Katalin Gosztonyi
email: kgosztonyi@caesar.elte.hu

Conceiving teaching trajectories in the form of series of problems: a step for the theoretical reconstruction of the Hungarian Guided Discovery approach

Keywords: Series of problems, Guided Discovery approach, Reverse engineering, Teaching trajectories, Hungarian mathematics education

published or not. The documents may come

Conceiving teaching trajectories in the form of series of problems: a step for the theoretical reconstruction of the Hungarian Guided Discovery approach

Introduction

In Hungary, there exists a tradition of mathematics education through problem solving and mathematical inquiry called "felfedeztető matematikaoktatás" that I translate as "Guided Discovery" (GD) approach. This approach was originally developed and practiced for the education of gifted students, but the reform movement of T. Varga, realized in the 1960s and '70s, transformed it into a basis of general mathematics education and a guiding principle of the official curriculum for primary and lower secondary school [START_REF] Halmos | Change in mathematics education since the late 1950's-ideas and realisation[END_REF]. The approach is still recognized by Hungarian specialists of mathematics education as relevant and coherent with modern educational trends. At the same time, it is practiced in ordinary classes only by a narrow circle of teachers, and its dissemination poses problems. Furthermore, the approach exists only "in act;" its theoretical description has never been developed. Although Varga actively participated in the international discourse on mathematics education in the '60s and '70s, and developed his conception at the same time as Brousseau or Freudenthal 1 , his publications were mostly focused on experimentation and curriculum development, and less on the theoretical description of the underlying conception. This article treats a step of a longer research project aiming to reconstitute the underlying theory of Varga's approach. I consider this reconstruction necessary for several reasons. On the one hand, it is an inevitable first step for its large-scale implementation, for the development of more adapted resources and teacher education programs. On the other hand, it is necessary to introduce Varga's work to the current international discussion on Inquiry Based Mathematics Education: to enrich both the international discourse by the specificities of Varga's approach and the Hungarian mathematics education by the dialogue with other approaches.

In this paper, I focus on one important aspect of the Hungarian approach: the long and complex teaching trajectories shaped by series of problems. The concept of Series of Problems appears to be crucial in this approach, both for curriculum and resource-developers and for teachers (Gosztonyi, 2015a). However, a theoretical understanding of this phenomenon is missing. Originally, the term "series of problems" was introduced in the frame of an interdisciplinary history of science project as a more or less undetermined methodological tool to analyze the structure of historical texts [START_REF] Bernard | Les séries de problèmes, un genre au carrefour des cultures[END_REF]. In this specific Hungarian context, by "series of problems" (SoP) I mean a list of problems with a conscious ordering which is relevant for an (or several) educational purpose(s). In what follows I will discuss the relevance of this notion for the Hungarian context, the difficulties to develop theoretical frames and methodological tools for the analysis of SoP, and the research process of this development.

In the first part of the article, I briefly summarize how a combined, historical-epistemologicaldidactical analysis of Varga's reform contributed to reveal the main principles of the GD approach. In the second part, I present an ongoing analysis of nowadays teachers' work, developed in the frame of a larger research project focused on Varga's legacy (MTA-ELTE project)2 . I suggest to describe the methodology of this analysis as reverse engineering3 : a progressive elucidation of the hidden principles of this "theory in act", starting from teacher's actual designs.

Back to history

The first step for the reconstruction of the GD approach was to go back to its origins, in the frame of a comparative study between Varga's reform and the contemporary French "Mathématiques modernes" reform, carrying out their combined, historical, epistemological and didactical analysis, based on written sources [START_REF] Gosztonyi | Traditions et réformes de l'enseignement des mathématiques à l'époque des « mathématiques modernes[END_REF]. The analysis of the historical context of the reforms pointed out, among other things, the crucial role of mathematicians in defining the epistemological foundations of mathematics and its teaching: a "bourbakist" epistemology in the French case, and a "heuristic" one in the Hungarian case. The didactical analysis of different sources (curricula, textbooks, teachers' handbooks) showed the epistemology be one of the main driving forces for realizing the reforms.

One of the theoretical frameworks used for the didactical analysis was Brousseau's Theory of Didactical Situations (1998). This theory proved to be powerful to reveal important characteristics of each reformbut also insufficient to explain some specificities of Varga's one. One of the most important aspects is the long teaching trajectories shaped by SoP.

Brousseau developed his theory from the 1970s, in the French context of the debates following the "Mathématiques modernes." Thus, the emergence of the theory goes back to the historical context I studied. The comparison of an early experimentation of Brousseau with Varga's works contributed to understand their conceptual differences.

Characteristics of the Hungarian Guided Discovery approach and the importance of SoP

The analyses described above helped to identify some main characteristics of the GD approach. In the background of Varga's reform project, a quite coherent "heuristic" epistemology of mathematics and of mathematics education can be outlined in the 20 th century's Hungarian mathematical culture [START_REF] Gosztonyi | Mathematical Culture and Mathematics Education in Hungary in the XXst century[END_REF], represented by mathematicians living in Hungary as well as by Pólya and Lakatos. These thinkers considered mathematics to be a continuously progressing and changing human creation, and as a social, dialogic activity. The development of mathematics appeared as a collective process based on a diversity of experience, driven by problems, attempts of solutions, and new questions emerging from these attempts. They posited that mathematical knowledge only could be attained through the experience of mathematical creation, and thus students should be guided through similar discovery processes.

The idea of organizing problems into ordered series appears often in the writings of these authors. For example, R. Péter's book popularizing mathematics, Playing with infinity [START_REF] Peter | Playing with Infinity[END_REF] is written as an interesting inquiry story where one question invites the other. Pólya's work, especially later books like Induction and analogy (1954), also present inquiry processes shaped by series of problems, and analyses their structure. This epistemology deeply influenced Varga's "Complex Mathematics Education Reform" conception [START_REF] Gosztonyi | Traditions et réformes de l'enseignement des mathématiques à l'époque des « mathématiques modernes[END_REF]. The coherence of his curriculum is assured by a parallel treatment of different mathematical themes. Elements of mathematical knowledge are introduced through long, progressive processes (taking often several years), based on numerous small, concrete problem situations. Problems crossing different themes contribute to the variety of experiences. In the treatment of these problem situations, collective discussions led by the teacher play a crucial role. The use of various manipulative tools and representations, playfulness, students' autonomy and creativity are also in the focus of the reform project.

In the teacher's handbooks related to Varga's reform, researchers can find numerous commentaries on the importance of organizing problems into series. Primary school teachers' handbooks describe a number of activities with ideas for their repetitions, variations, and ordering. However, the principles of the organization are dispersed in the books and rarely generalized. The handbooks insist on the importance of teachers' autonomous work on conceiving of long teaching processes on their own. In middle-school textbooks, new knowledge is often introduced by fictive dialogues of students around short series of problems.

Analysis of published examples

Although SoP play a central role in Varga's conception, they are seldom theorized. Problem collections used to be presented without commentaries about tasks' connections and their possible roles in a teaching process. Furthermore, mostly fragments of SoP are published; long-term series are rarely presented. Some specific historical documents, however, present more detailed examples and furnish enough commentary to understand the logic and the purpose of their construction.

One example is the above mentioned book of R. Péter, describing in one of the chapters a classroom situation driven by a dialectic of problems, attempts of solutions and new problems motivated by these attempts. The text is written in a literary style; thus, its structuration by problems is not obvious at the first glance, but the rich meta-discourse makes possible to identify the problems and to understand the rationale behind the structuring of the text (Gosztonyi, 2015a).

In the documentation of Varga's reform, one of the most transparent examples is the introduction of combinatorics in primary school. Here, a double structuration of the series can be identified. On one hand, a sub-series is related to a given material, varying several variables and progressively introducing restrictions (e.g., building colored towers, building colored towers with a given height, finding all towers with a given height, varying the number of colors and levels, using a color only once, using the same color several times but not for neighbor levels, and so on). On the other hand, same or similar tasks are repeated with different materials (e.g., building towers with Cuisenaire rods, coloring flags or drawing houses, creating words with given letters, writing melodies with musical notes, and so on). The ordering is not strict here, but a progression in the abstraction should be respected by teachers. The observation of the analogies between these tasks serves as a basis for generalization and abstraction. The comparison of this example with Brousseau's above mentioned teaching progression revealed important differences between the two authors' conceptions, and contributed to understanding the role of SoP in the Hungarian case. For Brousseau, the progression is based on one "fundamental problem situation," and a decontextualization (through phases of institutionalization) is necessary for the passage to other situations. For Varga, the progression is based on a diversity of problems and their convenient ordering to favor progressive generalization through the recognition of links between the problems. [START_REF] Gosztonyi | Understanding didactical conceptions through their history: a comparison of Brousseau's and Varga's experimentations[END_REF].

In summary, the analysis of written documents representing the heritage of the Hungarian GD approach, confirms the importance of SoP in the planning of teaching by GD, identifies some key principles concerning the role and the structuration of SoP, and highlights tools to analyze and represent SoP (e.g., by graph representations or the structuration by main and sub-series).

Analysis of teachers' work designing Series of Problems

Subsequent research steps involve analyzing the GD approach of expert teachers, drawing on the key principles and analytic tools described above. Several of these teachers reacted vividly when I proposed the term "Series of Problems" to them. It resonated with the kind of planning they considered crucial. However, they rarely make these practices explicit.

The documents created by these expert teachers typically contained a list of problems, with or without solutions. But the documents are only a skeleton of their planned teaching process. Essential elements remain implicit: the main teaching goals, the organization of classroom work, the planned discussions, and choices and adaptations they consider to make 'on the spot', depending on the students' reactions (e.g., change of order, omission of some problems or introduction of new ones). Furthermore, these projects are conceived as long-term developments, even for several years, and only parts of them appear in written documents. Expert teachers confirm having a complex network of problems in mind that they can mobilize in order to create problem series and to adapt them to their aims and to the students' needs or interest-during preparation or in the classroom.

Gaining access to these complex implicit networks is necessary but quite challenging. Its reconstruction is an iterative process, combining two approaches: (1) an a priori analysis on the principles revealed by the analysis of historical documents, and (2) a study of the expert teachers' documentational processes (in the sense of Gueudet & Trouche, 2010).

For the MTA-ELTE project, a mixed group of researchers, teacher trainers and expert teachers of the GD approach was created in order to fulfill a double aim: analyzing the expert teachers' practices (with a special focus on SoP), and fostering dissemination of the approach by developing resources and teacher education programs. However, complex negotiations were necessary to meet the interest and expectations of each participant. While I felt necessary to focus first on the analysis of teachers' practices, this was not motivating and relevant enough for the participating teachers. Thus, we decided to develop a collection of commented examples of SoP. The purpose of this collection under preparation was not to offer ready-made resources for teaching, but generic examples, which introduce the reader into the practice of using and preparing SoP. In addition, preparing commented examples also served the project's research aim: for teachers to explicate their teaching purposes and principles of structuration of SoP. I call this procedure reverse engineering. Starting from teachers' existing designs of SoP, researchers reconstruct tacit principles driving teachers' choices during the design of these trajectories.

Network of problems

+ A PRIORI ANALYSIS OF SERIES OF PROBLEMS ANALYSIS OF TEACHERS' DOCUMENTATIONAL PROCESS

Figure 1: Analysis of Series of Problems (the visible elements are marked with bold, the hidden ones with broken outline)

In the first step, teachers were asked to provide examples from their own practice of what they consider SoP. The received examples were of quite different nature: SoP of short and long-term sequences; SoP to introduce new mathematical notions, SoP to explore heuristic strategies, and SoP to serve diagnostic purposes. Confronting the new and previously analyzed examples led to discussion on various questions: classification of purposes, different structuration principles, and forms of representation. These discussions evoked new examples. For example, most SoP consisted of short sequences at the beginning, because this was easier to document for teachers; but during the discussions, they all agreed about the importance of long-term planning, leading to the description of long trajectories built by SoP.

In the next step, teachers were asked to write commentaries to their SoP, addressing colleagues unfamiliar with the GD approach, focusing on the didactical aims of their trajectories and on explaining the SoP-structure. The group is currently comparing these first writings in order to define the main elements to be included in each commentary.

In the further phases of the project, the MTA-ELTE project team will conduct experiments; giving commented examples to non-expert teachers, and asking them to use the examples as inspiration to develop their own teaching trajectories, and offer them consultation with expert teachers of the group. This part of the project will give evidence of how well the created resources may mediate the GD-approach and provoke further exploring of the expert teachers' practices.

Eszter's case

Eszter is an expert teacher of our group, a high school teacher for 20 years. As a long-term example, she proposed an example taken from geometry4 , built around the concepts of incidence and distance in the Euclidean plane/space, equidistant sets (loci) of points, and especially a "core" problem: finding the equidistant points from three different lines in the plane. This series covers a very long period: It starts at the first year of high school, and some parts of it concern the last year's curriculum. Although some notable theorems of higher level mathematics appear in the series (in a problematized form), the aim of the SoP, according to Eszter, is not to learn these theorems, but to familiarize students with the above mentioned concepts, related mathematical techniques and problem solving skills through the exploration of interesting geometrical questions.

The first version consisted of a "core" series of four problems with increasing complexity about the loci of points (red dots in Figure 2). In this "core sub-series", the order of the problems is strict. This order is intended to make the latter, more difficult problems progressively available for students. Several other sub-series are conceived, based on this core, exploring different mathematical themes (blue dots), including space analogies, angles and similarities, and the boundaries of geometry and combinatorics. Eszter immediately proposed a graph to represent the structure of the SoP. Her arrows showed complex mathematical or didactical interdependences between the problems, but also illustrated the flexibility of the structure which is not visible in the simple numerated list of problems. Some problems or sub-series are sequential, while for other problems, the order can be freely chosen. The representation of this structure helps the reader to understand which problems could be deleted without impacting a longer trajectory, or where new problems could be inserted.

After a first discussion, we agreed that it would be useful to complete this SoP with a preparatory part which treated the lower secondary school level prerequisites. Thus, Eszter added a new part to her graph (green dots in Figure 2). As she explained, the problems of this part serve for diagnostic purposes in her high school work. They also fill possible gaps in students' knowledge. The problems presented illustrate different types of tasks which have to be explored in order to prepare students for the "core." More tasks can be inserted, depending on the students' needs. In order to make this "typology" more accessible, Eszter added a coding system to the graph (e.g. D stands for distance, L for loci of points, S for space analogies). Some subgroups of problems are related to some of these codes, but many problems have multiple codes. The network of problems is less hierarchic here than in other parts leaving more flexibility to the teacher. However, the discussion also revealed some further hidden interdependences of the ordering. This suggests that Eszter's didactical choices are even more complex than what was in her current version of the graph. Another iterative process of discussions and the preparation of representations is necessary to reveal implicit choices. 5The discussions with Eszter led to a progressive exploration of her design and its guiding choices, while the graph representation plays the role of a mediator. Her case also shows individual specificities beyond the common principles shared with the colleagues (e.g. Eszter accords even more importance to the flexibility in her SoP than most of her colleagues).

Conclusion

This paper presented a research process aiming to make explicit a "theory in act" existing in the Hungarian context of mathematics education and to develop theoretical bases and methodological tools to analyze the work of the teachers who follow this approach. The first phase of this process consisted in the analysis of the historical sources of the approach with complex research tools from history, epistemology and didactics, while the current second phase is focused on teachers work. The process is essentially iterative; the analysis of the epistemological background offered some first steps to reveal main principles of the hidden theory. The use of existing didactical theories in the analysis of Varga's reform documents helped not only to reveal its characteristics but also to point out the limits of the use of those theories, specificities of the GD approach which needs new theoretical development to be analyzed. These studies helped to choose the aspects to focus on while analyzing teachers' work-the structuration of teaching trajectories by SoP being one of them. The principles revealed by these analyses and the tools developed by case studies of representative written examples contribute to interpret the teachers' designs but also to provoke teachers to make their guiding principles explicit. Collective discussion and interpretation of the teachers' designs led to further elucidation of the fundamental principles of the approach and to the development of tools to investigate their design process. The term "reverse engineering" is used to describe the iterative process of exploring teachers' designs and describing their guiding design principles. Through reverse engineering, "theories in act" about SoP will be gained, so they may contribute to the discourse on Inquiry Based Mathematics Education [START_REF] Artigue | Conceptualizing inquiry-based education in mathematics[END_REF].

Varga and Freudenthal met quite late when both of their conceptions were already well developed, but they mutually recognized each other's work and perceived conceptual proximities[START_REF] Varga | Komplex matematikatanítás. Kandidátusi alkotás ismertetése[END_REF].

MTA-ELTE Complex Mathematics Education Research Group, working in the frame of the Content Pedagogy Research Program of the Hungarian Academy of Sciences (ID number: 471028).

I am grateful to Arthur Bakker for suggesting this term during the conference. See also[START_REF] Calderon | Application of reverse engineering activities in the teaching of engineering design[END_REF].

The complete collection of problems can be found here: https://drive.google.com/open?id=1Nhe5DbX3Wgs29l75GhTlrGN8_5rs1-ylFigure 2: Eszter's graph (Simplified version)

A more detailed version of the graph was presented in form of a poster at CERME 11.


Georgia Bampatsikou

Triandafillos A Triandafillidis

Stefanos Asimopoulos

Kostas Hatzikiriakou

Students' learning paths about ratio and proportion in geometry: an analysis using Peirce's theory of signs

Keywords: Peirce, Semiotics, Classes of Signs, Ratio and Proportion, Elementary School

In this paper we use Peirce's theory of signs to analyze the conceptions of four twelve-year-old students on ratio and proportion while solving geometric problems of the stretchers/shrinkers type. The students participated in a series of interdisciplinary activities based on the principle of the camera obscura. We will present the learning paths of these students as chains of signification described in terms of Peircean semiosis. We will emphasize the challenges that they encountered during the reconstruction of their initial ideas and we will classify their conceptualizations in terms of Peirce's classification of Signs.

Introduction

Ιn the 1990s semiotics has started to gain momentum as an analytical tool among researchers in the field of mathematics education [START_REF] Presmeg | Introduction: Signs of signification: Semiotics in mathematics education research[END_REF][START_REF] Sáenz-Ludlow | Constructing knowledge seen as a semiotic activity[END_REF]. By and large semiotics is the study of how signs come to signify, a theory of "signification" (Presmeg, Radford, Roth, & Kadunz, 2016, p. 1). Mathematics, as a highly symbolic practice, calls for a study of the nature of sign systems in order to understand the processes of its teaching and learning [START_REF] Sáenz-Ludlow | Learning Mathematics: Increasing the Value of Initial Mathematical Wealth[END_REF]. In the same decade, mathematics education research was marked by the development of theoretical constructs that acknowledged the social origins of knowledge and consciousness. In these paradigms, knowledge and consciousness are seen as products of communication embedded in experiences that take place in historically-, culturally-and geographically-inscribed contexts [START_REF] Lerman | The social turn in mathematics education research[END_REF]. Knowledge and experience, then, are coconstructed, as the one partakes in shaping the other. As this process is replete with signs, semiotics may elucidate the production of mathematical meaning by learners, as well the shaping of contexts that support learners' conduct with mathematical content and practices.

In our paper, we will use Peircean semiotic theory to analyze instances of the development of four twelve-year-old students' ideas concerning ratio and proportion, as they participated in a series of interdisciplinary tasks based on the principle of camera obscura. Charles Sanders Peirce was an American philosopher, logician, mathematician, scientist, and semiotician. Contrary to other semioticians' claim for an essentialist dyadic structure between a 'signifier' and a 'signified', Peirce suggested that a Sign signifies only when it is interpreted in one's mind. He argued for a triadic structure for the process of semiosis among the Sign or Representamen (signifier), the Object (signified), and the Interpretant. The Interpretant, the most innovative and distinctive feature in Peirce's account on semiosis, refers to the interpretation and nature of the Sign and Object relation. The Interpretant, as we will have the chance to see later on, has the potential to generate a new Sign, thus moving the signification process towards a higher level of interpretation and generalization [START_REF] Sáenz-Ludlow | Constructing knowledge seen as a semiotic activity[END_REF].

By analyzing students' participation in these activities, we will explore how their learning paths constitute chains of signification described in terms of Peircean semiosis [START_REF] Sáenz-Ludlow | A collective chain of signification in conceptualizing fractions[END_REF], as Signs transform to more mature thoughts that become a kind of new knowledge. We suggest that Peirce's ten classes of Signs may constitute an analytical 'tool' that can be used by researchers and teachers alike for deconstructing and classifying students' conceptualizations, not only in ratio and proportion tasks, but in other instructional situations as well. We present this analytical tool in the section that follows, along with an outline of Peirce's semiotics emphasizing the process of semiosis.

Theoretical considerations

Even though he provided various definitions, Peirce defined "a Sign as anything which is so determined by something else, called its Object, and so determines an effect upon a person, which effect I call its Interpretant, that the latter is thereby mediately determined by the former" (Peirce, 1998, pp. 478). The Interpretant, coming as the most groundbreaking and distinguishing characteristic of Peirce's account on semiosis, refers to the understanding that we have of the Representamen and Object relation. According to Peirce signification is not a dyadic relation, as a Sign signifies only when it is interpreted in one's mind. The Interpretant does not refer to the user but to the interpretation, the meaning that is generated for a user [START_REF] Atkin | Peirce's theory of Signs[END_REF].

The importance of Interpretant is clearly visible in the process of semiosis, in which Signs transform to other thoughts that are more mature and become a kind of knowledge. When the relation of an Object (1) , a Representamen (1) and an Interpretant (1) is established, the Interpretant (1) is potentially converted to a Representamen (2) for a new Object (2) and determines a new and more advanced Interpretant (2) , and so on. The conceptual process of semiosis is theoretically unlimited, as the chain of meaning-making continues to new Signs interpreting a prior Sign or set of Signs (Figure 1). However, in practice this process becomes limited by force of habit [START_REF] Parker | The continuity of Peirce's thought[END_REF].

Peirce suggested that each of the three interrelated components of a Sign, namely Representamen, Object, and Interpretant, can be separated into a trichotomy. Representamens signify their Objects not through all their features, but in virtue of qualities, existential facts, or conventions and laws. In the first trichotomy, Signs then can be classified as Qualisigns, Sinsigns, and Legisigns respectively. The Representamen is the form in which the Sign appears, the spoken or written form of a word for instance, whereas the Sign is the whole meaningful ensemble. Peirce also believed that the nature of the Object constrains the nature of the Sign in terms of what is required for a successful signification. He thought about three categories of Objects, then, which were qualitative, existential or physical, and conventional or law-like in nature, constituting the second trichotomy. Thus, when a Sign reflects qualitative features of an Object, then the Sign is an Icon. When there is physical connection to its Object, then the sign is called an Index, and when there is a rule or law between it and its Object, then the sign is called a Symbol. The icon is an intuitively familiar trichotomy, as portraits, paintings, or diagrams fall under the Icons category. Indices may include fingers pointing to something or somewhere and proper names, while Symbols comprise broad speech acts like assertion and judgment. Last, but not least, Peirce thought that Signs could be classified in terms of their relation with their Interpretant. In this third trichotomy, he suggested three categories by identifying qualities, existential facts, or conventional features as the basis for classifying the Sign in terms of its Interpretant. If the Sign determines an Interpretant by focusing our understanding of the Sign upon the qualitative features it employs in signifying its Object, then the Sign is classified as a Rheme. If a Sign determines an Interpretant by focusing our understanding of the sign upon the existential features that it employs in signifying an Object, then the Sign is a Dicent. Finally, if a Sign determines an Interpretant by focusing our understanding on some conventional or law-like features employed in signifying the Object, then the Sign is a Delome, or as Peirce frequently-but confusingly-called it, an Argument [START_REF] Atkin | Peirce's theory of Signs[END_REF][START_REF] Colapietro | Peirce's approach to the self: A semiotic perspective on human Subjectivity[END_REF][START_REF] Freadman | Peirce's second classification of signs[END_REF].

These three previous trichotomies produce combinations that are called 'classes'. In order to categorize a Sign in these classes, we have to ask three questions: 1) What is the relation of the Sign with itself? (1 st trichotomy), 2) What is the relation between the Sign and its Object? (2 nd trichotomy), and 3) What is the relation between the Sign and its Object for its Interpretant? (3 rd trichotomy). The kind of relation that answers the first question qualifies the second, which in turn qualifies the third. Once though we characterize a Representamen as, for example, a Qualisign, which is at the first level of the first trichotomy, the relation of the Sign to its Object and its Interpretant in the second and third trichotomy has to be at the first level as well, as the Sign cannot be related to an Object and an Interpretant from a higher level.

According to these restrictions, then, the classes of Signs are ten (see Table 1). Even though the classes are hierarchically related, every link in the chain of signification does not necessarily constitute a more mature conceptual construct than those in the preceding links of the chain. Thus, in the highest level of Representamen, the Interpretant may be a Rheme (classes V, VI and VIIΙ). Signs of this kind appear often in the mathematics classroom, as students may be familiar with representations of mathematical ideas but their interpretations about them may still be vague or confined to these representations. The chain of signification as outlined in the beginning of this section, along with Pierce's taxonomy of Signs, may allow us to analyze the continuum of students' conceptualization (i.e. learning path) of 'complex' concepts like ratio and proportion [START_REF] Sáenz-Ludlow | A collective chain of signification in conceptualizing fractions[END_REF]).

An experimentation on ratio and proportion

We implemented the analytical tool that we outlined in the previous section in a data set produced from a research study that aimed to highlight students' ideas about ratio and proportion in geometry.

Students' ideas about these concepts were produced through their participation in a series of tasks that were part of an interdisciplinary experimentation. In particular, we wanted to address the two following questions: a) what are the conceptualizations of twelve-year-old students about ratio and proportion related to geometrical shapes; and b) how these conceptualizations could be modified through their participation in a series of interdisciplinary tasks. According to [START_REF] Lamon | Ratio and proportion: Connecting content and children's thinking[END_REF], the stretchers/shrinkers type of ratio and proportion problems, which have reference to geometry, pose difficulty for students in terms of recognizing their multiplicative structure. The students' strategies for solving ratio and proportion problems in a geometric context are the nonconstructive strategies of 'avoiding', 'visual or additive', and 'pattern-building'; the 'preproportional, 'qualitative proportional, and 'quantitative proportional' types of reasoning, which are considered as constructive strategies. The common ineffective strategy that students use in order to solve similar problems is the 'visual or additive strategy', which stresses the aforementioned students' difficulty with the multiplicative structure [START_REF] Lamon | Ratio and proportion: Connecting content and children's thinking[END_REF].

Students worked in pairs for a total of seven 90-minute sessions over a period of 3 weeks. In this paper we will discuss the learning paths of four out of the eight students who participated in the study, those of George, Helen, Bill, and Peter. The tasks that comprised our research were based on an experimental construct that followed the principle of the rectilinear propagation of light in the camera obscura. We used this device in order to depict proportional geometrical shapes. In most tasks students had to investigate the similarity of rectangles formed by tiny led light sources placed on their vertices and their image on a screen placed behind the pinhole on a plane parallel to that of the rectangle. The use of this model gave us the opportunity to negotiate in a non-typical way the concepts of multiplicative relationships, internal and external ratios and also connect fractions with ratio and proportion. According to the Science and Mathematics programs of study, the students were familiar with the concepts at hand. Research data was comprised of the teacher-researcher's fieldnotes (first author), the fieldnotes of a 'non-participant' observer, the transcripts of the recordings in each pair, as well as the students written work.

In the tasks of the initial diagnostic assessment, which was silent and individual, we asked students to draw the parents of a little boy fish, whose body and tail were isosceles right-angled triangles of different size (students were familiar with this fish as it is the main character in a widely-read series of children's books in Greek). The only direction we gave to the students was that the parents needed to be similar to the boy but larger. In the second session of the experimentation we asked students to find the relation between two rectangles that were similar. Students were only aware that one of the rectangles was bigger in size than the other. In the third session, in one of the tasks, we gave students a number of rectangles of various sizes and we asked them to find which of these were similar. In the fourth session of the experimentation, we gave students a rectangle (7 × 3 cm) and we asked them to draw a similar one knowing only the length of one of its sides (11.2 cm). In a summative assessment during the final session, which was silent and individual as the initial diagnostic assessment, we asked students to investigate regular hexagons in terms of their similarity.

Students' chains of signification

In the task of the diagnostic assessment, the four students spontaneously measured the length of the sides of the triangles that comprised the little boy fish. They concluded that the triangles were isosceles, making no reference though to the measure of the angles in the two triangles. While drawing the parents, five of the students followed an additive strategy, without paying attention to the proportion between the size of the triangles of the body and tail of the boy fish and those of his parents. Helen and Bill tried to ensure that the triangles were isosceles to begin with, and that the measure of the interior angles matched roughly the ones in the boy's body and tail. So they drew the two equal sides of the body and the tail of the parents by increasing their length by the same amount. For example, in drawing the father, one student increased the two sides from 5 to 8cm for the body and from 2.5 to 4.7cm for the tail. While drawing the third side of the triangles, though, the students realized that they could not enlarge this third side in same the manner that they had enlarged the other two. So, they increased the length of the third side as much as was needed in order to complete the triangles in their drawing. George designed one parent by doubling the sides of both triangles, thus paying attention to the interior angle in their drawings, but used an additive strategy for the other parent since they thought that they could multiply with a whole number (Qualisign, Icon, Rheme; in this case, the Sign constructed by the student is interpreted by the student as being of the same nature and quality with the given fish boy with O1: the multiplicative structure in the stretchers/shrinkers problems, R1: the students' drawings, and I1a: the use of multiplication and addition interchangeably and that ratio needs to be a whole number). Peter used multiplication for both parents, choosing the numbers 3 and 2 as operators for the father and the mother respectively (Sinsign, Icon, Rheme; the Sign is interpreted by the student as possibly standing for its Object with O1: the multiplicative structure in the stretchers/shrinkers problems, R1: the drawings and I1b: that multiplication as probably related to ratio and proportion).

In the second session, the four students spontaneously measured the lengths of their sides and Helen and Bill suggested that they should "find the difference between their lengths" (Qualisign, Icon, Rheme; the Sign is interpreted by the student as being of the same nature and quality with the given Representamen, with O2: the multiplicative structure of stretchers/shrinkers problems, R2: oral sentence, I2a: the relation between the two similar rectangles can be expressed by subtraction of their lengths). George, who had used a multiplicative strategy in drawing one parent and an additive in the other in the diagnostic assessment, disagreed by saying that "we need to divide to find out how many they are" (Sinsign, Index, Rheme; the Sign is interpreted by the student as possibly standing for another event-index, with O2: the multiplicative structure of stretchers/shrinkers problems, R2: oral sentence, I2b: division is related to ratio and proportion). At this point, Peter, who had used a multiplicative strategy in drawing both parents, disagreed, by saying "since we say twice or three times, we have multiplication" (Legisign, Icon, Rheme; is a type/law, a 'regularity of the indefinite future' that is interpreted by the student as possibly standing for its Object, with O2: the multiplicative structure of stretchers/shrinkers problems, R2: oral sentence, I2c: multiplication is related to ratio and proportion). Taking advantage of these statements, the researcher suggested that they should try both ideas, subtraction and division. In their first attempt, having completed the subtractions, they did not reach to a conclusion concerning the relation between the two rectangles.

In their second attempt, instead of multiplication, which is directly related to the multiplicative strategy, the students decided to use division by dividing the larger by the smaller number. Having observed that the two quotients were equal when rounded to one decimal point, the four students concluded that we could ascertain whether two rectangles are similar by using division.

In the third session, the four students measured the sides of the rectangles and circled the similar ones on their sheet showing preproportional reasoning. Only George chose to express the relation between the similar rectangles by making a clear reference to the operator that constituted them as similar, showing qualitative proportional reasoning (Legisign, Index, Rheme; is a type interpreted by the student as possibly standing for its Object -another event, with O3: the multiplicative structure of stretchers/shrinkers problems, R3: written expression, I3: division is related to ratio and proportion and the concept of unit plays a role to the result).

At the first part of the fourth session, the four students used the internal ratio (length 1 ÷ length 2 = breath 1 ÷ breadth 2) in their calculations, and after they had drawn the desired rectangle they used the external ratio (length 1 ÷ breadth 1 = length 2 ÷ breadth 2) simply to check their answer without realizing that both ratios could be used interchangeably in order to express the desired relation between the rectangles (Legisign, Icon, Rheme; is a type/law, a 'regularity of the indefinite future' interpreted by the student as possibly standing for its Object, with O4: the internal and external ratio of similar shapes, R4: arithmetic operations, I4: the relation of similarity can be found with operations between lengths and widths, widths and lengths, and widths-widths and lengths-lengths and also ratios are pure numbers, without measurement). In the second part of this session, in an attempt to enhance the qualitative proportional reasoning through the use of mathematical terminology in the 'stretchers/shrinkers' tasks, we asked the students to express division in an alternative way. So expressions of the kind "seven divided by two" transformed to "seven over two," especially from George and Peter, implicating thus a fractional terminology in the discussion (Legisign, Symbol, Rheme; is a type interpreted by the student as possibly standing for its Objectlaw, with O5: fractions are directly related to ratios, R5: oral sentences, I5: division can be alternatively expressed with a fraction and their terminology). All of the students though tended to always select the largest number as the numerator, as it was more convenient in the subsequent calculations. So we gave them a similar rectangle on the screen that was reduced in size and asked them to express the reduction with a fraction. All of them claimed that the answer for the reduction was "3.5 over 1.5, because in this way we begin from what it was (i.e. the length of the rectangle formed by the led light sources) and then we have the 'reduced' one" (i.e. the length of the rectangle on the screen)." The researcher then made a reference to maps their relation to the size of the actual land that they represent, and the four students immediately recognized that the ratio in the legend can be as small as 1 over 50000 (Legisign, Symbol, Delome/Argument; is a type interpreted by the student as semiotically standing for its Object-law, with O6: the role of numerators and denominators in ratios, R6: oral sentences, I6: flashback to maps and the various ratios in order to express the reduction may follow the same principles as in this situation). After this discussion, Bill, George, Helen, and Peter concluded that "the numerator expresses the representation on the map and the denominator the physical representation," and therefore the fraction that expresses the relation between the rectangles can be larger or smaller than 1 when it represents a stretch or a shrink (Legisign, Symbol, Rheme; is a type interpreted by the student as possibly standing for its Object-law, with O7: the role of numerators and denominators in fractions, R7: oral sentences, I7: there is a fixed way to express the enlargement or zooming out and numerators and denominators determine the result).

In the tasks of the summative assessment, Bill, George, Helen, and Peter were able to transfer successfully the understanding that they had built in relation to similar rectangles in the sessions of the experimentation(Legisign, Symbol, Dicent; is a type interpreted by the student as physically standing for its Object-law, with O8: the multiplicative structure of stretchers/shrinkers problems and the role of numerators and denominators in ratios, R8: written sentences and written expressions, I8: the appropriate way to express ratios for enlargement or reduction, without measurements).

As we can see, the four students started the sessions (diagnostic assessment) with vague ideas about ratio and proportion in geometrical shapes. Only Peter used multiplicative strategy in all tasks while the other students used the additive strategy. In the instructional sessions that followed, it became evident that students produced arguments in order to support their statements. As a result of this effort some of the students constructed conceptualizations that fall in the upper levels in Peirce's list of Sign classes. In the summative session they managed to transfer their knowledge to a new context. The development of the four twelve-year-old students' conceptualizations concerning ratio and proportion while working on tasks that required the stretching and shrinking of rectangles and regular hexagons, were the Signs that the students constructed. Following Pierce's classification of Signs and the strategies students used while working on these tasks, the criteria used to classify the students' Signs can be summarized as below (

Closing Remarks

The learning process is affected by many factors, such as students' ideas, the interaction between them, the context, the tasks, the representations involved, etc. This means that the learning process is neither rectilinear nor individual; it is 'fluid' and infinite, and includes a number of conceptualizations. Following Peirce's suggestions on how to make our ideas clear, the meaning of a concept or a sign consists in the habit to which it gives rise in the interpreter, be it either a habit of action or a habit of thought. This habit guides future actions, and is forever on trial in the light of upcoming experiences [START_REF] Triandafillidis | On "How to make our ideas clear": A pragmaticist critique of explication in the mathematics classroom[END_REF]. We see students' chains of signification side by side to their activity sequences while they engage with tasks in an elaborated Hypothetical (Teaching) Learning Trajectory that a teacher/researcher has designed. By using Peirce's classes of Signs then, we expand the work of other researchers who have used Peirce's theory of Signs to analyze the learning process based on the triadic nature of a Sign's three components. Our work may also assist a teacher/researcher to concretize students' produced Signs, the ideas that are involved, to what extent students become familiar with the concepts, and also mark those characteristics of the tasks that turn students' conceptualizations to higher level Signs, i.e. habits of action or thought that would endure future trials. In other words, it may help a teacher/researcher to apprehend turning points in students' activity sequences and the distance of a student's Sign from the mathematical goal(s) that she has set, as well as to take appropriate instructional actions in order to support the construction of Signs that are more mature and more relevant to the mathematical ideas at hand.

Figure

  Figure 1: The infinite process of semiosis . . .

Table 1 : Peirce's classes of Signs

 1

		S = R, O, I
	I	Qualisign, Icon, Rheme
	II Sinsign, Icon, Rheme
	III Sinsign, Index, Rheme
	IV Sinsign, Index, Dicent
	V Legisign, Icon, Rheme
	VI Legisign, Index, Rheme
	VII Legisign, Index, Dicent
	VIII Legisign, Symbol, Rheme
	IX Legisign, Symbol, Dicent
	X Legisign, Symbol, Delome

  Table 2):

	Sessions of Experimentation	Students' Signs according to ten classes of Peirce
	1 st : diagnostic assessment with	Qualisign, Icon, Rheme (I)	Bill, George, Helen
	individual tasks	Sinsign, Icon, Rheme (II)	Peter
	2 nd : conservation of properties in	Qualisign, Icon, Rheme (I)	Bill, Helen
	stretcher/shrinker problems	Sinsign, Index, Rheme, (III)	George
		Legisign, Icon, Rheme (V)	Peter
	3 rd : stress of the multiplicative	Legisign, Icon, Rheme (V)	Bill, George, Helen
	strategy	Legisign, Index, Rheme (VI)	Peter
	4 th : enhancement of multiplicative	Legisign, Symbol, Rheme	Bill, George, Helen, Peter in plenary
	strategy by similar rectangles and	(VIII)	session
	ratios of lengths and widths
	division related to fraction,	Legisign, Symbol, Delome (X)	George, Peter
	terminology, role of numerator and	Legisign, Symbol, Rheme	Bill, George, Helen, Peter in plenary
	denominator, legend of maps	(VIII)	session
	5 th : summative assessment with	Legisign, Symbol, Dicent (IX) Bill, George, Helen, Peter
	individual tasks

Table 2 : Signs constructed by the students

 2


Yvonne Liljekvist
email: yvonne.liljekvist@kau.se

Jorryt Van Bommel
email: jorryt.vanbommel@kau.se

Ann-Christin Randahl
email: ann.christin.randahl@kau.se

Cristina Olin-Scheller
email: christina.olin-scheller@kau.se

Multi-theoretical approach when researching mathematics teachers' professional development in self-organized online groups

Keywords: Functional grammar, mathematics teachers, PCK, professional development, social media

 framework. Most posts received responses and this response is in line with the expected response pattern. The speech functions 'Questions' and 'Offers' were most common. Further, most posts addressed subject specific knowledge. The multi-theoretical approach used when researching mathematics teachers' professional development in self-organized online groups showed that these large Facebook groups facilitated professional learning.

Introduction

Studies have shown that Swedish mathematics teachers, to a large extent, participate in selforganized online groups both while working as well as during their free time (e.g., [START_REF] Van Bommel | Facebook and mathematics teachers' professional development: Informing our community[END_REF][START_REF] Van Bommel | Teachers' informal professional development on social media and social network sites: when and what do they discuss? ERME-topic conference: Mathematics teaching, resources and teacher professional development[END_REF]. Moreover, participation in online communities is a global phenomenon: teachers worldwide are using social media as a professional development resource (e.g., [START_REF] Bissessar | Facebook as an informal teacher professional development tool[END_REF][START_REF] Patahuddin | Facebook as a learning space: An analysis from a community of practice perspective[END_REF][START_REF] Tour | Teachers' self-initiated professional learning through Personal Learning Networks[END_REF]. Hence, in studying social media as 'a place' for teachers' professional development, we investigated large Facebook groups with themes connected to teaching and learning in compulsory schools. It is well known that the subject being taught is in the centre of teachers' work (e.g., [START_REF] Kansanen | The didactic relation in the teaching-studying-learning process[END_REF][START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF]. Subject-specific Facebook groups (e.g., mathematics education) in an online teacher community may therefore imply a higher potential for professional development, and the subjectspecific theme (e.g., mathematics for primary school) of some of the teacher self-organized online groups make these groups particularly interesting to study.

If we want to understand the conditions for informal professional learning in online communities better, we have to acknowledge the change of the setting and develop theories that take such change into account [START_REF] Liljekvist | Mathematics teachers' re-sourcing and use of social media: can the 'prosumer' concept convey what's going on?[END_REF][START_REF] Liljekvist | Professional Learning Communities in a Web 2.0 World: Rethinking the conditions for professional development[END_REF]. Our study investigate these online communities at three levels: descriptive level [START_REF] Van Bommel | Teachers' informal professional development on social media and social network sites: when and what do they discuss? ERME-topic conference: Mathematics teaching, resources and teacher professional development[END_REF], interaction level [START_REF] Randahl | Svensklärare på sociala medier: En ämnesdidaktisk analys av tre svensklärargrupper på Facebook[END_REF], and the motivational level. These three levels give us an opportunity to answer the questions: how and what do teachers discuss in the groups. However, this paper goes beyond such a question and addresses the theoretical frameworks we used to capture a variety of aspects of this practice, in order to identify signs of a professional approach [START_REF] Talbert | Professional learning communities at the crossroads: How systems hinder or engender change[END_REF] in these communities. Further, the paper addresses the need to go beyond a specific theory when researching this phenomenon and a brief description is given of how the analytical framework was adapted to the context and empirical findings.

According to [START_REF] Little | Locating learning in teachers' communities of practice: Opening up problems of analysis in records of everyday work[END_REF], locating professional development in teachers' communities of practice directs attention to three considerations: (I) the representations of practice (e.g., how school practices become known and shared); (II) the practice orientation (e.g., whether or not the teacher community improves teaching, and how this interaction advances or impedes teacher learning); (III) the interaction norms (e.g., how participation and interaction are organized, and how this organization supports teacher learning and practice reform).

Hence, to understand how these practices become known and shared, the interaction in these groups was analysed within the framework of systemic functional grammar [START_REF] Halliday | Introduction to functional grammar[END_REF]. Due to Facebook's construction with posts and comments, language (in a broad sense) is used to initiate exchanges in all conversations, for giving and demanding information, goods, and services. In addition, when analysing the practices represented and to reveal knowledge known and shared by teachers as a community [START_REF] Little | Locating learning in teachers' communities of practice: Opening up problems of analysis in records of everyday work[END_REF], we have used [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF] framework. These two frameworks allowed us to categorize teachers' professional communities both in a general way and in relation to subject-specific education. Combining the two frameworks gives us more than two single descriptions: it provides us with a possibility to deepen the analysis of our data and reveals other aspects of mathematics teaching.

Analytical framework

Speech functions as enacting interpersonal relations

People use language to interact with others or to express their view about the world and how they interpret it. Within the framework of Systemic Functional Linguistics (SFL), this is described as the meta functions of the language [START_REF] Halliday | Introduction to functional grammar[END_REF]. Halliday and Matthiessen distinguish between three meta functions of language, that is, its ideational, interpersonal, and textual functions. The ideational function is a matter of expressing how one sees the world and how one interprets this experience. The interpersonal function concerns how the text creates a relationship between speaker and listener. Finally, the textual function concerns aspects that make language relevant, which is sometimes looked upon as a "help-function" to organize and mediate.

Dealing with relation-making processes in online groups, we concentrate on the interpersonal function. In a conversation, the speaker not only plays an initiating role in exchanging information, but also requires something of the listener. "Typically, therefore", Halliday & Matthiessen (2013, p. 135) conclude, "an 'act' of speaking is something that might more appropriately be called an interact: it is an exchange, in which giving implies receiving and demanding implies giving a response" (bold in original). In the semantic system of speech function, anyone can initiate either a giving or a demanding exchange in the conversation. That is, we can give or ask for information and we can offer someone something or ask someone to do something. These four primary speech functions: statement, question, offer, and command (see, e.g., [START_REF] Halliday | Introduction to functional grammar[END_REF] were mapped on the posts in a group. It is therefore possible to analyse what the teacher wanted to accomplish when posting in the group by asking what speech function was used.

Further, the response given can be analysed as either expected or discretionary responses [START_REF] Holmberg | Grammatik med betydelse: en introduktion till funktionell Grammatik [Grammar with meaning: An introduction to functional grammar[END_REF]. For instance, the desired response to a question is an answer and the discretionary response is to neglect to answer. Hence, it is possible to uncover interactional patterns in online groups. For example, when making a statement, a member of a group can acknowledge as well as contradict a particular matter, indicating that statements may initiate discussions. Another example: if posts mainly evoke the desired responses, this could indicate a community of trust. Finally, when posting a question, a teacher expects someone in the group to have the answer as well as being willing to give it. In this way, questions can reveal the knowledge base of the group.

Disclosing the shared professional knowledge

In the knowledge base framework, [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF] outlines seven categories that he suggests constitute the teacher's understanding needed to promote comprehensive student learning. His framework incorporates teacher awareness of various educational aspects, such as content, pedagogy, and organization: content knowledge (CK); general pedagogical knowledge (PK); curriculum knowledge (CuK); pedagogical content knowledge (PCK); knowledge of learners (KoL); knowledge of educational context (KEC); and, knowledge of educational ends (KEE).

Together these categories serve as a way to visualize the shared professional knowledge in these Facebook groups, organized by and for teachers. However, we learnt from our empirical material that three of the categories (i.e., CK; PCK; and KoL) were difficult to differentiate from each other [START_REF] Van Bommel | Teachers' informal professional development on social media and social network sites: when and what do they discuss? ERME-topic conference: Mathematics teaching, resources and teacher professional development[END_REF][START_REF] Randahl | Svensklärare på sociala medier: En ämnesdidaktisk analys av tre svensklärargrupper på Facebook[END_REF]. These categories appeared simultaneously in many posts, hence, we decided to use an overarching category (PCK+) in further analysis, as we did not need a finer-grained analysis in this part of the study. By doing so, we adopt the framework to the Scandinavian/German educational tradition where the relation between the teaching, studying and learning activities in the classroom "the didactic relation" (e.g., [START_REF] Kansanen | The didactic relation in the teaching-studying-learning process[END_REF]) is considered the professional knowledge base.

Method

The data were collected from large Facebook groups. The groups are formed and maintained by teachers, that is, self-organized and not initiated by schools, the Ministry of Education, researchers, etc. It is plausible that every member in the group has a connection to the education field, since the administrator of the group checks a member's affiliation to a school when entering the group. Through membership, one can post and comment. At the time of data collection, the groups were open for the general public to read. In this paper, we include the results from the analysis of three strategically selected Swedish Facebook groups with a theme connected to mathematics education, each with, at time of data collection, between 2000 and 15 000 members. The groups are labelled MA1-3.

By scrutinizing the activity pattern in the groups, a stratified random sample of posts could be conducted based on time of the school year (for a detailed description see van Bommel, [START_REF] Liljekvist | Professional Learning Communities in a Web 2.0 World: Rethinking the conditions for professional development[END_REF]). This method makes it possible to draw conclusions of the interaction on the groups as a whole. Eighty-four to ninety-eight initial posts were collected per group with a total of 1762 responses to these initial posts. The average amount of responses per initial post differed from 3.1 in MA3 to 10.1 in MA2. The interaction sometimes contained images (77 in total), documents (13 instances) or links to other sites (229 instances).

The posts were categorized using the analytical framework developed from speech functions [START_REF] Halliday | Introduction to functional grammar[END_REF] and the knowledge-base framework [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF]. Coding reproducibility was ensured via a multi-step process, where both inter and intra coder reliability was examined. All four authors where involved in the coding process, and all data was coded by at least two persons. Furthermore, the responses on the posts were analysed to confirm the coding of speech functions, as the response validates the speech function (e.g., [START_REF] Holmberg | Texters interpersonella grammatik [Interpersonal grammar in texts[END_REF]. Finally, the internal consistency of the categories was controlled for by scrutinizing all posts in each category, to ensure that all posts coded in each category were consistently coded. The procedure described, in which we recorded every step in the sampling, coding, and categorization, made it possible to backtrack, retrace, and correct errors made in the coding process.

Data

Below, four examples from our data are given (translated into English for this paper), including one or two of the responses to give some insight in the direction of the conversation. The examples will be referred to in the section results and the categorizations for each framework are stated in brackets, along with the number of responses.

Results

Interactional patterns exposed by speech functions

Each post was analysed with a focus on the interpersonal function, which concerns the relationshipmaking process (Table 1). The four speech functions, that is, statement, question, offer, and command, convey what the teacher wants to accomplish when posting in the group. The speech functions 'Questions' and 'Offers' dominated as shown in Table 1. To examine the interactional patterns and exchanges of the teachers posting in these groups, the responses to each post were analysed. The results show that 85.3% of all posts get responses. Most responses (80 %) are in line with the expected response pattern, that is, what the teachers wanted to accomplish when posting in the groups. In the examples above, we see that the questions received numerous responses (Example 1 and 2) whereas the statement (Example 4) did not receive any responses at all.

Knowledge addressed

To indicate the practice known and shared [START_REF] Little | Locating learning in teachers' communities of practice: Opening up problems of analysis in records of everyday work[END_REF], [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF] framework was used for the analysis. Using Shulman's categories, the content of the initial posts could be analysed giving a descriptive overview of the content addressed by the teachers in these communities. These descriptive results were then combined with the previously described results regarding the interaction patterns in each group. As Questions and Offers dominated, we focus on these two speech functions (Table 2).

The analysis showed that most Questions deal with PCK+, that is, issues regarding subject-specific teaching and learning. Example 1 illustrates such posts. This overall pattern is similar in all groups (see Table 2). However, few questions were raised in MA3 so the pattern is not as distinct as in the other groups.

Further, the results regarding the Offers in the posts show that, similar to the category Questions, mainly subject specific knowledge (PCK+) were shared (Table 2) as illustrated by Example 3. In MA3, a few more posts offered content knowledge (CK), similar to Example 2. MA3 also distinguished itself from the other groups in the number of posts unrelated to Shulman's framework (38.2%). Many of these posts contained mathematical jokes or humorous video clips related to mathematics.

Summary

The interaction that is the foundation of the Facebook groups is centered on asking for and sharing subject-specific knowledge (PCK+). Most posts received responses, and the responses were of the expected kind. The online groups were therefore mainly used as forums for making subject-specific education known and shared, but, broadly speaking, not as forums for discussion. Having used a stratified random sample, conclusions can be drawn regarding typical interaction patterns in large self-organized groups.

Discussion

In this paper, we have shown how two different frameworks enlighten different aspects of our data. When combining these frameworks, a new dimension of the data becomes discernible in terms of the way in which the mathematics teachers use self-organized online groups to advance their professional collaboration and professional development. More specifically, we were able to analyse what teachers wanted to accomplish when engaging in such groups. Further, we could see which parts of the teaching practice became visible. This is an important result, as it is an empirical question of whether the characteristics of groups of teachers working together are those of a community where professional development may occur. Since teachers' professional community is extended to social media [START_REF] Liljekvist | Professional Learning Communities in a Web 2.0 World: Rethinking the conditions for professional development[END_REF]Macià & García, 2016;[START_REF] Tour | Teachers' self-initiated professional learning through Personal Learning Networks[END_REF], we aimed at empirically investigating teachers' interaction in self-organized groups, addressing subject-specific knowledge, in order to understand how the groups are used as resources for professional development. The multi-theoretical approach used shows that these large Facebook groups display the characteristics of groups of teachers that merits professional development.

The teachers' online interaction is oriented towards practice to enhance students' learning; therefore, as [START_REF] Little | Locating learning in teachers' communities of practice: Opening up problems of analysis in records of everyday work[END_REF] illustrates in her studies of within-school teacher communities, the issues raised are closely tied to the teachers' everyday professional lives. The subject-specific themes of the groups seem to structure the interaction towards the PCK+, which could be described as the centre of teachers' knowledge base, or more specifically, the "didactic relation" between the teaching-studying-learning activities [START_REF] Kansanen | The didactic relation in the teaching-studying-learning process[END_REF]. The teaching practice that becomes known and shared [START_REF] Little | Locating learning in teachers' communities of practice: Opening up problems of analysis in records of everyday work[END_REF] in the three Facebook groups is characterized by a focus on pedagogical content knowledge, learners, and the material and methods used in the classroom. In forthcoming studies, it is now possible to go further and inquire into posts and comments to reveal the individual themes within the PCK+ category, for instance, what kind of subject is at play, and if and how teachers learn in these communities.

The representations of practice in the groups display the same pattern in what teachers want to accomplish when posting; they request knowledge from and offer knowledge to colleagues, and illustrate and share practices. This is in line with our understanding regarding professional learning communities in schools (cf. [START_REF] Little | Locating learning in teachers' communities of practice: Opening up problems of analysis in records of everyday work[END_REF]. By combining systemic functional grammar and the knowledge-base framework, we can go beyond the context of social media interaction and explain how teachers use the resources in self-organized groups to develop their collective knowledge on pedagogical and subject-specific issues related to their everyday practice. Furthermore, these three Facebook groups are used in only a limited way as an arena for debating and discussing school policy matters. This is indicated by the small proportion of statements (22 of the 258 relevant posts), by the fact that most statements get responses that merely confirm the content in question, and by the educational aspects focused on in the posts. Here, a focus on educational context and educational ends would have indicated an interest in such matters. Discussions around such issues might occur in other Facebook groups with such themes. We want to conclude that the willingness to respond and to share knowledge and teaching methods in the self-organized groups facilitates rather than impedes professional development.

  Example 1: (Question, PCK+, 23 responses, 2 likes) Initial post: Hi, I would need some tips for a thematic theme on length. The students are in grade 2 and have not previously reviewed cm, dm, m or how to measure. Anyone who has some good / cool lesson tip / tasks I could use. Thanks in advance! Response: Estimate how long things are, then measure… Response: We take one-meter-robes and dm-robes out to the forest and try to find objects of such length. Example 2: (Question, CK, 13 responses, 4 likes) Initial post: We discussed the definition of a circle in our team -What would you say, does a circle have one side or no side at all? Response: No sides, closed curve Example 3: (Offer, PCK+, 6 responses, 163 likes) Initial post: We measure outdoors! An elephant of 3.5m, a whale of 15m and a crocodile of 6m [Several images of outdoor work included] Response: Fun! What a cute crocodile! Example 4: (Statement, PCK+, 0 responses, 7 likes) Initial post: Problem solving increases student performance in mathematics. [link to blog]

Table 1 : The distribution of speech functions in each group

 1

		Statement	Question	Offer	Command	Total
	MA1 (n=90)	5	48	34	3	90
	MA2 (n=84)	9	44	25	6	84
	MA3 (n=84)	8	12	56	8	84
	Total (N=258) 22	104	115	17	258

Table 2 : Knowledge addressed in Questions and Offers

 2

				Questions				Offers
	Knowledge base	MA1 MA2 MA3 Total	MA1 MA2 MA3 Total
	PCK+	33	23	6	62	22	12	16	50
	CK	2	6	4	12	0	2	11	13
	PK	2	8	0	10	4	2	2	8
	KEC	4	5	2	11	1	0	1	2
	KEE	3	2	3	8	3	3	4	10
	Total	44	44	15	103	30	19	34	83
	Unrelated	7	5	0		3	5	21

Acknowledgment

This work was supported by the Swedish Research Council under Grant number 2015-01979, and by the research group ROSE (Research on Subject-specific Education). Magnus Kallin supported in the data collection and Jari Appelgren in the data analysis.


Christos Itsios
email: christos.itsios@uni-due.de

Bärbel Barzel
email: baerbel.barzel@uni-due.de

Understanding powers-individual concepts and common misconceptions

Keywords: Grundvorstellung, conceptual change, understanding, mental models, powers

The topic of powers and exponents is generally considered one major obstacle for learners in schools and universities. Our main goal is to answer the question: How can we bridge the gap between individual conceptions in students` minds and the relevant mental models (Grundvorstellungen) to conceptualize powers? With reference to this learning object, two tasks from our test instrument and initial results are presented, which are part of our diagnostic tool to enable the uncovering of students´ individual images and misconceptions in the area of powers, by means of comparing the prescriptive with the descriptive level of relevant mental models.

Introduction

Even though the subject of power expressions and its modern exponential notation has been dealt with for a long time [START_REF] Euler | Vollständige zur Algebra[END_REF], it seems that students of all ages still face difficulties when confronted with them, as well as with exponentiation in general. That has been documented over the years, mainly in school settings (e.g. [START_REF] Confrey | Exponential functions, rates of change, and the multiplicative unit[END_REF][START_REF] Pitta-Pantazi | Secondary school students' levels of understanding in computing exponents[END_REF][START_REF] Avcu | Eight graders' capabilities in exponents: making mental comparisons[END_REF]. The key issue seems to be the lack of understanding of basic rules and algorithms used to manipulate powers. To guard against such occurrences, there is a need for a conceptual change in students´ minds (vom Hofe & Blum, 2016, p. 237), in order to gain a deeper understanding on the conceptual level. [START_REF] Hahn | Bestand und Änderung-Ein Beitrag zur Didaktischen Rekonstruktion der Analysis[END_REF] have proposed that opportunities for conceptual change should even precede the introduction of rules on the procedural level, or as [START_REF] Sfard | The development of algebra: Confronting historical and psychological perspectives[END_REF] puts it: "Operational thinking must be replaced by structural" 1 . This proposition considers the distinction of knowledge into conceptual and procedural knowledge [START_REF] Hiebert | Conceptual and procedural knowledge in mathematics: An introductory analysis[END_REF]. Taking the above into consideration, we pose a twofold question: Which mental models (Grundvorstellungen, vom Hofe & Blum, 2016) should students form for a confident and flexible manipulation of power expressions, and which individual models do they have? To answer this, we first concern ourselves with the identification of adequate mental models through a subject matter analysis and an expert survey. On the basis thereof, we carry out a design research study, where we go from a first pilot study to developing test items, over to a number of three cycles, to aid us in achieving a final version of our diagnosis instrument. Our goal is to find out what students' individual mental models are and whether they are consistent with the desirable normative models. In the present article, after setting the theoretical background, we will provide some insights into our methodology and evaluate the potential explanatory power on a small item sample of the test instrument.

Theoretical Background

In this section we will present the theoretical approaches used in the study to operationalize understanding. First, an outline of the construct of mental models or Grundvorstellungen [START_REF] Vom Hofe | Grundvorstellungen" as a Category of Subject-Matter Didactics[END_REF] together with related terminology used in international literature will be given. Furthermore, the various transformations between representations will be displayed, since they play an essential role for successfully dealing with mathematical concepts. This sets the basis for the generation of the items used in the test instrument.

The concept of Grundvorstellungen (GVs)

Particularly when modeling with the help of mathematics, one needs to activate the suitable GV(s) to translate between the "real word" and the "world of mathematics", which in turn-when successfully performed-means, that the topic has been understood (Bossé, Adu-Gyamfi, & Cheetham, 2011, p. 117). The concept of Grundvorstellungen, or mental models, has a long tradition in German subject-matter didactics. It is used to describe the connection between mathematics and reality on a conceptual level [START_REF] Vom Hofe | Grundvorstellungen" as a Category of Subject-Matter Didactics[END_REF]. Three aspects of GVs can be distinguished. The normative aspect constitutes the basic idea of the concept; it is to be understood prescriptively as the desired status to be reached when understanding a concept. The descriptive aspect characterizes the individual perceptions and notions of learners; it is, therefore, a representation of the students' conceptualization. Finally, the constructive aspect compares the normative and the descriptive aspect to provide some insight concerning students' difficulties of understanding, as well as offer solutions for overcoming misconceptions [START_REF] Hefendehl-Hebeker | Subject-Matter Didactics[END_REF].

Considering that the concept has also been referred to as a theory, our view is the one offered by Radford, who mentions: "… Mogens [START_REF] Niss | Aspects of the nature and state of research in mathematics education[END_REF] contends that a theory in math education has two goals. First, it entails a descriptive purpose, aimed at increasing understanding of the phenomena studied. Second, it has a normative purpose, aimed at developing instructional design." [START_REF] Radford | On the growth and transformation of mathematics education theories[END_REF], p. 4). Moreover, Hefendehl-Hebeker et al. (2019) add: "… the approach of the GV concept combines normative and descriptive methods with a constructive aim. In this sense, analyses of students' work based on the GV concept typically do not remain at the descriptive level but lead to indications of a constructive 'repairing' of the analyzed problems." (p. 35).

To differentiate and clarify the concept from internationally used terms we should mention the theoretical proximity of GVs to concept image and concept definition [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF]. Although both approaches refer to understanding of mathematics using Piaget's cognitive psychology (vom Hofe 1998) and constructivism [START_REF] Hahn | Bestand und Änderung-Ein Beitrag zur Didaktischen Rekonstruktion der Analysis[END_REF] as background theories, GVs are mostly understood in their normative level, as a didactic category for teachers to make explicit which mental models are appropriate for understanding the specific mathematical concept or operation. To differ clearly between the normative mental models according to the mathematical concept and the descriptive mental models in students` minds, some authors replace the descriptive aspect with the term individual mental models (e.g. [START_REF] Prediger | The relevance of didactic categories for analysing obstacles in conceptual change: Revisiting the case of multiplication of fractions[END_REF]. Vom Hofe (1998) uses the term basic model as a prescriptive notion and individual image as a descriptive notion, explaining his integrative view as a "holistic view of mathematical thinking and doing," offering "methodical and theoretical possibilities for combining normative didactic conceptions with descriptive and interpretive working methods" (p. 326).

Furthermore, Soto-Andrade and Reyes-Santander (2011) compare Grundvorstellungen with Lakoff & Nunez's (2000) conceptual metaphors. The similarity of both is constituted by a similar mechanism of transmission: Real experience is the source domain allowing for concept formation, the goal is to provide abstract concepts with meaning. However, they differ in the aspect of their implementation in practice and in that through GVs, preexisting mathematical concepts are represented, rather than constructed. In this paper we will use the terms normative and descriptive aspect of GVs to mean mental models for the former and individual models for the latter, as used in [START_REF] Prediger | The relevance of didactic categories for analysing obstacles in conceptual change: Revisiting the case of multiplication of fractions[END_REF], according to Fischbein's definition of a mental model as a "meaningful interpretation of a phenomenon or a concept" (Fischbein, 1989, p. 12). Following the conceptual change theory, we will refer to the descriptive aspect in a competence-oriented meaning, rather than adhere to a deficit-oriented analysis. The formation of GVs is the key element of concept formation, thus, also entails conceptual change [START_REF] Kleine | With a focus on 'Grundvorstellungen'. Part 1: theoretical integration into current concepts[END_REF].

Grundvorstellungen of Powers

From the normative point of view, there are four aspects that can be extracted and considered as mental models for powers with natural numbers as both the base and the exponent. Following a literature review2 , a subject matter analysis and an expert survey our preliminary results were presented in a first draft [START_REF] Itsios | Potenzen und Potenzrechnung-eine Herausforderung[END_REF]. After further expert talks, the identified mental models were redrafted and rephrased. These are: (GV1) Repeated Multiplication (RM): this mental model can appear in two different ways-a more dynamic one (RM-d), describing a temporal-successive process where new objects emerge, e.g. the repeated doubling of a number of bacteria per hour, and a more static one (RM-s), focusing on the end result of a repeated action, as a whole or an object, describing the end result of a repeated action, e.g. a binary tree.

(GV2) Combinatory Conception (CC): this refers to the number of possible permutations with repetition (number of n-tuples), e.g. there are 10 5 possible combinations for a lock with five cylinders, each using a digit from 0 to 9.

(GV3) Stretching by the same factor (SF): this represents a magnification without the emergence of new objects, e.g. 5 2 can mean zooming into an image twice with a five-fold magnification.

Transformations among Different Representations

When learning a mathematical concept, students should engage in three actions as proposed by [START_REF] Lesh | Representations and translations among representations in mathematics learning and problem solving[END_REF]: recognition of the concept in multiple representations, flexible manipulation within these representations and translation between different representations. A stable use of multiple representational systems can be a guarantee for an easier development of concepts in mathematics and demonstrating mathematical understanding can mean being competent in the manipulation of these systems. These systems can vary from equations, graphs and tables to diagrams, charts or even verbal and written forms (Goldin, 2002, p. 197;Pape & Tchoshanov, 2001, p. 120). There is even a stipulation in the specific content of powers, that conceptual connections between representations is a prerequisite for deeper understanding of concepts (Nataraj & Thomas, 2012, p. 557). These transformations among representations set the basis for the design of our test instrument, which will be introduced in the next section.

Design and Methodology

A first draft of the test instrument was designed using 20 items in total, including three open-ended questions and 17 calculation tasks. For the construction of the items, the aforementioned GVs, as well as most appropriate transformations among representations were taken into consideration, to make the test as comprehensive as possible. This was then tested with a sample of 117 subjects (87 civil and mechanical engineering and 30 university mathematics students) and after a first assessment all responses were categorized according to the frequency of their occurrence. Then, together with the right answer, the three most frequent wrong answers were used as possible choices for each item. The final version comprises the initial three open-ended questions, one calculation task and 16 single-choice items. Following a baseline diagnosis, a total of 176 Grade 10 students from two different schools3 were tested in the main study, with the purpose of determining individual conceptions as well as to detect which GVs can be activated. For the purposes of this article we will present two items regarding the transformations from symbolic to graphical, as well as from symbolic to verbal.

Item Sample

Figure 1 shows two of the three open-ended questions (items A2 and A3), in which the diagnostic potential of individual mental models becomes evident. With them, we can explore the ideas that are prevalent in learners' minds about their conceptual understanding of power expressions and their connection to everyday life situations. Item A2 requires the translation from symbolic to graphical and we expect that mainly Grundvorstellung RM-s will be activated. By requiring a brief explanation, we add a second transformation that students must perform, namely graphical to verbal. Since practically none of the test subjects' answers includes an explanation, we will consider item A2 to address only the transformation symbolic to graphical. Item A3 requires the translation from symbolic to verbal and the Grundvorstellungen RM-d, CC and SF are expected to be activated. In case RM-s is activated, the translation should still be considered to be under symbolic to verbal, not symbolic to graphical, since the task requires a verbalization of the situation, regardless of the fact that the given situation could describe a spatial arrangement.

Some results

Our assumption concerning the results is the following: if an appropriate GV is activated, then there will be a plausible answer given. We make no claims to the opposite, namely, if no reasonable answer is given, as to whether the corresponding GV is present in the student's mind or not. We analyzed the data by categorizing each response to a corresponding pattern, within a small expert group to ensure interrater agreement. In Table 1, the results to item A2 are presented together with the respective response frequencies and in each case an example is given, followed by the relevant GV that most likely could be supported due to the proximity to an individual model. Only a total of 51 Grade 10 students answered this question, including isolated cases with answers that are not hereby listed, because we could not diagnose the structure of those responses. In these drawings we often encountered indications of repeated multiplication (RM), with correct answers as well as misconceptions. 16 answers corresponded to a more static view (RM-s, 1 st row), where 2 3 is misunderstood as multiplication, presented in a spatial-simultaneous arrangement. We also saw substantial dynamic versions, for example a sequence (2 nd row), which we see either as figured numbers or just as cardinality of the relevant numbers. We also attributed RM-d, when the variables were given as symbols, where we can say that the calculation is perceived correctly, but without any hints as to whether the meaning can be properly elaborated (3 rd row). What can be extracted from these results is that learners have not yet adequately shown that they can activate and use the necessary GVs to deal with the given task. However, their individual models, albeit not wrong, remain confined within the world of mathematics and symbols, rather than extending to real-life applications. Table 2 shows the answers given to test item A3, with a total of 46 Grade 10 students answering this question. Some isolated cases have not been listed under the response patterns, showing-again-no structure or diagnostic value. The last column shows, as previously in Table 1, the GV that in our opinion has to be activated, in order for the initial presented idea to make sense in the context of the given power expression.

Here we encountered cases where 3 2 is misunderstood as multiplication, presented in a temporalsuccessive way (1 st row). The most common given response that can be considered feasible is shown in the 2 nd row, where the power expression is viewed as a 3-by-3 spatial-simultaneous arrangement. Very few students (3) tried to present a situation, such that the Combinatory Conception could be identified, but failed to present it accurately (3 rd row).

Response pattern N Example GV

Type of calculation misconception (multiplication) 17 I am ill. I have to take pills. Two in the morning, two at noon and two in the evening.

RM-d

Area 7 I want to find out the area of a square-shaped room to lay a new carpet. Each side of the room has a length of 3m. I calculate 3 2 = 9, then I know that I need a 9m 2 carpet.

RM-s

Combinatorics 3 I own 3 dresses, 3 handbags and 3 pairs of shoes that I can combine. Looking at the first row of Table 2, we can identify a misconception that could be used as an opportunity for supporting learning. Further support, e.g. interviews, could be structured in a way that students are asked to compare the result of 3 2 with their own result (six), leading to a cognitive dissonance. This, in turn, could be the starting point of a conceptual change within their individual images to fit into the normatively set mental models.

CC

Discussion

In all presented cases, we can observe that multiplicative thinking is predominant, with more than 50% of given answers showing that the two variables in question are combined in a multiplicative fashion, mostly assuming commutativity4 . That can be seen in the first example of Table 2, where the numbers 2 and 3 of the expression are combined in a way that results in a multiplication context. After categorizing the response patterns for items A2 and A3 we identify the main problem to be a fixation on previous arithmetic operations of a lower level (multiplication). This problem encompasses depicting a number raised to the third power (see Table 1, first example given) as an array or a two-dimensional area, rather than a three-dimensional object, for example a cube.

Looking at the results of items A2 and A3 we can assume, that the main problem overall is a misconception of the mathematical object of exponents and powers. Confusing multiplication with exponentiation seems to be a principal reason for the demonstrated findings. A possible explanation for these findings is the lack of context variety offered to students when introducing the concept, leading to a false notion of what exponentiation is and when it can be applied, even if plain calculation tasks can be performed faultlessly.

Regarding this TWG, our contribution is twofold. First, by expanding the range of Grundvorstellungen to include the normative GVs for powers, as mentioned above. Second, with our example we have shown how the concept of GVs can help to bridge the gap between individual models in students` minds and the relevant mental models which are necessary for the specific conceptualization. The method of connecting and comparing preexisting ideas and misconceptions with the normative level of GVs shows promising prospects to adequately describe our results, potentially leading to an accurate diagnosis and treatment of these misconceptions.

Figure 1 :

 1 Figure 1: Test items A2 and A3

Table 1 : Results and response frequencies to test item A2

 1

Table 2 : Results and response frequencies to test item A3

 2

Although we do not adopt the theoretical framework of combining embodied mathematics with conceptual metaphors, we acknowledge the work done in Mowat (2010), as this offers a database of mathematics teachers' concept images on exponentiation.

One comprehensive school (Gesamtschule) and one secondary school (Realschule).

Responses to item A3 included both cases, where 2 and 3 were the multiplier and the multiplicand, respectively, and vice versa.