Nina Ullsten Granlund
email: nina.ullsten@gmail.com

Implementing theories for preschool teaching with play-based pedagogies

Keywords: Preschool mathematics teaching, play-based environment, professional development

The play-based Swedish preschool is facing new challenges due to demands from the Swedish Schools Inspectorate to evaluate how the teaching of mathematics is carried out in practice. This demand has evoked a dilemma for the preschool teachers that find it unclear what teaching in a preschool means. A professional development project was set up, aiming to implement theoretical tools from research into preschool mathematics teaching. The project started in 2016 with 45 teachers over a duration of seven months. An integral part of the project is to conduct research on the impact of the preschool teachers' conceptualization of mathematics as a subject, and what counts as a mathematical situation. The participating preschool teachers collected video recordings of mathematical situations from their respective practice. The professional development is an ongoing project, the impact of the implementation of theories are currently being researched.

Introduction

The Swedish preschool is facing new challenges, it has a tradition of caring and educating and is built upon a playful custom of will on behalf of the child. The preschool is not organized in formal lessons and its curriculum does not define any subjects like mathematics or biology, but suggests a thematic way of arranging the practice so that the learning goals found in the curriculum can be met in line with the principles of social pedagogy (Bennet, 2005). Yet, since the national curriculum for preschools was introduced in 1998, some of the goals have been obviously concerned with mathematical matters and these goals were sharpened in 2010 and has now again in 2018 been slightly revised [START_REF] Skolverket | Läroplan för förskolan Lpfö98 reviderad 2018[END_REF](Skolverket, , valid from 2019)).

Having particular goals for mathematics means that the children's learning cannot just be expected to happen through general everyday activities, but must be planned for by the preschool teachers. In previous national curricula, there has been no explicit mentioning of teaching, despite that teaching has been the overarching term used in the Education Act to describe the goal directed processes aiming for learning and development. Since 2018 however, teaching is an explicitly mentioned concept in the revised curricula [START_REF] Skolverket | Läroplan för förskolan Lpfö98 reviderad 2018[END_REF]. The changes in terminology, and perhaps also policy, came after the Swedish Schools Inspectorate expressed that there is a problem with an overall unclarity about what teaching in preschool means and how it should be done [START_REF] Skolinspektionen | Förskolans pedagogiska uppdrag-Om undervisning, lärande och förskollärares ansvar [Preschool's educational assignment-About teaching, learning and preschool teachers´responsiblility[END_REF]. The dilemma is that in the play-based tradition, the organization of learning is quite different from a typical school setting. For example, children's own initiatives are much more foregrounded than the time children spend in situations that are guided by a teacher [START_REF] Helenius | Explicating professional modes of action for teaching preschool mathematics[END_REF]. When the concept of teaching is imported from school to preschool, the playbased tradition is endangered. Voices have been raised, indicating that the teaching of mathematics might be particularly prone to act as a trojan horse for importing values and beliefs on teaching from school that might not necessarily fit very well in a play-based preschool practice [START_REF] Fosse | Mathematics as the Trojan horse in Norwegian early childhood policy[END_REF].

The present paper reports on a professional development (PD) project aiming to offer the preschool teachers1 theoretical tools for how they can think about the mathematics taught and how to think about the teaching in a structured and explicit way, that still respects the play-based tradition. Research on the effects on the teachers is an integral part of the project.

Implemented theories in the PD project

When the play-based pedagogy of Swedish preschool works at its best, it integrates a play-based pedagogy with learning of mathematics in accordance with the curriculum goals. This can not only be done by planning regular teaching activities, since it would affect the preschool activities in a direction away from play in favor of formal schooling. Teaching must thus also be organized by means of planning the environment and using spontaneous situations in deliberate ways. A theory of play-based preschool teaching must account for that. At the same time, preschool teachers have difficulties verbalizing what constitutes teaching acts. Preschool teachers tend to not regard themselves as teaching teachers [START_REF] Skolinspektionen | Förskolans pedagogiska uppdrag-Om undervisning, lärande och förskollärares ansvar [Preschool's educational assignment-About teaching, learning and preschool teachers´responsiblility[END_REF]. Therefore, a theory for play-based teaching should help the teachers to structure the type of work that is involved in play-based teaching by explicating particular components of such teaching [START_REF] Helenius | Explicating professional modes of action for teaching preschool mathematics[END_REF]. A structured way of thinking about teaching can be a tool for teaching in a more deliberate way. This is a way to give teachers concepts and words for discussing with colleagues and plan their teaching in a play-based environment.

Theorizing preschool teaching

A structured way of viewing the complexity of mathematics preschool pedagogy is through defining three dimensions of teacher action, with respect to mathematical pedagogy: pedagogical explication, teacher participation and situational planning (for more information see [START_REF] Helenius | Explicating professional modes of action for teaching preschool mathematics[END_REF]. They are not suggestions of normative categories, but examples of how the complex pedagogical practice of teaching preschool mathematics could be explicated.

First, in a dimension of pedagogic explication of the teaching and learning of mathematics, two different ways of including mathematics can be identified. One way is a classical teaching situation, where the participants know that the activity concerns the learning of some specific mathematical content. Following [START_REF] Walkerdine | The mastery of reason: cognitive development and the production of rationality[END_REF], such situations are called pedagogical with respect to the mathematical content [START_REF] Helenius | Analysing instrumental and pedagogic situations in preschools using the didactic space[END_REF][START_REF] Walkerdine | The mastery of reason: cognitive development and the production of rationality[END_REF]. The other way is when the situation is fundamentally about something else, but some mathematical activity is included in the situation. Such situations are called instrumental. In this theorization the child's experience of a situation as instrumental rather than pedagogical does not mean there are not pedagogical intentions from the teacher's point of view. For example, as exemplified in [START_REF] Helenius | Explicating professional modes of action for teaching preschool mathematics[END_REF], a teacher's decision to make a robot available to children have pedagogical intentions, but children playing with it are not thinking of it as a situation which they are supposed to learn mathematics. Teachers can make an active choice of explicating the mathematical content of a situation or not. This choice can be made beforehand, in planning, or on the fly, when a teacher observes a mathematical situation that she chooses to either act on by shifting the children's focus to the learning of some mathematics, or not. Similarly, when children act on their own, with no teacher involvement, in a situation that contain mathematics, the learning may or may not be visible for the children. Thus, when the children perceive the situation as being about learning something mathematical, that situation is 'pedagogical'.

Second, a dimension of teacher participation is quite obvious for children in preschool because they spend a lot of time conducting activities without any teachers involved. They know when the teacher is arranging activities, or not. The unarranged situations can provide children with opportunities to use mathematical knowledge and participate in mathematical activities. Therefore, time spent in preschool with no pedagogue involved can contribute to children's mathematical development in important ways. The same is true for time spent outside preschool, but the difference is that teachers can influence the mathematical content as well as children's experience of mathematics in situations in the preschool environment. One obvious example is planning specific activities that the teachers ask the children to do. But, there are also more subtle ways, such as planning and modifying the environment, in and around the preschool facilities so that it stimulates mathematical activities. For example, children can play with a programmable toy robot, only available to children for their free play.

This third dimension of situational planning relates to what [START_REF] Ginsburg | Mathematics education for young children: What it is and how to promote it[END_REF] calls "teaching moments" or "acting in the moment", in contrast to formally planning learning situations in Swedish preschool mathematics pedagogic discourse. It is of course important for teachers to use the option to plan situations where they participate themselves. Planning has a distinct advantage in giving the teachers time to predict what may happen in a situation, and prepare for different possibilities. But mathematically interesting situations can also occur outside planning. It is important to acknowledge that situations that are not directed by the teacher also allow for play in situations where children may use mathematics. Moreover, preschool teachers can affect what happens during "free time" in many ways. This is also a part of the planning. However, preschool teachers have choices concerning what situations they want to plan and which to deliberately leave unplanned.

The Schools Inspectorate evaluation of the preschool 2015-2017 points out that preschools see that children learn a lot from playing but they also found that teachers have a strong belief that children should not be disturbed when playing. This can lead to overlooking possibilities for learning and development. The evaluation also observes that mathematics mainly occurs in planned activities lead by an adult. The theory presented above attends to this dilemma by using the tool of documentation. If a teacher notices interesting activities in free play, these can be noted and documented and the teacher can later plan for how the experiences gained in the documented situations can be reused later. This can be done either by retrospectively challenging the children that were in the original situation, but also by using ideas from the documented situation for a planned activity with other children.

Theorizing mathematical activities

One aspect of the theory above is using mathematics instrumentally. Therefore there is a need for a conceptualization of mathematics that views mathematics as something that can come out of practical problem situations, that is, a conceptualization that does not only view applications of mathematics as something you do with mathematics that is already learned or known. Moreover, a conceptualization of mathematics that is to be useful for preschool also needs to bridge everyday mathematics with formal mathematics, preferably in a rather seamless way. It has previously been argued that one theory that fulfils these requirements is Bishop's theory of mathematical activity [START_REF] Bishop | Mathematical enculturation: A Cultural Perspective on Mathematics Education[END_REF][START_REF] Helenius | Explicating professional modes of action for teaching preschool mathematics[END_REF][START_REF] Helenius | Analysing instrumental and pedagogic situations in preschools using the didactic space[END_REF][START_REF] Johansson | Perceptions of mathematics in preschool: "Now we have a way of talking about the mathematics that we can work with[END_REF].

The background for Bishop's work is anthropological. By studying the use of mathematics in different cultures, he concluded that mathematics, just like rituals and cooking, occurs in all cultures. He also found that mathematics is different depending on where and in which culture it is evolved. And yet, certain regularities also occur in these studies and Bishop (1988a;[START_REF] Bishop | Mathematical enculturation: A Cultural Perspective on Mathematics Education[END_REF] characterizes these in terms of six distinct mathematical activities that he claims are found in every culture and can be seen as the roots of mathematical thinking. The descriptions below are from Bishop (1988a;[START_REF] Bishop | Mathematical enculturation: A Cultural Perspective on Mathematics Education[END_REF]):

• Counting: The use of a systematic way to compare and order discrete phenomena.

• Measuring: Quantifying qualities for the purposes of comparison and ordering, using objects or tokens as measuring devices with associated units or 'measure-words'.

• Locating: Exploring one's spatial environment and conceptualizing and symbolizing that environment, with models, diagrams, drawings, words or other means.

• Designing: Creating a shape or design for an object or for any part of one's spatial environment. It may involve making the object, as a 'mental template', or symbolizing it in some conventionalized way.

• Playing: Devising, and engaging in, games and pastimes, with more or less formalized rules that all players must abide by.

• Explaining: Finding ways to account for the existence of phenomena, be they religious, animistic or scientific.

The purpose of basing the view of mathematics on a theory is that through an increased awareness and knowledge about the role of mathematics in everyday context, the teacher can in a more deliberate way consider several options when planning for situations where children can develop knowledge in mathematics. This may lead to using a wider class of situations in a more systematically way, adapted to children's interests, needs, experiences and qualifications. The project's theoretical framework is focused on a mathematics based on the six mathematical activities of Bishop (1988a;[START_REF] Bishop | Mathematical enculturation: A Cultural Perspective on Mathematics Education[END_REF], which is well established in the curriculum [START_REF] Skolverket | Läroplan för förskolan Lpfö98 reviderad 2018[END_REF].

Implementation of the theories through professional development

The project is now running for the third time in a medium-sized Swedish city, where you can find approximately 40 preschools for children aged 1-5 years. The first round was carried out from May to December 2016. Two preschools participated. The participants were 45 teachers, with different educational backgrounds. The second round run with four preschools during August 2017 to March 2018.

Both theories that the PD is built on have the potential to be normative in the sense of putting restrictions on what preschool teachers believe are the sanctioned ways of teaching mathematics. This is a dilemma from an implementation point of view, because if a point of both the theories and the PD is that it should be helpful for describing and improving existing preschool practice rather than to introduce normative ways of thinking about this practice.

To avoid such normativity, we tried to make the PD itself very tightly tied to the existing practice of the participating teachers by using videos produced by the participants of their practice as a basis for discussion. The recordings were done as a response to loosely formulated tasks concerning, for example, what mathematics might mean or what type of mathematics the teachers should focus on.

As an example, one of the early tasks was: Videotape one planned mathematical situation and one situation that is not planned but where you think there is some mathematics going on. The recording from this task was then analyzed by a researcher before the next session with the aim of explicating examples of Bishop's activities as well as examples of the three dimensions of preschool teaching described above. The submitted material represented choices made by the teachers themselves. Moreover, the loose phrasing also created a large variation in the interpretations of what was considered interesting mathematical activities to submit. This variation was intended, and when the videos were discussed at the next session, the discussion leader (the researcher involved in the project) could ground the presentation of Bishop's six activities in the variety of mathematical activities already present in current preschool practice (even if all of the activities did not show up in the videos, and some therefore had to be pulled out of Bishop's hat).

Moreover, the set of videos could also be used as a basis for discussion when planned and unplanned situations were good, if the mathematics was pedagogical or instrumental and how teacher participation might have affected the situations. In other words, even these first set of videos could be used as an introduction to all three dimensions in the theory of play-based teaching.

In subsequent assignments, the teachers were asked to do more and more pedagogically complex things, like planning for a situation that characterized by a particular combination of the dimensions in the theory of play-based teaching.

Other questions that were discussed in the sessions concerned progression: Watch a videotaped situation and analyze what mathematics occurred in the situations. Think about this in terms of progression. Are there more basic ways of dealing with the mathematical ideas and concepts that are visible in the situation? What would constitute a more advanced way of dealing with those concepts? How an activity that would bring children toward a more advanced way of dealing with the same mathematical ideas could look?

In total, for each PD group, the PD included three such sessions and associated tasks distributed for half a year. This is not much, but it has to be taken into account that preschool teachers have much less time for both planning and competence development than school teachers. In addition to the PD-pedagogy described above, in between the sessions with the researcher, the preschools also got support from a local school developer and the teachers were also given time to discuss the different issues of the project on every workplace meeting (approximately once a month).

Researching the effect of the implemented educational theories

Discussions with the teachers indicate that they find the six activities of Bishop to be both easy to understand and helpful for identifying different mathematical activities. The framing of activities as planned and unplanned, but still being mathematical situations, may at a first glance be a relief for the teachers. The project is all about creating mathematical situations that stimulate and challenge the children with different kind of questions and it is built on the curiosity and interest of the children. The teachers seem to be happily provided with teaching tools for framing their profession.

Still, these indications must be consolidated in a proper study. To investigate if the teachers' perception of teaching and mathematical activities has been improved by the project, a systematic data collection was carried out during the fall of 2018. The methodology is built upon using documentation from the PD rounds, new submitted video material, and interviews with teachers that will be analyzed in relation to the theoretical perspectives used in the PD. During the first two rounds of the PD project, we have collected data in the form of videotapes from planned and unplanned mathematical activities that the teachers encounter between the monthly meetings. We are now about to analyze the impact of the implementation of theories for theorizing preschool teaching and for theorizing mathematical activities. Because the objective of the PD was to help the teachers develop an elaborate and structured way to view (think and talk about, analyze, plan, etc.) the play-based mathematics teaching in preschool, the research question I ask is: Do the teachers that have completed the PD display an elaborate way of discussing mathematics learning activities in preschool? The term 'elaborate' will here be taken to mean using concepts (though not necessarily terminology) from the two theories that form the basis of the PD when discussing preschool activities. To evaluate this question, the following method will be used. A sample of five teachers will be asked to submit new videos and documentation responding to the same set of tasks that were given during the PD. Using a stimulated recall methodology, the teachers will be asked to discuss both the old and new videos. Stimulated recall is a method very well suited for situations where you record the activity, using sound or video [START_REF] Calderhead | Stimulated recall: A method for research on teaching[END_REF]. The participants are given the chance to see or listen to what was recorded and are given the chance to make comments, either freely or guided by interview questions. The recorded material is a way of stimulating, or challenging, the respondent's way of thinking when it comes to the filmed or recorded episode [START_REF] Haglund | Stimulated Recall. Några anteckningar om en metod att generera data. [Stimulated Recall. Some notes about a method to generate data[END_REF]. By using the pedagogical documentation from the project, which consists of the filmed planned and unplanned educational activities, as well as notes taken during the activities, the purpose is to go back and investigate teachers' own perception of their conceptualization about mathematics, and the teaching of mathematics in preschool. By also including new material, teachers are given the chance to also reflect on developments of their practice. Progress will be coded both in terms of if teachers after the program can analyze and talk about a depicted situation in a deliberate way in line with the theories used in the PD, and in terms of if teachers can suggest a larger and more relevant spectrum of alternate teacher actions, or ways of challenging the children in depicted situations. This part of the methodology has previously been suggested by [START_REF] Lembrér | Preservice teachers recognising and responding to young children's engagement with mathematics[END_REF].

Concluding reflection

The quality of the pedagogy is of great importance for the learning of children. Pedagogical quality depends on how the teachers structure their work. By clarifying the special ingredients of the mathematics in Swedish preschools, as well as the role of the preschool teacher, I claim that quality can be improved. This is the intention of the described PD project.

In the PD project, participating preschools teachers can work with how to think about teaching and how to think about mathematics as a subject to be taught in preschool. The hypothesis is that an increased awareness of what mathematics is and what can be categorized as mathematical situations will be used by the preschool teachers for developing the teaching. As described above, the implemented theoretical tool for framing teaching preschool mathematics is the six mathematical activities of Bishop (1988a;[START_REF] Bishop | Mathematical enculturation: A Cultural Perspective on Mathematics Education[END_REF], for sorting, planning and connecting different elements in the different activities. A theoretical tool for framing the planned and unplanned situations is given by Helenius' (2018) three dimensions for theorizing preschool teaching. This paper has described a project aiming to implementing theories from research into preschool teaching practice. Next, we will evaluate the impact of our PD project by conducting research on the participating preschool teachers' conceptualization of mathematics as a subject, and mathematical situations as planned or unplanned activities. Data will be collected after the third round of the PD project, making research on the implementation of theories an integral part. By the time of the conference CERME 11, we will be able to report on the impact of the project.

We include all employees who work with teaching in the participating preschools, although not everyone has a formal teacher education.


Athanase Papadopoulos
email: athanase.papadopoulos@math.unistra.fr

A NOTE ON NICOLAS-AUGUSTE TISSOT: AT THE ORIGIN OF QUASICONFORMAL MAPPINGS

Keywords: AMS Mathematics Subject Classification: 01A55, 30C20, 53A05, 53A30, 91D20 Quasiconformal mapping, geographical map, sphere projection, Tissot indicatrix

was a French mathematician and cartographer. He introduced a tool which became known among geographers under the name Tissot indicatrix, and which was widely used during the first half of the twentieth century in cartography. This is a graphical representation of a field of ellipses, indicating at each point of a geographical map the distorsion of this map, both in direction and in magnitude. Each ellipse represented at a given point is the image of an infinitesimal circle in the domain of the map (generally speaking, a sphere representing the surface of the earth) by the projection that realizes the geographical map.

Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane, and he also developed a theory for the distorsion of mappings between general surfaces. His ideas are close to those that are at the origin of the work on quasiconformal mappings that was developed several decades after him by Grötzsch, Lavrentieff, Ahlfors and Teichmüller.

Grötzsch, in his papers, mentions the work of Tissot, and in some of the drawings he made for his articles, the Tissot indicatrix is represented. Teichmüller mentions the name Tissot in a historical section in one of his fundamental papers in which he points out that quasiconformal mappings were initially used by geographers.

The name Tissot is missing from all the known historical reports on quasiconformal mappings. In the present article, we report on this work of Tissot, showing that the theory of quasiconformal mappings has a practical origin.

Darboux, starts his 1908 ICM talk whose title is Les origines, les méthodes et les problèmes de la géométrie infinitésimale (The origins, methods and problems of infinitesimal geometry) with the words: "Like many other branches of human knowledge, infinitesimal geometry was born in the study of practical problems," and he goes on explaining how problems that arise in the drawing of geographical maps, that is, the representation of regions of the surface of the Earth on a Euclidean piece of paper, led to the most important developments in geometry made by Lagrange, Euler, Gauss and others.

The theory of quasiconformal mappings has its origin in the problems of drawing geographical maps. Teichmüller, in the last part of his paper Extremale quasikonforme Abbildungen und quadratische Differentiale (Extremal quasiconformal mappings and quadratic differentials), published in 1939 [35], which is the main paper in which he develops the theory that became known as Teichmüller theory, makes some comments on this origin, mentioning the work of the French mathematician and geographer Nicolas-Auguste Tissot (1824-1897). Grötzsch, in his paper Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche (On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions), published in 1930 [START_REF] Grötzsch | Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche[END_REF], mentions several times the name Tissot, referring to the Tissot indicatrix which he represents in the pictures he drew for his article. The directions of the major and minor and minor axes of this ellipse constitute are important element in some of his results. A geographical map is the image of a mappinghenceforth called a projection-from the surface of the Earth, considered as a sphere or spheroid, onto the Euclidean plane. The Tissot indicatrix is a device introduced by Tissot, who called it the indicating ellipse (ellipse indicatrice, which was used by geographers until the middle of the twentieth century. It is a field of ellipses drawn on the geographical map, each ellipse representing the image by the projection-assumed to be differentiable-of an infinitesimal circle 1 at the corresponding point on the sphere (or spheroid) representing the surface of the Earth. Examples of Tissot indicatrices are given in Figure 1.

In Figure 2, we have reproduced drawings from a paper of Grötzsch in which he represents the Tissot indicatrix of the maps he uses.

Although the work of Tissot is closely related to the theory of quasiconformal mappings, his name is never mentioned in the historical surveys of this subject, and the references by Grötzsch and by Teichmüller to his work remained unnoticed. In this note, I will give a few indications on this work.

Before surveying the work of Tissot in §3, I will give, in §2, a short biographical note on him.

Biographical note on Tissot

Nicolas-Auguste Tissot was born in 1824, in Nancy, which was to become, 26 years later, the birthplace of Henri Poincaré (whom we shall mention soon). Tissot entered the École Polytechnique in 1841. He started by occupying a career in the Army 2 and defended a doctoral thesis on November 1 The expression "infinitesimal circle" means here, as is usual in the theory of quasiconformal mappings, a circle on the tangent space at a point. In practice, it is a circle on the surface which has a "tiny radius." In the art of geographical map drawing, these circles, on the domain surfaces, are all supposed to have the same small size, so that the collection of relative sizes of the image ellipses becomes also a meaningful quantity. 2 The reader should note that the École Polytechnique was, and is still, is a military school. 17, 1851; cf. [START_REF] Tissot | Mouvement d'un point matériel pesant sur une sphère. Suivi de Sur la détermination des orbites des planètes et des comètes. Thèses présentées á la faculté des sciences de Paris[END_REF]. On the cover page of his thesis, he is described as "Ex-Capitaine du Génie." Tissot became later a professor at the famous Lycée Saint-Louis in Paris, and at the same time examiner at the École Polytechnique, in particular for the entrance exam. He eventually became an assistant professor (répétiteur ) in geodesy at the École Polytechnique.

After having published, in the period 1856-1858, several papers and Comptes Rendus notes on cartography, in which he analyzed the distortion of some known geographical maps (see [START_REF] Tissot | Sur la détermination des latitudes au moyen de la méthode de M. Babinet[END_REF], [START_REF] Tissot | Sur les altérations d'angles et de distances dans le développement modifié de Flamsteed[END_REF], [START_REF] Tissot | Sur le développement modifié de Flamsteed[END_REF]), Tissot started developing his own theory, on which he published three notes, in the years 1859-1860, [START_REF] Tissot | Sur les cartes géographiques[END_REF][START_REF] Tissot | Sur les cartes géographiques[END_REF]43], and then a series of others in the years 1865-1880 [START_REF] Tissot | Sur la construction des cartes géographiques[END_REF][START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques[END_REF][START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques[END_REF][START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques[END_REF][START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques[END_REF][START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques[END_REF][START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques[END_REF]. He then collected his results in the memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF], published in 1881, in which he gives detailed proofs. In a note on p. 2 of this memoir, Tissot declares that after he published his first Comptes Rendus notes on the subject, the statements that he gave there without proof were reproduced by A. Germain in his Traité des projections des cartes géographiques [START_REF] Germain | Traité des projections des cartes géographiques : Représentation plane de la sphère et du sphéroııde[END_REF] and by U. Dini in his memoir Sopra alcuni punti della teoria delle superfici [START_REF] Dini | Sopra alcuni punti della teoria delle superfici[END_REF]. He notes that Germain and Dini gave their own proofs of these statements, which are nevertheless more complicated than those he had in mind and which he gives in the memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF]. He also writes that Dini showed that the whole theory of curvature of surfaces may be deduced from the general theory that he had developed himself. In fact, Dini applied this theory to the representation of a surface on a sphere, using Gauss's methods. Tissot also says that his ideas were used in astronomy, by Hervé Faye, in his Cours d'astronomie de l' École Polytechnique [START_REF] Faye | Hervé Cours d'astronomie de l' École Polytechnique[END_REF]. The texts of the two Comptes rendus notes [43] and [START_REF] Tissot | Sur le développement modifié de Flamsteed[END_REF] of Tissot are reproduced in the Germain's treatise [START_REF] Germain | Traité des projections des cartes géographiques : Représentation plane de la sphère et du sphéroııde[END_REF].

Besides his work on geographical maps, Tissot wrote several papers on elementary geometry. We mention incidentally that several preeminent French mathematicians of the nineteenth and the beginning of the twentieth century published papers on this topics. We mention Serret, Catalan, Laguerre, Darboux, Hadamard and Lebesgue; see e.g. [START_REF] Tissot | Démonstration nouvelle du théorème de Legendre sur les triangles sphériques dont les côtés sont très-petits relativement au rayon de la sphère[END_REF][START_REF] Tissot | Sur les hélices. Nouvelles annales de mathématiques[END_REF][START_REF] Tissot | Formules relatives aux foyers des coniques[END_REF].

On the title page of Tissot's memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF] (1881), the expression Examinateur à l' École Polytechnique follows his name, as he was in charge of the entrance examination. In his Éloge historique de Henri Poincaré [8], Darboux relates the following episode about Tissot, examining Poincaré: 3Before asking his questions to Poincaré, Mr. Tissot suspended the exam during 45 minutes: we thought it was the time he needed to prepare a sophisticated question. Mr. Tissot came back with a question of the Second Book of Geometry. Poincaré drew a formless circle, he marked the lines and the points indicated by the examiner, then, after wandering long enough in front of the blackboard, with his eyes fixed on the ground, he concluded loudly: "It all comes down to proving the equality AB = CD. This is a consequence of the theory of mutual polars, applied to the two lines." Mr. Tissot interrupted him: "Very good, Sir, but I want a more elementary solution." Poincaré started wandering again, this time not in front of the blackboard, but in front of the table of the examiner, facing him, almost unconscious of his acts; then suddenly he developed a trigonometric solution. Mr. Tissot objected: "I would like you to stay in Elementary Geometry." Almost immediately after that, the examiner of Elementary Geometry was given satisfaction. He warmly congratulated the examinee and announced that he deserves the highest grade. 4 Poincaré kept a positive momory of Tissot's examinations. He expresses this in a letter to his mother sent on May 6, 1874, opposing them to the 10-minute examinations (known as "colles") that he had to take regularly at the École Polytechnique and which he said are pitiful. He writes: 5 "When I think about the exams of Tissot and others, I can not help but take pity of these 10 minutes little colles where one puts in danger his future with an expression which is more or less exact or a sentence which is more or less well crafted, and where a person is judged upon infinitesimal differences." 6

On the work of Tissot on geographical maps

Tissot studied at the École Polytechnique, an engineering school where the students had a high level of mathematical training and at a period where the applications of the techniques of differential geometry to all the domains of science were an integral part of the curriculum. His work is part of a well-established tradition where mathematical tools are applied to the craft of map drawing. This tradition passes through the works of preeminent mathematicians such as Ptolemy [START_REF] Ptolémée | Traité de géographie, translated from the Greek into French by l'Abbé de Halma[END_REF][START_REF] Berggren | Ptolemy's Geography: An Annotated Translation of the Theoretical Chapters[END_REF], Lambert [START_REF] Lambert | Beiträge zum Gebrauche der Mathematik und deren Anwendung[END_REF][START_REF] Lambert | Anmerkungen und Zusätze zur Entwerfung der Land-und Himmelscharten[END_REF], Euler [START_REF] Euler | De repraesentatione superficiei sphaericae super plano[END_REF][START_REF] Euler | De proiectione geographica superficiei sphaericae[END_REF][START_REF] Euler | De proiectione geographica Deslisliana in mappa generali imperii russici usitata[END_REF] Lagrange [START_REF] De Lagrange | Sur la construction des cartes géographiques[END_REF][START_REF] De Lagrange | Sur la construction des cartes géographiques[END_REF], Gauss [START_REF] Gauss | Allgemeine Auflösung der Aufgabe: die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden, daß die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird[END_REF], Chebyshev [START_REF] Chebyshev | Sur la construction des cartes géographiques[END_REF][START_REF] Chebyshev | Sur la construction des cartes géographiques. Discours prononcé le 8 février 1856 dans la séance solennelle de l'Université Impériale de Saint-Pétersbourg[END_REF], Beltrami [START_REF] Beltrami | Risoluzione del problema: Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette[END_REF], Liouville (see the appendices to [START_REF] Monge | Applications de l'analyse à la géométrie, 5e édition[END_REF]), Bonnet [START_REF] Bonnet | Thèse d'astronomie: Sur la théorie mathématique des cartes géographiques[END_REF] Darboux [START_REF] Darboux | Sur la construction des cartes géographiques[END_REF][START_REF] Darboux | Sur un problème posé par Lagrange[END_REF][START_REF] Darboux | Sur une méthode de Tissot relative à la construction des cartes géographiques[END_REF], and there are others. It was known since antiquity that there exist conformal (that is, anglepreserving) projections from the sphere to the Euclidean plane. 7 But it was noticed that these projections distort other quantities (length, area, etc.), and the question was to find projections that realize a compromise between these various distorsions. For instance, one question was to find the closestto-conformal projection among the maps that are area-preserving. Hence, the idea of "closest-to-conformal" projection came naturally. Among the mathematicians who worked on such problems, Tissot came closest to the notion of quasiconformality. 4 Avant d'interroger Poincaré, M. Tissot suspendit l'examen pendant trois quarts d'heure : le temps de préparer une question raffinée, pensions-nous. M. Tissot revint avec une question du deuxième Livre de Géométrie. Poincaré dessina un cercle informe, il marqua les lignes et les points indiqués par l'examinateur; puis, après s'être promené devant le tableau les yeux fixés à terre pendant assez longtemps, conclut à haute voix: Tout revient à démontrer l'égalité AB = CD. Elle est la conséquence de la théorie des polaires réciproques, appliquée aux deux droites. "Fort bien, Monsieur, interrompit M. Tissot; mais je voudrais une solution plus élémentaire." Poincaré se mit à repasser, non plus devant le tableau, mais devant la table de l'examinateur, face à lui, presque inconscient de ses actes, puis tout à coup développa une solution trigonométrique. "Je désire que vous ne sortiez pas de la Géométrie élémentaire," objecta M. Tissot, et presque aussitôt satisfaction fut donnée à l'examinateur d'élémentaires, qui félicita chaleureusement l'examiné et lui annonça qu'il avait mérité la note maxima.

5 [START_REF] Poincaré | La correspondance de jeunesse d'Henri Poincaré: Les années de formation[END_REF], letter No. 62. 6 Quand je pense aux exams de Tissot et autres, [. . . ] je ne puis m'empêcher de prendre en pitié ces petites colles de 10 minutes où on joue son avenir dans une expression plus ou moins exacte ou sur une phrase plus ou moins bien tournée et où on juge un individu sur des différences infinitésimales. 7 Ptolemy, in his Geography, works with the strereographic projection, see [START_REF] Berggren | Ptolemy's Geography: An Annotated Translation of the Theoretical Chapters[END_REF]. See also Ptolemy's work on the Planisphere [START_REF] Sidoli | The arabic version of Ptolemy's Planisphere or Flattening the surface of the sphere: text, translation, commentary. Dual English-Arabic text[END_REF].

Let us summarize a few of his results on this subject. An important observation made by Tissot right at the beginning of his memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF] (p. 1) is that finding the most appropriate mode of projection depends on the shape of the region-and not only its size, that is, on the properties of its boundary,. Finding maps of small "distorsion" (where, as we mentioned, this word has several possible meanings) was the aim of theoretical cartography. Tissot discovered that in order for the map to minimize an appropriately defined distortion, a certain function λ, defined by setting

dσ 2 = (1 + λ) 2 ds 2 ,
must be minimized in some appropriate sense, where ds and dσ are the line elements at the source and the target surfaces respectively. The minimality of λ may mean, for example, that the value of the gradient of its square must be the smallest possible.

In fact, Tissot studied mappings between surfaces that are more general than those between subsets of the sphere and of the Euclidean plane. He started by noting that for a given mapping between two surfaces, there is, at each point of the domain, a pair of orthogonal directions that are sent to a pair of orthogonal directions on the image surface. Unless the mapping is angle-preserving at the given point, these pairs of orthogonal directions are unique. The orthogonal directions at the various points on the two surfaces define a pair of orthogonal foliations preserved by the mapping. Tissot calls the tangents to these foliations principal tangents at the given point. They correspond to the directions where the ratio of lengths of the corresponding infinitesimal line elements attains its greatest and smallest values.

Using the foliations defined by the principal tangents, Tissot gave a method for finding the image of an infinitely small figure drawn in the tangent plane of the first surface. In particular, for a differentiable mapping, the images of infinitesimal circles are ellipses. In this case, he gave a practical way of finding the major and minor axes of these ellipses, and he provided formulae for them. This is the theory of the Tissot indicatrix.

From the differential geometric point of view, the Tissot indicatrix gives information on the metric tensor obtained by pushing forward the metric of the sphere (or the spheroid) by the projection mapping.

We recall that in modern quasiconformal theory, an important parameter of a map is the quasiconformal dilatation at a point, defined as the ratio of the major axis to the minor axis of the infinitesimal ellipse which is the image of an infinitesimal circle by the map (assumed to be differentiable at the give point, so that its derivative sends circles centered at the origin in the tangent plane to ellipses). The Tissot indicatrix gives much more information than this quasiconformal dilatation, since it keeps track of (1) the direction of the great and small axes of the infinitesimal ellipse, and (2) the size of this ellipse, compared to that of the infinitesimal circle of which it is the image.

Darboux got interested in the work of Tissot on geography, and in particular, in a projection described in Chapter 2 of his memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF]. He wrote a paper on Tissot's work [START_REF] Darboux | Sur une méthode de Tissot relative à la construction des cartes géographiques[END_REF] explaining more carefully some of his results. He writes: "[Tissot's] exposition appeared to me a little bit confused, and it seems to me that while we can stay in the same vein, we can follow the following method [. . . ]" 8 Tissot showed then how to construct mappings that have minimal distortion.

Tissot's work was considered as very important by cartographers. The American cartographer, in his book Flattening the earth: two thousand years of map projections [START_REF] Snyder | Flattening the Earth: Two Thousand Years of Map Projections[END_REF], published in 1997 and which is a reference in the subject, after presenting the existing books on cartography, writes: "Almost all of the detailed treatises presented one or two new projections, they basically discussed those existing previously, albeit with very thorough analysis. One scholar, however, proposed an analysis of distorsion that has had a major impact on the work of many twentieth-century writers on map projections. This was Tissot [. . . ]."

Modern cartographers are still interested in the theoretical work of Tissot, see [START_REF] Laskowski | The Traditional and Modern Look at Tissot's Indicatrix[END_REF].

We mentioned several preeminent mathematicians who before Tissot worked on the theory of geographical maps. From the more recent era, let me mention Milnor's paper titled A problem in cartography [START_REF] Milnor | A problem in cartography[END_REF], published in 1969. The reader interested in the theory of geographical maps developed by mathematicians is referred to the papers [START_REF] Papadopoulos | Euler and Chebyshev: From the sphere to the plane and backwards[END_REF], [START_REF] Papadopoulos | A link between cartography and quasiconformal theory[END_REF] and [START_REF] Papadopoulos | Quasiconformal mappings, from Ptolemy's geography to the work of Teichmüller[END_REF] which also contain more on the work of Tissot. Index geographical map, 2, 4, 6, 7 Grötzsch, Herbert (1902-1993)

Figure 1 .

 1 Figure 1. Four geographical maps on which the field of ellipses (Tissot indicatrix) are drawn. The maps are extracted from the book Album of map projections [33]. These are called, from left to right, top to bottom, the stereographic [33, p. 180], Lagrange [33, p. 180], central cylindrical [33, p. 30] and equidistant conical projections [33, p. 92]. The first two projections are conformal and not area-preserving. The last two are neither conformal nor area-preserving.

Figure 2 .

 2 Figure 2. Two figures from Grötzsch's paper Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche [19]. Grötzsch drew the Tissot indicatrices of his quasiconformal mappings. (In each drawing, the major and minor axes of the ellipses are shown.)

Poincaré entered the École Polytechnique in 1873. In the French system of oral examinations, which is still in use, a student is given a question or a set of questions which he is asked to prepare while another student (who had already been given some time to prepare his questions) is explaining his solutions at the blackboard, in the same room. Thus, it is not unusual that at such an examination, some students listen to the examinations of others.


Charalompos Sakondis
email: xsakonid@eled.duth.gr

Reidar Mosvold
email: reidar.mosvold@uis.no

Ove Gunnar Drageset
email: ove.drageset@uit.no

Nic Siún

Rukiye Didem Mhuirí

Taylan
email: tayland@mef.edu.tr

Tay- Lan Didem

Charalampos Sakonidis

Siún Nic Mhuirí
email: siun.nicmhuiri@dcu.ie

Rukiye Didem Taylan

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction to the papers of TWG19: Mathematics Teaching and Teacher Practice(s) Introduction

Thematic Working Group 19 (TWG19) emerged as a result of the splitting of a TWG called "From a study of teaching practices to issues in teacher education" into three sub-groups after CERME8. One of the new groups targeted teacher education and professional development (TWG18), another focused on teacher knowledge, beliefs and identity (TWG20), and TWG19 targeted "Mathematics teacher and classroom practices". Since its conception, TWG19 has been in a process of developing its identity. In discussions, core concepts have been taken for granted, without explicit or shared definitions. This became apparent at CERME10, where participants called for a development of common groundsor at least for efforts to increase awareness about differences in conceptualizations, theories and methods, and of the implications of these differences. After CERME10, two changes have been made to move forward TWG19's work: 1) the name of the working group has been adjusted, and 2) an initiative of offering sets of shared data for analysis and use in the working group has been initiated.

This introductory paper discusses the ongoing development of TWG19's identity and where the continued development appears to be heading. Due to a large number of submitted papers, the group split into two (TWG19a and TWG19b) for CERME11, but we tried to maintain a sense of community by having three common sessions, and by discussing the same overarching questions in split sessions. In the final session, team leaders shared reports to the whole group and facilitated a concerted discussion around these questions. This joint introduction to TWG19a and TWG19b testifies to these efforts. There are three main sections in this introduction. The first section discusses emerging issues and critical considerations as reflected in the section name. The second section reports from the joint activity around the initiative of analyzing and using shared data, and the third section discusses patterns and trends in the individual papers that were presented. The introductory paper concludes with a discussion of the collective efforts made to deal with the emerging issues and some reflections on the way ahead.

Emerging issues to pursue: some critical considerations

After CERME10, the name of TWG19 changed from "Mathematics teacher and classroom practices" to "Mathematics teaching and teacher practice(s)". The change of wording is intentional and deserves elaboration. First, the change from teacher to teaching signals that the group targets teaching rather than teachers and their characteristics; teacher knowledge, beliefs and identity are the focus of TWG20. This shift reflects a development that [START_REF] Skott | Classroom practice and teachers' knowledge, beliefs, and identity[END_REF] described and reflected upon in their recollection of research on classroom practice, knowledge, beliefs and identity over ten biannual CERME conferences and corresponds with a general shift of focus in research on teaching. Although the shift from teachers to teaching has emerged over several decades, conceptualizations of teaching appear to be underdeveloped in our field. While discussing the problem of theorizing teaching, Jaworski (2006, p. 188) suggests that "the big theories do not seem to offer clear insights to teaching and ways in which teaching addresses the promotion of mathematics learning". Furthermore, Jaworski suggests that one of the problems lies in how learning and teaching are connected with practice, and this relates to a second change in the name of TWG19.

The shift from classroom practices to teacher practice(s) signals that the primary focus of the research in TWG19 is on practices of teachers and teaching rather than on any practices in classrooms. Use of the term practice has changed over the years. [START_REF] Jaworski | Theory and practice in mathematics teaching development: critical inquiry as a mode of learning in teaching[END_REF] discussed "teaching as learning in practice", whereas [START_REF] Schön | The Reflective Practitioner: How Professionals Think in Action[END_REF] consider teaching to be a reflective practice. The former suggests conceiving teaching as a social practice in which teachers are practitioners, whereas the latter offers a view of how teachers "act in the classroom as informed, concerned professionals and about how they continue to learn" (Lerman, 1998, p. 33). The discussions by these authors imply some of the different meanings that might be attached to the term practice. [START_REF] Lampert | Learning teaching in, from, and for practice: What do we mean[END_REF] distinguishes between four different meanings of the word practice, and three of those are relevant for TWG19. First, practice might refer to what teachers do in contrast to theory. Second, the word is used to describe a collection of practices, some of which might be more core or high-leverage than others. A third use is to consider teaching as a professional practice, like the practice of law or medicine. The word practice in the new name of TWG19 might encompass all three meanings.

The first joint session of TWG19a and TWG19b at CERME11 was devoted to discussing some of these foundational issues, and attempts were made to contribute to the discussion of the underdeveloped conceptualizations and theorization of mathematics teaching. To introduce the discussion, two of the presented papers drew on different theoretical foundations. In the first presentation, Watson drew on developments in cognitive psychology, neuroscience and ontology to provide a theoretical account of teacher decision making and how this influences practice. He argued that the momentary decisions influence the character of the lesson and proceeds to focusing on algorithmic reasoning, a conscious process making use of heuristics and pre-established routines and processes to reduce the demand on working memory. In mathematics teaching, in-the-moment decisions have to be immediate and thus well-practiced. The algorithmic mind quickly accesses routines and procedures that are learnt through participation and provide well-rehearsed and culturally-embedded approaches to respond to situations in the classroom. Watson concluded that it is important to further study teachers' decision making, to better understand the affective dimensions, the culturally embedded routines and strategies acquired through participation in the professional community. In the second paper, Sakonidis argued that mathematics teaching tends to be seen today as a professional practice shaped by the expectations and norms of the learning settings where is exercised. Sociocultural theories provide useful lenses for the relevant research and three of them appear to readily lend themselves to this direction: the theory of learning participating in a community of practice, Cultural-historical activity theory, and [START_REF] Skott | Understanding the role of the teacher in emerging classroom practices: searching for patterns of participation[END_REF] patterns of participation theory. Sakonidis concluded by highlighting the need for dynamic perspectives to disentangle and understand the ever-evolving outcome of individual and communal acts of meaning-making (by both teachers and students) characterizing the practice of mathematics teaching.

Following the presentation of these two papers, Hoover and Mosvold presented their reflections on the developments in the field of research on mathematics teaching. They started by reflecting on the different meanings of teaching, and how the field has proposed dichotomies like pedagogic versus didactic, art versus science, ambitious versus conventional, and profession versus occupation. Based on these initial reflections, they proposed a distinction between teaching as activity and teaching as work. The first interprets teaching as activities done by teachers, whereas the second considers teaching to be work to be done. Each of these understandings has different implications for studying teaching. Studies of teaching as activity tend to seek empirical descriptions of what teachers are doing, and they study methods of expert teachers in order to improve teaching. On the other hand, studies of teaching as work tend to investigate the demands of instruction by applying conceptual analysis, and they conceptualize professional practice for teacher education as a means to improve teaching. This distinction was taken up by participants in the group and stimulated productive discussions.

In search of answers to the emerging issues in a joint research activity

As another effort to move the group forward, TWG19 decided to prepare sets of shared data for CERME11. These shared data sets were made available for analysis and use in participants' papers, and they were also available for use in presentations and as a common reference point for discussions in the group. Four sets of data were prepared and uploaded to a secure server that all participants in TWG19 were given access to. Three of the data sets contained videos of mathematics teaching, whereas one data set contained transcripts only. Each data set was accompanied by a document that laid out policies for sharing and reuse, and a document with information about the person or institution that shared the data as well as with permissions and limitations for usage.

Videos are extensively used in research on teaching. In a presentation to the group about sharing data, Mosvold, Hoover and Suzuka suggested that study of a complex phenomenon like teaching requires good data, and that video data provides access to nuances in communication and interaction. Videos also provide more detail about context. While adding value, video data also introduce some challengesincluding practical as well as ethical concerns. Sharing and reusing video data provide benefits like better utilization of data, access to high-quality data, saved costs, as well as increased quality and scope of analysis. However, successful sharing and reuse of video data demand careful considerations around issues of usability and reuse of data. Many find the challenges entailed in sharing and reusing data so severe that they are reluctant to get involved.

Engaging in this initiative has provided us with useful experiences that we hope might help us as a field move forward towards overcoming the anticipated difficulties.

Having multiple eyes look at the same data opens up for richer and more nuanced interpretations.

For instance, five papers analyzed a Norwegian video where a teacher facilitates a group of students' presentations of their solutions to a problem of figuring out when the king of Norway was born, given that he celebrated his 80 th birthday on that particular day. Some participants focused on how the teacher was moderating the dialogue, how he elicited students' mathematical thinking (Drageset), and how he positioned students as mathematical authorities whose thinking were put on public display and collectively analyzed (Bass & Mosvold). Others emphasized how the teacher's use of revoicing might interrupt and even prevent further development of the discourse (Kleve, Solem, & Aanestad), and Nic Mhuiri followed up by observing that student interactions were always funneled through the teacher and never developed into discussions in whole class. Whereas most papers applied particular frameworks or theories in their analysis of this video, Hoover and Goffney used conceptual analysis to identify aspects of mathematics teaching that might disrupt patterns of inequity.

Another frequently analyzed video displays Deborah Ball teaching fractions on the number line.

From his analysis of turns in the conversation, Drageset noticed how the teacher often requested students to ask questions instead of entering a traditional pattern of initiation-response-evaluation. He also discussed how the teacher moderated the discourse while simultaneously guiding students' participation and attending to norms. Mosvold and Bjuland also mentioned attending to norms of participation in their use of the data set to unpack the role of positioning in the work of teaching mathematics. A couple of papers also observed how assigning students with agency and authority becomes visible in this instance of mathematics teaching (Bass & Mosvold; Nic Mhuiri). Again, the nuances provided by the sum of the analyses exceeds the contributions of each individual paper.

Engaging in analysis of shared data prompts some new questions: Why do we see different things when analyzing the same data? Can we reach agreement of some kind in our analyses? Is agreement even a goal? After all, our efforts to analyze and discuss these shared data sets did not lead to a shared understanding of mathematics teaching, but they stimulated productive discussions about different conceptualizations of teaching.

In search of answers to the emerging issues in individual participants' research

The papers reporting on studies of mathematics teaching and teacher practices that did not use the shared data sets can be placed in three groups: 1) teachers' practices, actions and resources before or during daily instruction; 2) teachers' implementation of teaching practices and actions developed in the context or in the aftermath of a particular purposeful professional development (PD) activity; 3) developing tools or practices of monitoring classroom instructional activity. Emerging issues from these groups of papers are discussed below.

Papers reporting on issues related to daily teaching practices, actions and resources apply established theoretical or conceptual frameworks to examine their functionality and effectiveness.

For instance, Pericleous used cultural-historical activity theory to investigate how the activity of proving was constituted in a primary classroom. She analyzed tensions that arose for the teacher and identified points of contradiction between the classroom micro-system and elements of the macrosystem, such as curriculum context. In a series of focus group interviews, Grundén used critical discourse analysis to interrogate teachers' planning practice(s) in mathematics. The results described how teachers construct their own discourse of mathematics teaching in resistance to the official discourse of the National Agency of Education. Grundén contended that critical discourse analysis is useful for attending to the complexity of mathematics teaching. Rudsberg, Sundhäll and Nilsson inquired into teachers' monitoring of students' meaning making. To this purpose, they adopted a framework of epistemological move analysis and a pragmatic perspective on learning to analyze lesson transcripts. The results suggested that epistemological move analysis might enhance our understanding of the relationship between students' meaning-making and the related actions of teachers.

Stouraitis and Potari studied a prospective secondary school teacher's first field experiences. Using activity theory, the authors focused on emerging contradictions considering prospective teachers' planning and enactment of a lesson plan, as well as what led to such contradictions and how contradictions could potentially influence prospective teacher learning. Calor et al. examined teachers' use of a model based on small-group work, drawing on socio-cultural notions such as collaborative learning and scaffolding as well as on cognitive ideas like Janvier's approach to representation systems. The analysis of a pre-and post-test on mathematical level raising shows that more mathematical level raising occurred in the small-group than in the control condition. Olsson and Teledahl described an investigation into how principles of feedback might be used by a mathematics teacher to encourage students' creative reasoning. Their analysis illustrates the challenges that this presents for teachers and highlights associated issues such as the existing classroom norms for interaction, as well as teachers' beliefs about the object of teaching.

Andrews explored what happens mathematically when teachers and learners engage in betweendesk-teaching across different mathematics topics. From a multiple case study, he focused on the case of one teacher who used between-desk-teaching practice with a strategic purpose that could potentially support student understanding. Taylan and Esmer used the instructional actions framework to examine a novice teacher's instructional actions in response to unexpected classroom moments. Analyzing lesson planning meetings and observations as well as teacher interviews, they found out that responses to unexpected moments include both supporting and extending actions. Kayali and Biza studied a teacher's use of the available resources in relation to their teaching aim.

Based on the 'Knowledge Quartet' [START_REF] Rowland | Foundation knowledge for teaching: Contrasting elementary and secondary mathematics[END_REF], their analysis highlights different aspects of a teacher's classroom work and leads to suggesting replacing code 'instructional materials' to 'resources' and expanding the Knowledge Quartet by adding a code named 'connection among resources'.

The submissions related to teachers' implementation of instructional practices and tools developed in professional development contexts aim at providing rich mathematical classroom activity for all students. In particular, Büscher focused on teachers' designed tasks during a professional development course to reconstruct their categories of differentiation for percentage problems. The results show that teachers tended to differentiate in ways that exclude low-achieving students from conceptually rich learning opportunities due to their partitioning of their envisioned ideal-typical solution paths. Psycharis, Potari, Triantafillou and Zachariades investigated how secondary school teachers attempted to balance mathematical challenge and differentiation in whole class settings in the context of a professional development program which aimed to support teachers in engaging all learners while teaching challenging tasks.

Klothou, Sakonidis and Arsenidou used a multiple case study approach to investigate the use of learning trajectories in teaching fractions by three primary teachers participating in a PD project.

The analysis shows that all teachers tended to view learning trajectories primarily as a way of organizing mathematical content more than as an elaboration of the possible development of student thinking. The study by Medová, Bulková, and Čeretková was conducted in the context of a professional development program focusing on inquiry-based learning in upper secondary mathematics. Results indicate that students' independent learning was observable more often in the inquiry-based lessons than in regular lessons, whereas teachers played a more dominant role in the regular lessons. Maugesten explored Norwegian lower primary teachers' views about good mathematics teaching by using a focus group interview with teachers after a two-year school-based professional development. The results reveal that teachers focused on communication in classroom and use of representations, student thinking in and about the subject of mathematics, and resources, textbooks, and tasks that are related to everyday mathematics.

Finally, the papers related to developing tools or practices of monitoring classroom instructional activity concentrated on cognitive as well as social determinants of this activity. Thus, Nowińska reported on the development of a rating system for analyzing metacognitive activities: Metacognitive-discourse instructional quality. This is illustrated with examples of how one can identify different parts of metacognition related to planning, monitoring, and reflection. She emphasized that such a rating system needs to identify patterns that are stable across lessons and practice. Arnesen and Grimeland also presented an investigation of teachers' planning. They asked teachers to identify the mathematical ideas they would focus on and use Bloom's Taxonomy to classify the learning goals teachers had identified for lessons. Findings indicate that higher-order cognitive process categories were poorly represented in the learning goals and teachers' descriptions of mathematical ideas were vague or absent.

The study by Mellroth and Boesen provides a proposal for teachers to help them notice mathematically highly able students by illuminating aspects of students' problem-solving processes. The paper of Ableitinger, Anger and Dorner is unique in its efforts to attend to student voice. Student feedback was collected in post-lesson interviews and quantitative and qualitative methods were used to compare students' and researchers' choice of significant events in mathematics lessons. The analysis indicates differences between students' and researchers' choices of significant events. Even when both groups identified the same event as significant, different reasons were given for this choice.

Summarizing, all papers presented above highlight challenges of accessing interesting data in consistent ways within or across instructional settings. The adopted theoretical frameworks view teachers as professionals coordinating the instructional activity aiming at rich learning environments and maximum participation. These frameworks offer analytical tools and sometimes providecombined or coordinatednew theoretical lenses allowing useful insights into various aspects of mathematics teaching and teacher practice(s). The accompanying methodologies are mostly qualitative. Although these methodologies vary, they tend to concentrate on examining instructional routines, actions, tools and practice(s) by analyzing teachers' designs, classroom implementations or classroom excerpts and so on.

Towards a collective effort to deal with the emerging issues

Discussions in TWG19 at CERME11 often revolved around the core questions of what we mean by teaching and how we study it, and the distinction between teaching as activity and teaching as work was discussed frequently. The majority of participants in TWG19 tend to consider teaching as what teachers do, but there is considerable variation in this view of teaching as activity and how it is studied. Some consider teaching as what teachers do, but they also include teachers' thinking and emotions (e.g., Watson). Others focus more squarely on teacher actions and study patterns of such actions (e.g., Sakonidis). Yet others emphasize that teaching is a communicational activity (e.g., Nachlieli et al.). Among the papers viewing teaching as activity, many were concerned with the connection between teaching and learning (e.g., Ayan; Grundén, Serrazina et al.). Some consider the teacher as an agent who shapes new classroom practices (Nic Mhuiri). Others consider the teacher as a facilitator or an orchestrator (Baldry), who provides students with opportunities to explore (e.g., Pericleous Having the distinction between teaching as activity or work out there seemed to fuel the discussions. People started engaging with questions about the purpose of distinguishing between teaching as activity and teaching as work, and how the conceptualization of one might be independent of the other. The discussion of papers often targeted foundational questions about implications of our views on teaching, of applying different theoretical frameworks to analyze teaching, and about how different methods and types of data might or might not inform us about teaching and the demands of teaching. The effort of introducing and using shared data was another initiative that seemed to stimulate productive discussions in the group. Some reflected on how seeing different things in the same data must prompt careful reflection about use of different theoretical frameworks, whereas others suggested that seeing different things might be fruitful provide a broader and more complete image of teaching.

Conclusion

TWG19 is still developing its identity, but the group is taking some important steps forward. The name change stimulated some important foundational discussions about conceptualizations of teaching and methods of studying teaching. The distinction between teaching as activity and teaching as work was useful, but further conceptual work is needed. The participants' contributions, despite the polyphony identified in their attempts to conceptualize mathematics teaching as well as in the theoretical lenses and the methodologies employed to study its classroom manifestations, somehow highlight the issues to consider and the questions to pursue in the way ahead.

The initiative of sharing data was also interesting, and several participants were interested in contributing to and using such data sets in upcoming conferences. Sharing and reusing data provide opportunities for richer analysis, but it also appears to be a productive space for developing shared understandings and possibly also better consensus about the object of study. However, further efforts are needed to investigate what kinds of (video) data are most suitable for joint analysis of mathematics teaching, how to organize an even more productive space for such joint analysis, and how such analysis might be scaled up. It is also imperative to investigate ways of organizing analysis of shared data in order for it to be more than an interesting exercise in data analysis.

  ). Yet others consider the teacher as a transmitter of knowledge(Ableitinger et al.). Teachers' use of feedback is supposed to influence student learning, and Olsson and Teledahl investigated development of productive feedback practices.Other papers lean towards a conceptualization of teaching as work to be done. A couple of papers are explicitly directed towards such a conceptualization of mathematics teaching (e.g., Hoover & Goffney; Mosvold & Bjuland), whereas others point in this direction implicitly. For instance, Nachlieli et al. studied tasks that teachers are faced with, and Santos et al. investigated demands of teaching mathematics. It is also worth noticing that some papers appear to study what teachers do, but they still make claims about the work to be done (e.g., Baldry).