Eva Müller-Hill
email: eva.mueller-hill@uni-rostock.de

Explanatoriness as a value in mathematics and mathematics teaching

Keywords: explanation, mathematical value, proof, philosophy, mathematical practice

In this foremost philosophical paper, I propose a rather holistic idea of (re)establishing mathematical proof in school mathematics by explicating and enacting the particular merits of mathematical proof with respect to a variety of mathematical activities, e.g.: observation, systematic variation of example cases, informal argumentation, defining, hypothesising, inductive and abductive reasoning. Such activities are connected naturally to proving in practices of mathematics, with "practice" referring both to mathematics as a human endeavor in general, and to mathematics a (scientific) discipline. To this end, a concept of "mathematical value" from the philosophy of mathematical practice is employed. I focus exemplarily on the value of explanatoriness, which is both a core merit of good proof in scientific mathematical practice, and an important didactical goal of teaching proof.

Introduction

From the perspectives of mathematics education research and curriculum development, it is widely agreed to date that mathematical proving shall be encouraged and fostered in school mathematics throughout all content areas and grade levels. At the same time, at least in Germany, standards of rigor and formalization in classroom proving changed rapidly in comparison to, e.g., the traditional Euclidean paradigm of proof in elementary geometry, or to algebraic proving in the calculus, and developed into content-oriented, depictive, or intuitive ways of argumentation. In spite of the efforts to make mathematical proving an integrated core element throughout the curriculum, the demand for more proving in German mathematics classrooms frequently produces teachers' complaints: the curriculum is too dense, time is too short, and after all, proof is not that relevant for the final exams; pupils don't like proof, they are afraid they cannot handle it and fall into blockade, and after all, proof is not that relevant for students' future lives. This phenomenon can, probably and partly, be explained as an enduring symptom of a perceived "false dichotomy" [START_REF] Sierpinska | On practical and theoretical thinking and other false dichotomies in mathematics education[END_REF] of mathematical proving on the one hand, and other kinds of mathematical activities on the other. Such a dichotomy may have been over-emphasized by both the traditional university part of mathematics teacher education (at least in Germany), and by the former, strong curricular focus on proofs in the context of elementary geometry, with a sustainable effect on teaching attitudes and beliefs that cannot be simply suppressed by declarations of curricular intent (compare also [START_REF] Knuth | Teachers' conceptions of proof in the context of secondary school mathematics[END_REF] for similar aspects of teachers' proof conceptions and attitudes towards teaching proof in the U.S.). Hence, there appears to be an ongoing need for clarification of a reasonable interpretation of the call for more proving in mathematics classrooms. This need was already expressed by [START_REF] Stylianides | Proof and Proving in School Mathematics[END_REF], together with the proposal of defining a framing core concept of mathematical proof to be explicated and enacted in classroom (pp. 291f.), which can be coherently specified both with regard to different loops of the curricular spiral, and to different fields of mathematical content, and which allows for a clear distinction from inductive, "empirical" (p. 298) arguments. In recent years, the awareness of this need has also led to a deep discussion in mathematics education about diverging and common aspects of mathematical proof and argumentation [START_REF] Mariotti | Argumentation and proof[END_REF]. A substantial part of this debate is devoted to logical or cognitive aspects, including the role of deductive vs. other forms of inferential reasoning, of argumentation base, of language, symbolism, or other semiotic elements (ibid.), e.g., the analysis of structural and referential aspects of argumentations and proofs that provide cognitive unity in conjecturing-and-proving learning settings (Fiallo & Gutierrez 2017), or the examination of the coordination of different registers in the transition from argumentation to proving [START_REF] Duval | Cognitive functioning and the understanding of mathematical processes of proof[END_REF]; with a different reading of "epistemic value" than I will use in the following).

Focus and rationale of this paper

The approach I want to propose here is, in a way, a complementary one. One main difference is that it does not aim at a closed definition of what mathematical proof is, nor at an emphasis of a clearcut distinction of proving vs. non-deductive forms of mathematical activities, e.g., inductive or abductive argumentation based on variation of example cases. A second main difference is the idea that an investigation of mathematical proving and related activities that sticks to the logical and cognitive aspects, might not suffice to grasp all epistemologically relevant facets, which may partially conform to approaches employing Habermas' model of rational behavior (reported in [START_REF] Mariotti | Argumentation and proof[END_REF]. The term "false dichotomy" cited above refers to Michael Otte's view of complementary facets of proving in mathematics (Otte, 1990, p. 59ff.), and emphasizes intuitive, social, and metaphorical facets of proof which cannot be properly modeled with regard to the logic and cognitive functioning of proof or argumentation. Several much more recent studies from different fields suggest that, in a similar vein, issues of epistemic values essentially connected to mathematical proving, like explanatoriness, transcend the scope of logical and cognitive matters, due to a discursive character, to non-reducible pragmatic and affective aspects, and also to the metaphysics of mathematical proving; compare, e.g., [START_REF] Müller-Hill | Die epistemische Rolle formalisierbarer mathematischer Beweise [The epistemic role of formalizable mathematical proof[END_REF] on the epistemic role(s) of formalizability as a feature of discursive proving actions, [START_REF] Novaes | A dialogical conception of explanation in mathematical proofs[END_REF] on explanatoriness within a dialogical conceptualiziation of mathematical proof, [START_REF] Andersen | Acceptable gaps in mathematical proofs[END_REF] on a socio-empirically informed view on acceptable gaps in proofs, [START_REF] Johansen | An Empirical Approach to the Mathematical Values of Problem Choice and Argumentation[END_REF] on argumentation and problem choice as mathematical values, or [START_REF] Dawkins | Values and norms of proof for mathematicians and students[END_REF] on the integral role of values like "increasing mathematical understanding" for the "apprenticeship of students into proving practice" (p. 123). Thus, a focus on explanatoriness as an epistemic value does not mean to exclude other perspectives than the epistemological one, but is inclusive to, e.g., sociological, communicative, and pragmatic perspectives. The following sections aim at indicating first steps into the direction of a program which consists of (1) explicating the particular epistemic merits of the idea of proof that become manifest in the activity of proving in mathematical practice, (2) understanding these coherently in view of the whole process of hypothesizing, generating and communicating proof, (3) emphasizing aspects and features of what counts as proof in mathematical practice that play a particular role in achieving these merits, and (4) indicating elements of a variety of mathematical activities, particularly the generation, operation with, and variation of examples, that already correspond to these aspects and features of proof. In the next two sections, a number of useful concepts, conceptions and results have to be introduced in quite curtate descriptions. First, I provide conceptual considerations as well as exemplary socio-empirical evidence for an understanding of explanatoriness as a core epistemic value of scientific mathematical practice in the sense described above. This goes hand in hand with the stance that the epistemic nature of the scientific discipline of mathematics, understood as a human practice and endeavor which generates, communicates and imparts mathematical knowledge and skills, is relevant to the teaching and learning of mathematics for principal reasons. I then sketch a philosophically informed didactical model of mathematical explanation, developed in [START_REF] Müller-Hill | Eine handlungsorientierte didaktische Konzeption nomischer mathematischer Erklärung [An action-based didactical conception of nomic mathematical explanation[END_REF], and a model of the interplay of deduction, induction, and abduction in explanation processes. I shortly review the socioempirical insights on the background of these models, and finish with a glimpse on future work in enacting explanatoriness as a value in school mathematics.

Explanatoriness as an epistemic value in scientific mathematical practice

I employ a concept of value here that emerged in recent discussions within philosophy of mathematics (see [START_REF] Larvor | Mathematical Cultures. The London Meetings 2012-2014[END_REF]. I follow Ernest's understanding of value as something that is "expressed through the action of valuing", and "manifested in both the prizing of certain characteristics and in the making of fundamental choices" (Ernest, 2016, p. 190), including besides ontic, aesthetic, and ethical values also epistemic values (p. 191). Ernest discusses the epistemic values of truth, provability, universalism, objectivism, and rationalism (pp. 193ff.). Explanation is a well-known and frequently discussed function of mathematical proof, both from the point of view of philosophy and of mathematics education. Frequently proposed features of explanatory proofs are, e.g., surveyability [START_REF] Hill | The epistemic status of formalizable proof and formalizability as a metadiscursive rule in actual mathematical practice[END_REF], reference to characteristic features of the involved mathematical entities and structures [START_REF] Steiner | Mathematical explanation[END_REF], or width [START_REF] Hanna | The width of a proof[END_REF]. These are more or less intrinsic features of mathematical proofs as objects. Understanding explanatoriness as a fundamental value, which manifests itself particularly in activities related to mathematical proving, somehow changes the perspective: Explanatoriness is conceptualized rather dialectically by looking for traces of valuing actions (like prizing and choice-making) directed at proving in mathematical practice that may indicate this value. The issue of (epistemic) values in mathematics has, with exceptions like (Lorenzen, 1974, pp. 152ff.), rather been neglected by philosophy of mathematics until recently (Ernest 2016, p. 191) due to the methodological paradigm of classical analytical philosophy, which also dominated philosophy of mathematics for quite a long time. Accordingly, philosophy can contribute to an epistemology of mathematics by conceptually grasping epistemic features of proof through semantic analysis of, e.g., knowledge attributions, resulting in truthconditions. Concerning the question of epistemic values in actual mathematical practice, a different approach is needed that can take concrete valuing actions like prizing and choice-making of practicing mathematicians into account. Valuing actions directed to mathematical proving may include the way in which examples are chosen and exploited during a proving process, the prizing of alternative hypotheses, the choice of definitions and methods, the degree of formalization and abstraction, the representations used, or the fine-grainedness of the argumentation. Philosophy of mathematical practice, a transdisciplinary endeavor at the intersection of philosophy and sociology of mathematics, has developed conceptual and methodological frameworks that allow appropriate investigation of such issues.

Results from a socio-empirically informed epistemological study

In this paragraph, I will review exemplary empirical results from an interview study [START_REF] Müller-Hill | Die epistemische Rolle formalisierbarer mathematischer Beweise [The epistemic role of formalizable mathematical proof[END_REF] conducted with mathematicians of high standing, from various fields of professional specialization areas, under the lenses of explanatoriness as an epistemic value in mathematical practice (for further methodological details and results of this study, see also [START_REF] Hill | The epistemic status of formalizable proof and formalizability as a metadiscursive rule in actual mathematical practice[END_REF]). In general, the interviewees thoroughly point to a strong relation between explanatory proofs and understanding why the proven theorem holds (ibid.; particularly aspects (2)-( 4)). This is in line with the close relationship of explanation and understanding which is widely agreed upon in philosophical as well as in didactical discussions (see, e.g., the still prominent, well-differentiated discussion in [START_REF] Sierpinska | Understanding in mathematics[END_REF], or the very recent [START_REF] Dawkins | Values and norms of proof for mathematicians and students[END_REF]). Furthermore, selected quotes from (Müller-Hill, 2011) display that prizing and choice-making within processes of generating, communicating and assessing proofs in mathematical practice happens with respect to the robustness of hypotheses and arguments under local and global attacks, e.g., when "try[ing] to shoot holes in it" and "ask questions why it is true, why it works this way, not that way" (Interview 4,p. 246). This appears to be particular relevant in review processes, as "the right way of understanding the contribution that [a] new proof brings" is to ask "what is really the new idea" that "makes something work that previously did not work", and to find "the key argument that made the proof work", or instead, "where it breaks". If the proof is "non breaking" under attacks, one "would see the spark of [a] new idea that finally makes it work" (Int. 5, p. 246). It also includes variation of perspective or context, to look at a hypothesis or an argument "from different directions, through different angles" (Int. 4, p. 246). Also, "in the process of understanding it happens that you have a definition, and that you think you understand it, and that you come back to it for one reason, or somebody tells you something about it, and that you realize that you never understood it at all" (Int. 6, p. 246). In prizing and choice-making actions, the balance between establishing a "big picture" and "the details" is also valued. The "big picture" conveys "the real understanding of why it is true" (Int. 5, S. 199), a "global idea" of "why it works like that", an "overview of the proof" (Int. 4, S. 179), which also affects the consistency with already established results. On the other hand, there are "the details, to follow the proof from step to step, the logical consequences" (Int. 4, S. 179), and "the process of writing down these details", which are sometimes not valued as equally important to the global picture because "not all people are good in this" though they "really know mathematical truths" (Int. 5,pp. 171,199). A well-balanced proof, in this sense, also makes it easier to "be able to look at it at a later time, and still be able to reproduce the arguments" (Int. 4, p. 179), which is taken as an indicator for understanding (see also [START_REF] Hanna | The width of a proof[END_REF]). The different aspects have a contextdependent impact on concrete valuing actions: "So one thing: maths is in a sense something personal. […] When do I think I understand something?" (Int. 4, p. 246), which can be influenced by "deep personal involvement with [a] proof" (Int. 2, p. 313).

Explanatoriness within a didactical model of nomic mathematical explanation

Insights gained from the interview study appear to be in line with the concept of so-called nomic mathematical explanation [START_REF] Müller-Hill | Eine handlungsorientierte didaktische Konzeption nomischer mathematischer Erklärung [An action-based didactical conception of nomic mathematical explanation[END_REF]. In this view, an explanation provides a general pattern, expressed by a general conditional sentence, from which a phenomenon can be inferred as a regular consequence under certain conditions. Moreover, an explanatory, nomic pattern has to fulfill certain conditions on form (ibid., p. 181) and content.

Conditions on form: Basic structure of an explanatory pattern

Whenever manifestation condition(s)

would be fulfilled, the event would take place (as a manifestation of certain properties of the involved objects) and whenever manifestation condition (s) would not be fulfilled (but condition(s) instead), the event would not take place (but the event ).

The second part of the pattern is usually not formulated explicitly. An example pattern discussed in (ibid., p. 202) was reconstructed from a student's explanation of why the sum of three consecutive natural numbers is divisible by three: Whenever we would take three consecutive numbers a, b, c, and move from the biggest number to , we would get three times .

Conditions on content: Basic invariance criteria for an explanatory pattern

Some philosophical contributions on the concept of mathematical explanation (see [START_REF] Hanna | Reflections on Proof as Explanation[END_REF] provide ontic conditions, like reference to "a characterizing property" or "salient feature" of the involved mathematical entities, for an argument to qualify as explanatory. Such conditions serve to explicate the conceptual meaning of "explanation". The account of nomic mathematical explanation developed in (Müller-Hill, 2017) complements the ontic conditions by spelling out epistemic invariance conditions as search keys, test and quality conditions for explanatory patterns. It is to a good extent compatible with ideas of explanatory unification and causal explanation that are at the heart of well-established views of scientific explanation [START_REF] Kitcher | Explanatory unification and the causal structure of the world[END_REF]. Table 1 reports and illustrates conditions that are relevant to grasp the aspects of explanatoriness resulting from the interview study. Functional invariance is necessary for an explanatory pattern. However, there can be explanatory patterns with different degrees of functional invariance, which affects the pattern's informativity: the more differentiated the intervention relation, the more informative the pattern. Scope invariance has different specifities (object, contextual/representational, embedding), each of which can also

Functional invariance

Invariance under intervention at the manifestation conditions: pattern gives correct information, if is fulfilled or not (e.g., regarding the example pattern above: What if I intervene at by taking non-consecutive numbers a, b, c).

Scope invariance

Object invariance

Pattern also gives correct information when the involved objects are varied (e.g., by varying the chosen tripel ).

Contextual & representationnal invariance

Pattern also gives correct information when the used contexts and/or employed representations are varied (e.g., by changing from symbolic to figurative representations of .

Embedding invariance

Pattern is consistent with corresponding patterns for an embedded or expanded pool of related phenomena (e.g., with the transition from natural numbers a, b, c to integers).

occur with different degrees. The degrees of scope invariance affect a pattern's systematization power.

The process of explanation generation

The process of generating a nomic explanation can be described as an iterative cycle of phases of abduction, deduction and induction (Figure 1), starting with abducting a potential explanatory pattern.

The abduction may be creative, that is, on the basis of experience gained by working with and varying test cases, the subject identifies a (by her) formerly unknown pattern to explain the phenomenon (e.g., by exploring ). But it may also be rather selective, that is, the subject already knows one or several alternative patterns that are explanatory for a class of somehow related, familiar phenomena, and identifies (by partial analogy, e.g., to familiar opposite-change task formats on addition) one of these that could potentially also explain the phenomenon in question [START_REF] Magnani | Abduction, Reason, and Science. Processes of Discovery and Explanation[END_REF]. The cycle continues with choosing and varying test cases for the pattern (e.g., for "Whenever we would take three consecutive natural numbers a, b, c, and take from the biggest number and pass it to , we would get three times ".) Via hypothetical deduction, instances of the proposed pattern are produced for each test case (e.g., should be equal to ,…). These test instances are then compared with "reality", thereby inductively backing or weakening the pattern (e.g., ,…). If the pattern appears to be sufficiently backed up by inductive evidence, one can exploit the experiences already made to collect ideas for a deductive justification of the pattern's general validity (e.g., by replacing concrete test values for a, b, and c by generic figurative or general symbolic representations), or enter the cycle again with a variation of the former test cases to find a more general or more informative pattern (e.g., by trying out equidistant instead of consecutive number tripels). If not, one may iterate the whole cycle.

Reviewing the results from the interview study

We can now distinguish between explanatoriness in a situated sense, and explanatoriness in a systematic sense (see Table 2), which are complementary. Employing this distinction, the interview study results reported above suggest that prizing and decision making within mathematical activities related to proving in scientific mathematical practice happens with regard to explanatoriness both in the situated and in the systematic sense, with context dependent weighting. This includes, particularly, the balancing of informativity and systematization power, as two complementing aspects of explanatoriness.

Outlook and future work: Enacting explanatoriness in class

Regarding educational issues, the framework presented in this paper is meant both as a model and tool for researchers to reconstruct, analyze and understand explanation and valuing processes in mathematics and mathematics teaching, and as a basis for task design, and for initiating such processes in classroom. In this sense, the program formulated at the beginning of this paper entails at least two core messages: under an epistemic reading, "mathematical proof" in any strong sense is an ideal construct that is paradigmatic regarding the virtues and validity claims of the discipline of mathematics (that is one reason why we should still teach about it). However, various kinds of actual mathematical activities directed to providing proofs can and should already be driven and regulated with respect to these virtues and validity claims (that is why they work effectively in mathematical practice, and that is how we should teach them). The paper provides a rough impression of how these programmatic aims can be pursued, regarding exploratoriness as an epistemic value connected to proving. Though tersely presented, the reported results already indicate some opportunities and types of activities related to mathematical proving that can be appropriate to enact explanatoriness as a value in class. As a simple example, think of tasks like the following that could be tackled when introducing the concept of mean value (in German Gymnasium, e.g., in grade six): Starting with a given data set of natural numbers, think of ways to vary the set in order to increase, lower, or leave invariant its mean value. Justify your answers.

Pilot runs with corresponding learning environments in the form of written tasks produced data that we actually evaluate to revise the learning environment design. However, it needs well-trained teachers to exploit such opportunities properly (compare, e.g., Lesseig, 2016a;[START_REF] Lesseig | Conjecturing, generalizing and justifying: Building theory around teacher knowledge of proving[END_REF]. The next step in our work is to gain and analyse video-data from implementations of the revised learning environments. The results will also be used for developing a video-vignette based teacher training for secondary in-service teachers.

Figure 1 :

 1 Figure 1: Interplay of abduction, induction and deduction

Table 1 : Invariance criteria for explanatory patterns (Müller-Hill 2017)

 1

Table 2 : Situated and systematic sense of explanatoriness

 2


Andreas Moutsios-Rentzos
email: amoutsiosrentzos@aegean.gr

Eleni Plyta

Keywords: Proof, geometry, complexity, high school, shadow education

teaching and research institutions in France or abroad, or from public or private research centers.

A systemic investigation of students' views about proof in high school geometry: the official and shadow education systems in a school unit Andreas Moutsios-Rentzos, Eleni Plyta

A systemic investigation of students' views about proof in high school geometry: the official and shadow education systems in a school unit

Proof in high school geometry: official and shadow education systems Though proof and proving lies at the heart of modern mathematics, research findings suggest that the high school students experience limited opportunities to be engaged in rich proving practices [START_REF] Otten | The mathematical nature of reasoning-and-proving opportunities in geometry textbooks[END_REF]. The geometry course may potentially give the high school students opportunities to deal with different types of proof [START_REF] Hanna | Opportunity to learn proof in Ontario grade twelve mathematics texts[END_REF], but this feature seems not to be exploited in the geometry classroom, where the vast majority of the experienced proofs seem to be of the form: a deductive argument written in formal mathematical language [START_REF] Battista | Geometry and proof[END_REF]. Furthermore, though abductive reasoning is at the crux of the proving process [START_REF] Arzarello | Abduction and conjecturing in mathematics[END_REF], the students seem to struggle using an abductive argument in the formulation of a deductive proof (Pedemonte & Reid, 2010). Considering the language of proof, [START_REF] Weber | Mathematics majors' perceptions of conviction, validity, and proof[END_REF] differentiated the deductive arguments expressed in a narrative form (using everyday language) from the formal ones (using necessary mathematical relationships for the proof theorem and linking words without explanation steps or texts). Moreover, the students seem to mainly choose an argument that is expressed in a mathematically acceptable way (valid or not), in order to satisfy a teacher, rather than themselves [START_REF] Hoyles | Curriculum change and geometrical reasoning[END_REF].

Drawing upon the fact that a significant number of high school students in Greece attend private tutoring lessons [START_REF] Bray | The Challenge of Shadow Education: Private Tutoring and its Implications for Policy Makers in the European Union[END_REF]; which increases as the students approach the national exams to enter university), our study attempted to map the perceived effect to the students' views about proof in geometry, with respect to both the official education system and the shadow education system. 'Shadow education' refers to the private tutoring, which complements the official school teaching and is focussed on improving the students' performance in the official education system [START_REF] Bray | The Challenge of Shadow Education: Private Tutoring and its Implications for Policy Makers in the European Union[END_REF][START_REF] Mazawi | Beyond Shadows: Equity, Diversity, and Private Tutoring[END_REF][START_REF] Mori | The origin of universal shadow education: What the supplemental education phenomenon tells us about the postmodern institution of education[END_REF]. In this study, we employ the term institutional discourse to refer to all the sources of mathematical authority that are visible to the students, which may communicate to the students what a high school acceptable proof in geometry should be. The school teachers and the textbook communicate what the students perceive as an officially acceptable proof [START_REF] Bieda | Enacting proof-related tasks in middle school mathematics: Challenges and opportunities[END_REF], but the effect that an un-official source authority (such as the teachers of the shadow education system) would have on their views about an acceptable proof in geometry remains an open question.

In line with our previous studies [START_REF] Moutsios-Rentzos | Opseis ths apodeixhs sto scholiko biblio ths Algebras ths A' Lykeiou [Aspects of proof in the school Algebra text book of first grade of the Greek Highschool[END_REF][START_REF] Moutsios-Rentzos | Is this an acceptable mathematical proof?" A systemic investigation of high school students' proof beliefs and evaluations[END_REF], we focus on the school unit as an open system interacting with the broader educational system: official and shadow. Considering that the elements and the subsystems of a system are in a reciprocal relationship, constantly changing themselves and the system, we posit that the students in a school classroom who attend private tutoring lessons constitute a sub-system of the school unit, affecting and being affected by the official and the shadow institutional discourse, thus affecting the whole school unit. A similar role is played by the school textbook, which constitutes a seemingly common reference point for both official and shadow teaching. Nevertheless, the teachers' personal views and practices are mediators of the textbook knowledge to the students [START_REF] Hanna | Opportunity to learn proof in Ontario grade twelve mathematics texts[END_REF]. This is a complex situation, since teachers (official or shadow) may choose to differentiate their teaching content from official textbooks (Tarr, Chavez, Reys, & Reys, 2006), whilst at the same time they are influenced by their own views or beliefs about geometry when it comes to teaching practices [START_REF] Stipek | Teachers' beliefs and practices related to mathematics instruction[END_REF].

Following these, we adopted a systemic approach to investigate the high school students' views about what constitutes an acceptable proof in high school geometry. We focussed on what is visibly communicated to the students as the institutional discourse about proof in geometry, in a threefaceted model: the school teacher practices and the textbook (representing the official education system discourse), as well as the private tutoring school teacher practices (representing the shadow education system discourse). In this study, our conceptualisation of the institutional discourse about proof includes the different proving methods and argument types employed in the proving argument, as well as the language employed in expressing the argument. In order to gain deeper understanding about the wholistic impact that the official and shadow education institutional discourses may have on the students' relationship with mathematics, we investigated the students' mathematical identity, contrasting their self-identification with their being identified by the others [START_REF] Abreu | Schooled mathematics and cultural knowledge[END_REF][START_REF] Kafoussi | Investigating parental influences on sixth graders' mathematical identity: the case of attainment[END_REF]. In this case, the significant 'others' were the school and private tutoring school teachers. The construct of mathematical identity allows us to obtain a measure of the qualitative relationship that the student has with mathematics, as well as of its links with the authority figures of the official and the shadow education system. Consequently, we address the following questions: a) What aspects of proof and proving practices are identified in the three-faceted institutional discourse (the school geometry textbook, the school teachers' practices and the shadow education teachers' practices)? b) How do the students relate these aspects of proof with the three-faceted institutional discourse? c) How do the students identify themselves with respect to mathematics and the three-faceted institutional discourses?

Methods and procedures

The study was conducted with the students attending the first grade (16 years old) of a public Greek "Lykeio" (high school). We included in the study only the students who took private tutoring lessons (N=47; out of 76), their geometry teachers (with the pseudonyms Petros, Katia and Michalis) and the private tutoring school teachers (with the pseudonyms Thaleia, Nikolas, Dionysis) of a private tutoring school that is chosen by the vast majority of the students. Finally, we considered the official geometry textbook (Argyropoulos, Vlamos, Katsoulis, Markatis, & Sideris, 2010). The data collection was conducted close to the end of the first high school year, including:

1) School textbook analysis with the task (of both theory and applications) being the unit of the content analysis. We drew upon [START_REF] Otten | The mathematical nature of reasoning-and-proving opportunities in geometry textbooks[END_REF] and [START_REF] Weber | Mathematics majors' perceptions of conviction, validity, and proof[END_REF] to investigate the following: a) proving methods including: direct proof, the Euclidean-type of proving in geometry analysis and synthesis (drawing from Pappus; see, for example, Heath, 1956, p. 442), and reductio ad absurdum; b) argument type including: deductive, empirical, an outline of the proof, explicit indication that is already proven elsewhere or that it will be proved later (past or future), no argument but explicitly left to be proven by the students, and no argument; c) deductive proof language (see Weber 2010): formal (containing mainly symbolic language) or narrative (containing mainly natural language). 2) Structured observations of the school teachers' teachings (the private tutoring school did not give us permission to conduct any observations), in order to obtain a mapping of the proving practices communicated in the school class by the teachers. We focused on the teachers' practices about proof reasoning and proof language in the school class. 3) Semi-structured interviews with the school and the shadow education teachers to investigate their proof teaching practices; based on [START_REF] Almeida | A survey of mathematics undergraduates' interaction with proof: some implications for mathematics education[END_REF], [START_REF] Bieda | Enacting proof-related tasks in middle school mathematics: Challenges and opportunities[END_REF], Moutsios-Rentzos and Korda (2018). 4) Students' views about proof in geometry questionnaire, to identify their views about which proving argument is more likely to appear in the school textbook, the school class or the shadow education. We chose a theorem from the school textbook (with no proof included) and five arguments (Balacheff, 1988;[START_REF] Weber | Mathematics majors' perceptions of conviction, validity, and proof[END_REF]: deductive-narrative, deductive-formal, abductive, reductio ad absurdum, empirical (see Figure 1). 5) Being good at maths questionnaire, investigating the way that the students' identity is constructed, contrasting self-identification ("I am … at maths") and being identified by the teachers of the two systems ("My school teacher considers me to be … at maths", "My shadow education teacher considers me to be … at maths") on a five-point Likert scale (with "3"= "average").

Mapping the official and shadow education systems in a school unit

The geometry textbook

The geometry school textbook includes mainly deductive arguments (see Table 1), which appear in the form of direct proof, with reductio ad absurdum and analysis-synthesis proofs having fewer appearances. Concerning language, students encounter deductive arguments in narrative form in the theory part of the book, while the applications are both in narrative and formal form. Thus, proof in the Greek Geometry textbook is communicated mainly as a direct proof, containing deductivenarrative arguments, which accords with the findings of the analysis of the Greek Algebra textbook of the same Lykeio grade (Moutsios-Rentzos & Pitsili-Chantzi, 2014).

The school teachers' proof teaching practices

Considering their teaching practices, as well as the type of arguments and expected students' activity, deductive reasoning came up with the highest appearance in the school classroom. The teachers were mainly asking students about deductive, yet isolated, arguments rather than a complete proof, which later were collected and written by the teacher to constitute the proof. In case of the students' having difficulties in finding arguments or synthesising the proof, the teachers asked additional questions, feeding the students' abductive argumentation. In general, abductive arguments were quite common, but they were not present in the written proof, as they did not include reasoned conjectures [START_REF] Herbst | Interaction with diagrams and the making of reasoned conjectures in geometry[END_REF]. No reductio ad absurdum or empirical arguments were included in the teachings we observed: neither by the teachers, nor in the students' responses. Considering the language of the arguments, we noticed a divergence between verbal and written arguments: the teachers constructed the proof in a more narrative form, while the arguments written were closer to formal. The teachers justified this choice, as a result of the limited instruction time within the classroom. Moreover, considering types of reasoning, all three teachers stressed the importance of abductive reasoning in their teaching, as abductive arguments help students to focus, think and find the "right solution". Nevertheless, their reluctance to include them in the written proof is evident, as, for example, Katia stated that she prefers to use the "formal way", indicating the type of arguments used in the official textbook. Commenting about the rest of deductive forms used in the school textbook, the school teachers noted that reductio ad absurdum alienates students, as it is not commonly found in the textbook and the teachers themselves don't use it in the classroom. Finally, all teachers agreed that empirical arguments are employed in the school classroom, but they all said that they emphasise to the students the fact that the idea "it' s right because it looks like so" is not acceptable in geometry.

Prove that parallel segments who have their extremes in two parallels lines are congruent.

Deductive-formal argument l1//l2 thus AB//DE. Also AD//BE. Using the definition, ABED is a parallelogram, thus AD=BE.

Working similarly for BCFE.

So: AD=BE=CF Deductive-narrative argument

We know that quadrilateral ABED has two pairs of parallel sides: AB//ED since l1//l2 and AD//BE (given as an assumption). So, we can conclude using parallelogram definition that ABED is a parallelogram, thus it has its opposite sides equals, meaning AD=BE (1). Working similarly, the quadrilateral BCFE is a parallelogram, since BC//EF(l1//l2) and CF//BE, so we can conclude that BE=CF (2). From ( 1), ( 2) we conclude: AD=BE=CF.

Reductio ad absurdum

Let segments AD, BE, CF be parallel but not congruent. The quadrilaterals ABED and BCFE are parallelograms (since they have two pairs of parallel sides) but they don't have their sides equals (from our assumption). The latter statement is unvalid, considering parallelograms properties, thus the segments AD, BE, CF are congruent.

Abductive argument

We notice that in order to prove that segments AD, BE and CF are congruent, we only have to prove that quadrilaterals ABED and BCFE are parallelograms, because afterwards using parallelogram properties we will be able to assume that the opposite sides are equals. But the latter statement is applicable since AB//DE and AD//BE (using the assumptions given) which means ABED is a parallelogram, and also BC//EF and BE//CF which means that BCFE is a parallelogram.

Empirical argument

We can notice using the diagram that quadrilaterals ABED and BCFE are parallelograms. Thus they have their opposite sides equals, so AD=BE=CF.

Figure 1: One statement, five arguments

The shadow education teachers' proof teaching practices

Considering the shadow education system, the private tutoring school teachers also chose to talk especially about abductive reasoning and the abductive arguments they use in the classroom. Referring to deductive proofs, they made a special reference to reductio ad absurdum and the difficulties the students face, because of their lack of knowledge of mathematical logic. All three agreed that they use this proof type only when it's absolutely necessary in a school textbook case and nowhere else. Finally, they also agreed with the school teachers that they make absolutely clear in their students that they must reject empirical arguments in geometry, because "it says prove that they are parallels, not see that they are parallels" (Nikolas). Considering the argumentation form, the teachers stated that while they accept all argumentation forms that their students would produce, they prefer to use more narrative and less formal arguments. They also expressed their preference for narrative arguments, commenting about textbook's proofs: "I write it the way I think of it at that time or the way my student uses…I want to keep close to the textbook's proof, but I think the language used is very formal and students don't understand it written this way" (Dionysis).

The students' views: the school teacher, the shadow education teacher, the textbook

Considering the students' evaluations about the five argument types, it seems that when focussed on the official discourse (school textbook and teaching), the students evaluate the presented arguments as not to statistically significantly differ from the theoretical neutral, with only the empirical argument getting a statistically significant lower evaluation (Mdn=2.0, P=0.005; One-sample Wilcoxon Signed Rank Test). Considering the unofficial institutional discourse, three arguments (Narrative, Abductive, Reductio ad absurdum) were evaluated as looking statistically significantly closer to what the students encounter in the private tutoring school teachings (respectively,Mdn=4.0,P=0.001,Mdn=4.0,P=0.006,Mdn=4.0,P=0.001;. When considering both official and shadow education, the Friedman's ANOVAs (followed by Wilcoxon Signed Ranks Tests with Bonferroni correction applied) revealed that the students considered three argument types (Abductive, Reductio ad absurdum, Empirical) to be statistically significant more like the ones they encounter in the shadow education teaching in comparison with the official system: Abductive (teaching and textbook; respectively, P=0.024, P=0.001), Reductio ad absurdum (only textbook; P=0.001), Empirical (teaching and textbook;respectively,P=0.497,P=0.003). No statistical differences were identified in the Deductive and Narrative arguments.

Concerning the students' mathematical identity about their 'being good at maths', the students thought that their school teacher's identifying them as being good at maths, would not statistically differ from "Average" (see Table 2). In contrast, their self-identification and their being identified by the shadow education teacher is statistically significantly higher than the "Average'. When considering both systems, it is revealed that the students' self-identification does not statistically significantly differ from their school teacher's identification, but it is statistically significantly lower than their shadow school teacher's identification.

Re-visiting the complexity in the school unit: concluding remarks

Considering the institutional discourse, the school teachers and the textbook seem to converge. The textbook analysis revealed the dominance of deductive, narrative arguments, in the type of direct proof, with few appearances of proof by analysis-synthesis or reductio ad absurdum. The same argument types are evident in the observed proving practices of the school teachers, though they state the importance of abductive arguments in the conjecturing phase of proving. Considering the shadow education teachers' reported practices (since no observations were conducted), we identified a bigger variety of the type of arguments and the language used. While they uphold the view of geometry proof as a deductive argument, they explicitly stated that most of the times the teacher has to add more ideas and to differentiate the argument or the language used, in order for the solution to be more transparent and accessible to the students. Furthermore, they agreed with the school teachers about the importance of abductive reasoning and argumentation in teaching, as well as about their stressing to the students that empirical arguments are not acceptable in geometry.

The students' views seem to converge with the institutional discourse, as the empirical and the reductio ad absurdum argument are the least likely to be chosen by the students. Furthermore, the fact that the institutional discourse favours the narrative form is evident in the students' views. The students' views also support the private tutoring schools claims for a richer proof repertoire, as three arguments types (Narrative, Abductive, Reductio ad absurdum) were evaluated as being statistically significantly looking closer to what they encounter in the private tutoring school teachings than the school class teaching. Though the empirical argument was not evident in our observations and the teachers strongly argued against its being a valid argument, the students' views revealed that for them it is an argument that was missing only in the school textbook.

The students' self-identification about their being good at math converges with their school teacher, even though they think that their private tutoring school teacher would think higher of them. Thus, though both teachers are sources of mathematical authority, the students' self-identification seems to be aligned with the teacher of the official system. The importance of this alignment lies on the general nature of the phrase "being good at math", which embodies the students' positioning about mathematics. Hence, the students identify themselves as being good at mathematics or not, in line with the school teachers' sociomathematical norms.

Consequently, the complex relationship of the two systems is evident on their links with the students' views about proof. The shadow education system seems to offer a richer proof experience, which remains essentially unofficial, in meaning it does not significantly affect the students' mathematics identity construction. Further research should be conducted to map these relationships, as they develop through the three Lykeio grades; especially in the last grade of Lykeio, where the national exams (required to enter university) are a third pole of authority that transcends all school units, thus affecting the authority of both the shadow education teacher and the school teacher.

Table 1 : Appearances of proof and proving in the school geometry textbook.

 1

	Appearances of proof and proving	Theory	Applications Total
	Argument type
	Deductive	73	32	105
	Empirical	6	0	6
	Outline	3	0	3
	None -Past or future	1	0	1
	None -Left for students	4	0	4
	None	35	0	35
	Proving method
	Direct	63	27	90
	Analysis-Synthesis	2	3	5
	Reductio ad absurdum	10	0	10
	Deductive proof language
	Narrative	45	18	63
	Formal	28	14	42

Table 2 : Student's identification about their "being good at math"

 2


Shogo Murata
email: smurata@human.tsukuba.ac.jp

Meta-knowledge about definition: The case of special quadrilaterals

Keywords: Defining, meta-knowledge, special quadrilaterals

Introduction

In proving processes, it would be important for us to focus not only on proofs but also on definitions. For example, if the same mathematical object can be defined differently, the chosen definition plays a crucial role in proving different results on properties of the object. The previous studies related to definition and defining focus on interactions between concept images and concept definitions [START_REF] Vinner | The role of definitions in the teaching and learning of mathematics[END_REF][START_REF] Zandieh | Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning[END_REF], or interactions between examples and definitions [START_REF] Zaslavsky | Student's conceptions of a mathematical definition[END_REF][START_REF] Zazkis | Exemplifying Figure 1: Classification of quadrilaterals definitions: A case of a square[END_REF]. While it is important for us to focus on the role of the example, it is also important to notice the Meta-Knowledge about Definition (MKD). MKD would be characterized as a controller of constructing definitions. Therefore, it is necessary to answer the question: How can Meta-Knowledge about Definition foster students' defining activities?

Method

The method is theoretical exemplification for prospective empirical investigation, which is the construction a framework and the use it through some examples. When we define mathematical objects, the definitions would be controlled by some criteria. Here, the definitions themselves are described by object-language and the criteria of definitions are described by meta-language. Van Dormolen & [START_REF] Van Dormolen | The many facets of a definition[END_REF] proposed the seven criteria of definition: hierarchy, existence, equivalence, axiomatization, minimality, elegance, and degenerations. Table1 is proposed by reorganizing these criteria as MKD. It is expected to characterize students' defining activities from the perspectives of definitions, examples, and MKD.

Purpose:

Define it relatively according to the purpose or the context.

Existence: Confirm the object to define exists in a system.

Equivalence:

Construct equivalent statements about the object to define.

Arbitrariness:

Define it by selecting a statement from the equivalent statements.

Coherence: Define it by using undefined terms and defined terms.

Non-circularity: Define it not to make definitions circular reasoning.

Non-contradiction: Define it not to be a contradiction to other definitions.

Minimality:

Define it by minimal conditions for the existence.

Elegance: Define it more nicely, more simply, or more applicably.

Exactness: Define it exactly to exclude examples not to expect.

Table1: The components of Meta-Knowledge about Definition

By using the components of meta-definition, we defined ten special quadrilaterals: (1) Square, (2) Rectangle, (3) Rhombus, (4) Parallelogram, (5) Isosceles trapezoid, (6) Kite, (7) Cyclic quadrilateral, (8) Trapezoid, (9) Ellipse quadrilateral, (10) Tangential quadrilateral (See Figure 1). The purpose of defining was to construct a hierarchical classification focusing on sides and angles. According to the elegance, we defined these quadrilaterals focusing on duality between sides and angles. The quadrilaterals of (3), ( 6), ( 9), and (10) were defined by focusing on equality of sides, while the quadrilaterals of ( 2), ( 5), ( 7), and (8) were defined by focusing on equality of angles. Furthermore, according to the exactness, the definition that "A kite is a quadrilateral in which two pairs of adjacent sides are equal" was not accurate. It was necessary to correct "two pairs" to "distinct two pairs" for exact definition. As a result, we constructed the hierarchical classification focusing on sides and angles, and discovered the ellipse quadrilateral which is an unfamiliar figure but should exist certainly.

It is concluded that students' mathematical defining activities could be fostered through some processes in which mathematical definitions are controlled by MKD. We could use the components of MKD as a framework for analyzing students' mathematical defining activities. The implication for mathematical practice is to pay attention not only to the interactions between example and definition but also to the role of MKD. It is important for teachers to prepare teaching materials that will facilitate students' activities of examining definitions at a meta-level.


Chrysi Papadaki
email: chrysi@uni-bremen.de

David Reid
email: dreid@uni-bremen.de

Christine Knipping
email: knipping@math.uni-bremen.de

Abduction in argumentation: Two representations that reveal its different functions

Keywords: Abduction, argumentation, proof, Toulmin-model, Global Argumentation Structure

Abduction can function in argumentation and proving processes in different ways. In this paper we focus on the ways in which abductive arguments, taking place in argumentation and proving processes, can be reconstructed in the frame of a Global Argumentation Structure (GAS). We propose two different ways to model abductive arguments within a GAS, with a forward or a backward flow, depending on the function of these arguments in the whole argumentation.

Introduction

Researchers in mathematics education have seen abduction either as process of reasoning backwards (a kind of reversed deduction) [START_REF] Pedemonte | Etude didactique et cognitive des rapports de l'argumentation et de la démonstration dans l'apprentissage des mathématiques[END_REF][START_REF] Knipping | Argumentation structures in classroom proving situations[END_REF] or as a process for creating hypotheses [START_REF] Peirce | Deduction, induction, and hypothesis[END_REF][START_REF] Pease | Five theories of reasoning: Interconnections and applications to mathematics[END_REF]. The purpose of this paper is to examine the ways in which abductive arguments have been modelled up to now and to propose two different ways of modelling them, depending on their function in the argumentation. We are interested in modelling abductive arguments in the frame of the whole Global Argumentation Structure [START_REF] Knipping | Argumentation structures in classroom proving situations[END_REF] of the argumentation or proving process taking place, rather than in isolation from the other arguments taking place (as done for example by [START_REF] Pedemonte | Etude didactique et cognitive des rapports de l'argumentation et de la démonstration dans l'apprentissage des mathématiques[END_REF].

Abduction in argumentation and proof

Abduction is an area of research that has received increasing emphasis in mathematics education, in reference to a number of different processes. Here we discuss abduction in mathematical argumentation. As we consider mathematical proof to be a kind of argumentation, our discussion includes also abduction in mathematical proof.

A simple way to distinguish between abduction and deduction is to consider how they connect a case to a result via a rule. An example often cited in the mathematics education literature comes from an article Charles Saunders Peirce published in Popular Science Monthly in 1878. At this time he referred to abduction as "hypothesis", and characterised the two processes as shown in Table 1 below. From this example it is evident that abduction can be thought of as a reversed deduction, reasoning from a Result via a Rule to a Case, rather than from a Case via a Rule to a Result.

Deduction.

Hypothesis. [Abduction.]

Rule.-All the beans from this bag are white.

Case.-These beans are from this bag. ∴ Result.-These beans are white.

Rule.-All the beans from this bag are white.

Result.-These beans are white. ∴ Case.-These beans are from this bag. [START_REF] Toulmin | The uses of argument[END_REF] provides a model that can be used to analyse arguments in general, not only deductive arguments. In his model (see Figure 1), an argument includes the following elements:

C (claim or conclusion) the statement of the speaker, D (data) are data justifying C, W (warrant) the inference rule that allows data to be connected to the claim, B (backing) is reason to believe W, Q (qualifier) expresses how much to believe C, R (rebuttal) tells you when belief in C is not supported Toulmin considered only two types of reasoning when he created his model, namely deduction and induction. There are however, several cases in the literature of researchers modelling abduction using Toulmin's model in different ways. In Table 2 we use Peirce's example and terminology as a common reference in order to describe and depict the different ways in which abduction has been modelled using Toulmin's terminology. [START_REF] Pease | Five theories of reasoning: Interconnections and applications to mathematics[END_REF] keep the forward flow (arrows pointing from left to right) of Toulmin's model but they characterise as a datum the Result that is known and needs no justification, and the claim as the Case that is inferred. They express the uncertainty of the claim by employing a qualifier (the word "probably") before the claim (see Figure 2). [START_REF] Pedemonte | Etude didactique et cognitive des rapports de l'argumentation et de la démonstration dans l'apprentissage des mathématiques[END_REF] was the first researcher in mathematics education to use Toulmin's model to describe abduction. She too models an abductive argument with a forward flow but reverses [START_REF] Pease | Five theories of reasoning: Interconnections and applications to mathematics[END_REF] placement of data and claim. She considers a claim to be the known target of the inference (Result) and the datum to be the unknown fact (Case), which must hold in order for the claim to stand. She denotes the status of the unknown datum with a question mark next to the letter D representing the datum. The difference between Pease and Aberdein's, and Pedemonte's modelling of abduction may arise because of the different functions they attribute to it. Pease and Aberdein see abduction as a process of generating a hypothesis, whereas Pedemonte sees it as a reversed deductive process, as "reasoning backwards". It is also important to note that these researchers model individual arguments, in isolation from other arguments. In contrast [START_REF] Knipping | Argumentation structures in classroom proving situations[END_REF][START_REF] Knipping | A method for revealing structures of argumentations in classroom proving processes[END_REF] describes a way of linking together arguments analysed using the Toulmin model to describe larger structures of argumentation. She reconstructs the whole argumentation of students, building a Global Argumentation Structure (GAS), which reveals a complete overview of the argumentation that took place. For example, Figure 3 shows her analysis of the argumentation in a lesson in which the Pythagorean theorem was proven. [START_REF] Knipping | Argumentation structures in classroom proving situations[END_REF], like Pedemonte, sees the function of abduction as "reasoning backwards". She reconstructs complex deductive argumentation processes, which may contain some abductive arguments. The flow of the overall GAS is a forward one, but the flow of the abductive arguments is a backward one (see Figures 3 and4). Like [START_REF] Pedemonte | Etude didactique et cognitive des rapports de l'argumentation et de la démonstration dans l'apprentissage des mathématiques[END_REF] The arrow in AS-X in Figure 3 marks the abduction. The students know they want to conclude that ABCD is a square, and they use abduction to go from this desired conclusion to the datum (BCD=90°) needed. The dotted line marks the shift in focus to finding an argument for the conclusion (BCD=90°). Through AS-2 and AS-3 they establish BCD=90° and combining this datum with the previous established statement ABCD is a rhombus, they deduce the desired conclusion, ABCD is a square.

Methodology of the data analysis

In the following we compare two Global Argumentation Structures developed following the procedures outlined in [START_REF] Knipping | Revealing structures of argumentations in classroom proving processes[END_REF]. They describe a three-stage process: reconstructing the sequencing and meaning of classroom talk; analyzing arguments and argumentation structures; and comparing argumentations to reveal their rationale.

Here we are chiefly interested in comparing two Global Argumentation Structures that emerged in two mathematics classrooms and which have some interesting differences regarding the abductive arguments taking place in them. We will describe each context and the abductions that occurred before comparing the two.

Ms James' classroom

The abduction described here occurred in Ms James' grade 9 (age 14-15 years) classroom in Canada. The class was trying to explain why two diagonals that are perpendicular and bisect each other define a rhombus. The students had discovered and verified this property while working in a Dynamic Geometry Environment (DGE).

Figure 5 shows the reconstructed argumentation stream that occurred after the class had identified the given information in the situation and had recorded it in the diagram Ms James was drawing on the board. She then asked the class what information was needed to show that a shape is a rhombus, given the definition. This stream is interesting because the statements that were made first chronologically, appear last, on the right hand side, and vice versa. This is similar to the abductive stream described by [START_REF] Knipping | Argumentation structures in classroom proving situations[END_REF] in that the argument goes backwards from the intended conclusions to the data needed to deduce that conclusion. It differs, however, as it involves several abductive steps connected in a stream.

There are linguistic markers of what is going on in the transcript. For example, Ms James says:

Ms James: If I can prove that that is the same length as that, is the same length as that, The phrase "If I can" marks this as an abduction. In a classroom proving process, where the main flow of the argument is deductive, it makes sense to diagram such abductions in this way, as if they were deductions, but marking them with arrows showing the flow of the argument in the opposite direction. This is shown in the GAS in Figure 6.

Axel and Dave

In the case presented here two 10 th grade students (Axel and Dave) in a German classroom worked together on a geometry task designed in a DGE. All the students in the classroom worked on the geometrical tasks in pairs and at the end all the tasks were discussed in a whole-class discussion with the focus being the argumentation the students provided to justify their answers. In the task the DGE window was divided into two sub-windows (see Figure 7). On the right sub-window (3D Graphics) there was a 3D coordinate system in which a solid had been constructed and hidden, and the plane xOy. On the left sub-window there were three sliders (h for height, n for tilt and d for spin) that the students could manipulate in order to move the invisible solid, and a two-dimensional depiction of the intersection of the solid with the xOy plane. The main question set to the students was: "What solid do you think this could be, judging from its cross-sections?". The students were provided with a worksheet, which included an Exploration Matrix with specific (h, n, d) sliderpositions they could use if they wanted. They were asked to justify their answers in writing. The students were not told how to justify their conclusions, only that they should attempt to explain the validity of their conclusions and to be as complete (leaving no doubts) as possible. There was no requirement that they follow a deductive structure, or that they produce a formal proof. The GAS presented here (Figure 8) represents Axel and Dave's argumentation that took place in their discussion during their pair work alone, before the classroom discussion. Axel and Dave used the positions in the Exploration Matrix for their explorations. They started by generating four hypotheses and during their exploration they refuted step-by-step three out of the four hypotheses.

The remaining hypothesis (H 16 ) became then a claim (Cl 23-25 ), which they would later provide an argument for, in order to verify it. After the verification the epistemic value [START_REF] Duval | Cognitive functioning and the understanding of mathematical processes of proof[END_REF] of the statement "The solid is a cone" changes and from a claim it becomes a conclusion (C 118.2-120 ). This GAS has two phases (Argumentation Streams). In the first phase (AS-1) belongs the process of generating hypotheses, during which the students suggest possible solutions to the problem, basing their decisions on the data they gather. This is then followed by the second phase (AS-2), during which more data are provided to support the claim (Cl 23-25 ). With the support of these new data, the claim is then treated by the students as a conclusion (C 118.2-120 ).

The abduction here (AS-1) is not represented as reasoning backwards, rather a process of suggesting multiple possible solutions to a problem all of which, at least initially, seem plausible.

Here the chronological flow of the statements made corresponds to their logical function in the argumentation. For example, the conclusion C 118.2-120 (which is usually the last inference drawn in an argumentation) comes last both chronologically (in the argumentation) as well as structurally (in the GAS). In such cases, in which speculation is required and the students engage in hypothetical reasoning, it makes sense to diagram abduction with a forward flow.

In the case of Axel and Dave, unlike in that of Ms James' classroom, there are no warrants in the GAS. This should not be interpreted as that the students had no warrants in mind when creating hypotheses or drawing conclusions. The absence of warrants is merely the result of our In Ms James' classroom abduction was used to reason backwards from a desired conclusion to the data needed to deduce it. The students know that the shape is a rhombus and they are seeking data that can connect that conclusion to what they know about the diagonals. They are using abduction in order to explain the known conclusion. Most of the arguments in the GAS are deductive, and it is important to note that the final 'official' proof omits the abductive steps. Having abduced the necessary data, they then establish it deductively from other data they already know, and then use it to deduce their desired conclusion. In the case of Axel and Dave, abduction is used to generate hypotheses. They are exploring to discover something new. Their hypotheses are then tested in the second phase of their argumentation, through comparison with data generated in the DGE.

The differences in the GASs in these two situations reveal an important difference in the way abduction functions in them. This shows how such an analysis can reveal not only the presence of an abduction but also the role it plays in the overall argumentation. The ways in which the abductions in Ms James' classroom and Axel and Dave's pair work were represented, as flowing backwards or forwards, reflect the functions of the abductions. When abductions are used to explain, the flow is backwards to the facts that explain the conclusion. When abductions are used to explore, the flow is forwards, from data that has been collected to possible results, and includes qualifiers to mark the epistemic value of the inference. The role of qualifiers is vital in order to keep a forward flow for the abduction in an overall abductive GAS.

AS-1

AS-2

These differences in function, in turn, depend on the nature of the mathematical tasks. In the case of Axel and Dave the geometrical object the students are asked to work with is unknown and they are not told that they need to prove anything. The focus of their argumentation is the generation and justification of hypotheses, rather than proof production. It is not even possible for them to deductively prove a hypothesis is correct; they can only disprove false hypotheses. In Ms James' case the task was explicitly to find a proof of a given statement. The status of the statement as a conclusion is already established, by the students empirical explorations earlier and by the teacher's authority. In such a proof production task it is clear that the overall structure must be deductive, and abductive argument is incidental, not essential to the argumentation. The final proof could have been produced without using abduction.

We have shown in this paper how [START_REF] Pease | Five theories of reasoning: Interconnections and applications to mathematics[END_REF] way of modelling abduction (forward flow of the abductive argument) can be embedded into GASs reconstructed using [START_REF] Knipping | Argumentation structures in classroom proving situations[END_REF][START_REF] Knipping | A method for revealing structures of argumentations in classroom proving processes[END_REF] methodology, when the function of the abduction is exploring. We are not claiming that the two ways that abduction has been modelled in the GASs we described are all the possibilities. Research in other contexts will no doubt reveal other ways in which abduction is embedded in argumentation structures. However, our describing and distinguishing these two ways, along with the associated functions of the abductions in the argumentation, is a contribution to research in this area.

Figure 1 :

 1 Figure 1: The Toulmin model of an argument (1958)

Figure 2 :

 2 Figure 2: Structure of an abductive argument, Pease and Aberdein (2011, p.12)

  she considers the claim to be the Result of the inference and the datum to be the Case, which must hold in order for the claim to stand. But she marks the abduction differently, by reversing the arrow to indicate the backward flow, and she does not use a question mark next to the letter D to indicate the unknown fact.

Figure 3 :

 3 Figure 3: GAS of discussion on the Pythagorean theorem, Knipping (2008, p. 437)

Figure 4 :

 4 Figure 4: Abductive argument reconstructed with a backward flow, Knipping (2003, p.8)

  is the same length as that [draws in the segments AB, BC, CD, and DA]. If I can prove that, I'm done. Rhombus.

Figure 5 :Figure 6 :

 56 Figure 5: The abductive argumentation stream from Ms James' class (the parenthetical comments after each statement indicate speaker and transcript line)

Figure 7 :

 7 Figure 7: Task in DGE -Position (h=0,55, n=90 , d=0 ) of hidden solid

  methodology according to which, what is depicted in the reconstructed argumentation is only what the students state explicitly, and not what they may have been thinking but have not said.

Figure 8 :

 8 Figure 8: The GAS in Axel and Dave's discussion

Table 1 :

 1

Peirce's syllogisms for deduction and abduction (1878, p. 472; CP 2.623)

Table 2 : Peirce's abduction modelled with Toulmin's terminology

 2


Helén Sterner
email: hse@du.se

Planning for mathematical reasoning -Surprising challenges in a design process

Keywords: Design Principles, Mathematical reasoning, Patterns

Introduction

Generalizing is a key element in mathematical reasoning as well in early algebra and algebraic thinking [START_REF] Kaput | What is algebra? What is algebraic reasoning[END_REF]. However, students find it difficult to understand the meaning of generalized arguments, which means that teachers need to support students to develop their generalized argument [START_REF] Stylianides | Research-based interventions in the area of proof: the past, the present and the future[END_REF]. A way to increase teachers' possibilities to support students' reasoning may be through intervention studies. [START_REF] Stylianides | Seeking research-grounded solutions to problems of practice: Classroom-based interventions in mathematics education[END_REF] write about the importance of research-based interventions of proof and proving in teaching. Since generalized arguments can be seen as a part of proof and proving [START_REF] Stylianides | Reasoning-and-Proving in School Mathematics, the Case of Pattern Identification[END_REF], intervention studies also about generalized arguments should be important. Similar reflections have emerged in the Argumentation and proof group at CERME where a lack of design-based studies promoting investigation in the classroom and a need to shift research focus from the learners to the teacher is emphasized [START_REF] Mariotti | Argumentation and proof[END_REF]. This poster will exemplify a part of a cyclically recurring intervention process by answering the question: What challenges do teachers meet when trying to understand a given Design Principle (DP) (McKenney & Reeves, 2012) and design and implement teaching based on it?

Mathematical reasoning and algebraic thinking

One part of mathematical reasoning and early algebra as well as of algebraic thinking is the structure and the understanding of relationships in quantities [START_REF] Kaput | What is algebra? What is algebraic reasoning[END_REF]; these relationships could be embedded in figural repeating patterns. Patterns and pattern identification could be seen as essential components in elementary school when working with the activity of reasoning-and-proving [START_REF] Stylianides | Reasoning-and-Proving in School Mathematics, the Case of Pattern Identification[END_REF]. Teaching patterns can provide students' understanding and support their arguments why things work the way it do. Mulligan, Mitchelmore, English and Crevensten (2013) make similar assumptions, pointing out that teaching and learning mathematics through patterns and structure as well as generalized approaches may provide students with a deep understanding of mathematics.

Methodology

The frame of this study is based on educational design research, as well as an intervention inspired by classroom design research [START_REF] Stephan | Conducting classroom design with teachers[END_REF]. In the implementation of the intervention, DPs will be used to address the specific foci in this case, strategies for mathematical reasoning and proving in algebra. Three mathematics teachers (grade 1 and 6) and the author are part of this intervention. DPs are used to guide the content of the interventions and the DPs are used as a framework in the initial analyses. The DPs will guide and give the intervention a theoretical foundation in the ongoing process (three recurring cyclical phases): first designing the intervention and the teaching, second the three teachers implementing the intervention and the third, the reflection after the implemented intervention. To stimulate mathematical reasoning, and proving in teaching and learning of pattern, six DPs have been created from previous research, and one of them is presented in this poster.

DP1

The students will have possibilities to identify a pattern, to structure the pattern and to generalize the pattern [START_REF] Mulligan | Reconceptualizing Early Mathematics Learning: The Fundamental Role of Pattern and Structure[END_REF][START_REF] Stylianides | Reasoning-and-Proving in School Mathematics, the Case of Pattern Identification[END_REF].

Preliminary results from a part of the design research process

Preliminary results show the complexity involved in teachers' understanding of teaching and learning patterns relating to DP1. The complexity becomes visible when teachers discuss and try to understand the concepts; identify, structure, and generalize patterns in relation to their own practice. Teachers try to exemplify what it means to identify, structure and generalize, and thereby become aware that the concepts overlap. Results also show that teachers' understanding of generalization develops during the discussion; from equalizing generalization with general formulas to generalizations as something possible to show with different representations such as describing with words or showing with concrete material and pictures. In the classroom, teachers meet challenges when trying to challenge students' understanding of a generalized pattern, for example by asking them to explain what is behind a general formula.