There are numerous problems associated with the teaching of Euclidean geometry at secondary schools today. Students do not see the necessity of proving results which have been obtained intuitively. They do not comprehend that the validity of a deduction is independent of the 'truth' of the initial assumptions. They do not realise that they cannot reason from diagrams, because these may be misleading or inaccurate. Most importantly, they do not understand that Euclidean geometry is a particular interpretation of physical space and that there are alternative, equally valid interpretations. A possible means of addressing the above problems is tbe introduction of nonEuclidean geometry at school level. It is imperative to identify those students who have the pre-requisite knowledge and skills. A number of interesting teaching strategies, such as debates, discussions, investigations, and oral and written presentations, can be used to introduce and develop the content matter. / Mathematics Education / M. Sc. (Mathematics)

Cette thèse traite de l’étude des objets reliés au codage de Bowen-Series du flot géodésiquepour des surfaces hyperboliques de volume fini. On démontre d’abord que le billard géodésiqueassocié à domaine fondamental even corners d’un groupe fuchsien cofini est conjuguéà une bijection du tore, appelée codage étendu, dont l’un des facteurs est la transformationde Bowen-Series. L’intérêt principal de cette conjugaison est qu’elle ne fait toujours intervenirqu’un nombre fini d’objets. On retrouve ensuite des résultats classiques sur le codage deBowen-Series : il est orbite-équivalent au groupe, ses points périodiques sont denses, et ses orbitespériodiques sont en bijection avec les classes d’équivalence d’hyperboliques primitifs dugroupe ; ce qui permet finalement de relier sa fonction zeta de Ruelle à la fonction zeta de Selberg.Les preuves de ces résultats s’appuient sur un lemme combinatoire qui abstrait la propriétéd’orbite-équivalence à des familles de relations qui peuvent être définies sur tout ensemble surlequel agit le groupe. Il est aussi possible de conjuguer le codage étendu à un sous-shift detype fini, sauf pour un ensemble dénombrable de points. Enfin, on prouve que les distributionspropres pour la valeur propre 1 de l’opérateur de transfert sont les distributions de Helgason defonctions propres du laplacien sur la surface, puis que l’on peut associer à toute telle distributionpropre une fonction propre non triviale de l’opérateur de transfert et que ce procédé admet uninverse dans certains cas. / This thesis focuses on the study of the objects linked to the Bowen-Series coding of the geodesicflow for hyperbolic surfaces of finite volume. It is first proved that the geodesic billiardassociated with an even corners fundamental domain for a cofinite fuchsian group is conjugatedwith a bijection of the torus, called extended coding, one factor of which is the Bowen-Seriestransform. The sharpest property of that conjugacy is that it always only involves a finite numberof objects. Some classical results about the Bowen-Series coding are then rediscovered : itis orbit-equivalent with the group, its periodic points are dense, and its periodic orbits are inbijection with conjugacy classes of primitive hyperbolic isometries ; which eventually links itsRuelle zeta function to the Selberg zeta function. The proofs of those results use a combinatoriallemma that abstracts the orbit-equivalence property to families of relations that can be definedon every set on which the group acts. The extended coding is also proved to be conjugated witha subshift of finite type, except for a countable set of points. Finally, it is shown that eigendistributionsof the transfer operator for the eigenvalue 1 are the Helgason boundary values ofeigenfunction of laplacian on the surface, plus that one can associate to each such eigendistributiona non-trivial eigenfunction of the transfer operator and that this process has a reciprocalin some cases.