would be required to establish such a link. Our work here adds to detail of why forms of shape thinking or covariational reasoning may be attributed to students.

Figure 1 :

 1 Figure 1: Exit pass given to students