Paolo Boero
email: boero@dima.unige.it

Fiorenza Turiano
email: fiorenza.turiano@gmail.com

Integrating Euclidean rationality of proving with a dynamic approach to validation of statements: The role of continuity of transformations

Keywords: Euclidean geometry, transformational reasoning, continuity principle, rationality

During a long-term teaching experiment aimed at developing 10 th grade students' culture of theorems through a pathway in Euclidean plane geometry, some students' autonomous reasoning moved towards non-Euclidean proofs based on continuity of transformation of geometric figures. Based on the use of existing analytical tools to analyze such episodes, the aim of this paper is to outline a wider scope for synthetic geometry in order to make it more suitable for students' approach to the culture of theorems. Through the introduction of a continuity principle to legitimate such extension, the paper suggests how to exploit students' potential in transformational reasoning, and to bridge the gap between synthetic geometry and analytic geometry rationalities in classroom work.

Introduction

Since 2016 we are engaged in designing and implementing in 10 th -grade classes a teaching and learning pathway in Euclidean geometry aimed at promoting students' approach to the culture of theorems [START_REF] Bartolini Bussi | Approaching and developing the culture of theorems in school[END_REF] i.e. at developing not only the knowledge of statements, proofs and their applications, but also autonomous proving and the awareness of crucial meta-knowledge about theorems (the role of hypotheses and thesis, the requirements of proof, etc.). Such aim is not easy to attain; the question that originated the study reported in this paper was: is it possible to lessen the students' difficulties with proving by legitimating some spontaneous ways of solving geometry problems and validating statements? Let us consider an episode, a single case of a wider phenomenon observed in the three classes where the experiments on our pathway have been performed. Its role is to put into evidence the potential inherent in students' dynamic approach to solving theoretical problems in geometry. Students have already learnt to construct a circle tangent to both sides of an angle, and to justify the construction. In interaction among them and with the teacher, students have learned to build up the circle with center chosen on the bisector and ray derived from the following construction: to draw the perpendicular line from the chosen center to one side of the angle; to consider its intersection with that side, and the segment joining it with the center as the ray of the circle. They have proved (with the help of the teacher) that the drawn circle is tangent to both sides of the angle through the steps of Euclid' validation of the same construction. Some weeks later they are asked to construct a circle tangent to the three sides of a given triangle. They have at their disposal a worksheet with a drawn triangle, the ruler and the compass.

One student (Ale) draws the bisector of the angle ABC, then he tries to choose a point on the bisector as the center of a circle by taking different points on the bisector and adapting the width of the compass to the sides of the triangle. Finally, he draws the circle S (the arrow and S are added to the original figure). Ale is not satisfied with his drawing. After a while he makes a free-hand drawing of a circle near to B, then other circles more and more near to AC. Finally, he stops and observes the worksheet for several seconds. Then he starts writing: "The solution of the problem is when the circle, which is tangent to the sides of the angle B, meets the third side". The participant observer (PO: the first author of this paper) starts an interaction with Ale: Ale: Yes, when I move the center on the bisector the circle is (pause) the circle becomes bigger and bigger, and remains tangent to the two sides of the bisector (pause) PO: Are you sure that it becomes tangent to the third side?

Ale: Because the circle (pause) if I continue moving the point on the bisector, one part of the circle will go outside the triangle (pause) Therefore there will be ONE (emphasis) point to get the contact, (pause) the tangency with the third side.

PO: And if the circle becomes bigger and bigger?

Ale: It will be no more tangent to the sides of the triangle, but… No, if I come back, the two intersections finally join in the tangency point.

Given that the circle exists (by such continuity considerations) it is easy to prove that its center belongs to the three bisectors -provided that students already know that if a point is equidistant from the sides of an angle, it belongs to the bisector of the angle, and that the tangent straight line is perpendicular to the ray in the point of tangency. Hence the ruler and compass construction may be easily made by using the intersection point of two bisectors and the perpendicular straight line from it to one side of the circle. In this case the method of construction derives from the reasoning used to prove the existence of the tangent circle and the knowledge of its properties. Note that performed exploration might result in the construction of a theoretical justification (cognitive unity of theorems: see later), once Ale's reasoning by continuity would be legitimated.

In Book IV of the Elements (prop.4) Euclid describes how to construct a circle tangent to the three sides of a given triangle and provides a theoretical justification for it.

PROPOSITION 4 (book 4, Heath's translation)

In a given triangle to inscribe a circle.

Let ABC be the given triangle; thus it is required to inscribe a circle in the triangle ABC.

Let the angles ABC, ACB be bisected by the straight lines BD, CD [I. 9], and let these meet one another at the point D; from D let DE, DF, DG be drawn perpendicular to the straight lines AB, BC, CA. Now, since the angle ABD is equal to the angle CBD, and the right angle BED is also equal to the right angle BFD, EBD, FBD are two triangles having two angles equal to two angles and one side equal to one side, namely that subtending one of the equal angles, which is BD common to the triangles; therefore they will also have the remaining sides equal to the remaining sides; For, if it cuts them, the straight line drawn at right angles to the diameter of the circle from its extremity will be found to fall within the circle : which was proved absurd; [III. 16] therefore the circle described with centre D and distance one of the straight lines DE, DF, DG will not cut the straight lines AB, BC, CA; therefore it will touch them, and will be the circle inscribed in the triangle ABC. [IV. Def. 5]. Let it be inscribed, as FGE.

Euclid's line of thinking is different from Ale's. Euclid describes how to solve the problem of the circle inscribed in a triangle by finding candidates to be the center of the circle and three of its rays, then a theoretical justification for the chosen solution follows, which relies on definitions and previously proved theorems, and results in a proof of the existence of the inscribed circle. No continuity or transformational considerations are made.

Based on the above episode and other episodes that will be shortly presented later, and with reference to some constructs in mathematics education literature, integrated with a principle of continuity to legitimate the widening of the scope of synthetic geometry, we will present and discuss the potential inherent in students' transformational reasoning (Simon, 2006) in the approach to the culture of theorems, once that principle is assumed.

Theoretical background Transformational reasoning, and the continuity principle

Transformational reasoning was defined by [START_REF] Simon | Beyond inductive and deductive reasoning: The search for a sense of knowing[END_REF] as:

The mental or physical enactment of an operation or set of operations on an object or set of objects that allows one to envision the transformations that these objects undergo and the set of results of these operations. Central to transformational reasoning is the ability to consider, not a static state, but a dynamic process by which a new state or a continuum of states are generated (p. 201).

In past research of our group this construct was already used, in particular, to characterize one of the types of generation of conditionality of statements [START_REF] Boero | About the generation of conditionality of statements[END_REF]. In this paper it will be used to describe processes of discovery of the reason why a statement is true, or strategies to solve a construction problem in Euclidean geometry.

In our toolkit we will integrate the construct of transformational reasoning with a principle of continuity. Continuity provides students not only with hints for the solution of a construction problem or a proving problem, but also, in some cases, substantial elements for the validity of the construction and the proof. For instance, in the episode presented in the Introduction, Ale "by continuity" proves the existence of the tangent circle. Thus a principle of continuity as a criterion for the epistemic validity of a conclusion derived through transformational reasoning might widen the scope of acceptable proofs, with positive consequences for the approach to the culture of theorems (see Discussion). How to formulate the continuity principle within the perspective of synthetic geometry (i.e. geometry theory based on constructions performed with ruler and compass)? Continuity axioms (including Archimedes' axiom) were among the axioms added by Hilbert to Euclid's axioms (Trudeau, 1987), but it is not easy to formulate a consequence of them which in simple operational terms accounts for the truth of the "existence theorem" proved by Ale. However, it is possible to provide a formulation of the continuity principle in operational terms by making reference to analytic geometry:

The continuity principle guarantees the solution of a synthetic geometry problem through transformational reasoning provided that the translation of the problem and the related transformational strategy into analytic terms (by using algebraic expressions) allows a treatment which brings to the solution thanks to the continuity of the set of real numbers.

As an example, consider the episode presented in the Introduction. As a generic example [START_REF] Mason | Generic examples: Seeing the general in the particular[END_REF] of the situation, we may assume B=(0,0), a represented by x=0, b represented by y=0, and the side AC represented by y=-2x+3. Then the system y=-2x+3 & (x-K) 2 +(y-K) 2 =K 2 , will represent the circle centered in the point (K, K) of the bisector, which is tangent to a and b. By varying K we represent the situations of no intersection with the straight line y=-2x+3, of tangency (two possibilities, including tangency from the exterior), and of intersection in two points.

We will use the above criterion as a provisional solution for the characterization of the continuity principle; further research (and the analysis of more episodes!) is needed to formulate it in more precise and effective terms.

Theorems

Mariotti (2001) defined a Theorem as a statement and its proof with reference to a theory (and related inference rules). With reference to Guala & Boero (2017), Mariotti's definition encompasses theorems related to various kinds of theories throughout history (e.g., Euclid's, as well as Hilbert's, geometry; graph theory, with its crucial reference to visual objects; 19thcentury classical probability theory as well as Kolmogorov's axiomatic theory, etc.), together with the different ways of considering proof since the Greeks (Grabiner, 2012) along with the cultures these ways came from (Siu, 2012) (p. 210).

In this paper we use the construct of Theorem to consider different, possible ways of proving the same theorem with reference to different theories and different inference rules.

Cognitive unity of theorems

After having found some cases of theorems (in geometry, and in elementary arithmetic) for which students behaved in a similar way, Garuti, Boero and Lemut (1998) defined "cognitive unity of theorem" what happens for some theorems when: during the production of the conjecture, the student progressively works out his/her statement through an intensive argumentative activity functionally intermingled with the justification of the plausibility of his/her choices. During the subsequent statement-proving stage, the student links up with this process in a coherent way, organizing some of the previously produced arguments according to a logical chain (p. 345).

The cognitive unity construct was also extended to the case of the relationships between the exploratory phase of proving a theorem, and the subsequent construction of a proof for that theorem [START_REF] Garuti | Cognitive unity of theorems and difficulty of proof[END_REF]: indeed, the exploratory phase of proving shares some common aspects with conjecturing (as re-construction of the meaning, and appropriation, of a statement; and identification of elements for its validity).

The cognitive unity construct may be used to account for what happens when students produce, thanks to transformational reasoning, arguments that may be re-arranged in a proof (be it Euclidean, or based on a continuity principle: see the episode in the Introduction).

Rationality

Many cultural activities (including mathematical ones) may be described as discursive activities sharing some common features: First, criteria to establish truth and falsity of propositions, and validity of reasoning. Second, strategies to attain the goal of the activity, which can be evaluated. Third, a specific language for social interaction and self-dialogue.

The rationality construct elaborated by [START_REF] Habermas | On the pragmatics of communication[END_REF] may be exploited to move from such a superficial description to a deeper treatment of discursive activities. According to Habermas' construct, rational behavior is characterized by: conscious taking in charge of truth and validity criteria (epistemic rationality), of strategies to attain the goal (teleological rationality), and of communication means (communicative rationality); and by dynamic links between knowing, doing and communicating in the rationality perspective (for a discussion of potential and limitations of Habermas' construct as it was adapted to mathematics education, see [START_REF] Boero | Habermas' construct of rational behavior in mathematics education: New advances and research questions[END_REF].

Where the episodes come from

We think that it is important to put into evidence some salient features of the long term teaching experiment that provided us with the elements for the theoretical elaboration of this paper (in particular, Ale's and other episodes like those presented here); in particular we agree with [START_REF] Simon | Beyond inductive and deductive reasoning: The search for a sense of knowing[END_REF] when he says that transformational reasoning is a natural way of thinking, which needs to be "nurtured" -thus suitable cultural and educational choices must be performed in order to allow students to develop it.

A teaching-learning pathway to the culture of theorems

We have chosen to base the approach to the culture of theorems in grade X of high school on the Euclid's plane geometry for the following reasons: First, in Italy, since the end of the XIX century and for several decades Euclidean geometry was the main subject intended to allow secondary students to meet theorems, proofs, proving. Second, Euclidean geometry offers the possibility to approach different kinds of proof (including proof by contradictionthe preferred indirect proof in Euclid's Elements). Third, through the alternation of "theorems" and "constructions to be validated" in Euclidean geometry the teacher is offered the opportunity of stressing the relevance of theoretical thinking in mathematicsin particular the distinction between a drawing and a geometric figure, and the need of moving from visual truth to theoretical truth (even if in Euclid's elements the goal of going beyond visual evidence to validate statements is not completely accomplished: see Trudeau, 1987. However, in our design we have not followed any path taken as such from Euclid's Elements; we have chosen only some constructions and some theorems. In the case of theorems with a thesis consisting of more than one claim we have chosen only one claim (this is in the case of the theorem considered in the second and third episodes below).

The educational context of the episodes

The episodes presented in this paper happened within a pathway to the culture of theorems in the domain of Euclidean geometry that had been experimented for the first time in the year 2016-17 in a 10 th -grade class of a scientific oriented high school, and in the year 2017-18 in two 10 th -grade classes of the same school, with two 50' lessons per week from October to May (out of 5 hours devoted also to other mathematics subjects: algebra, probability and statistics). Classroom activities were based on an alternation of individual activities, in some cases preceded by a constructive interaction with the teacher, and collective activities. Individual activities concerned six kinds of tasks: solution of construction problems, conjecturing, proving, analyzing, evaluating and improving some schoolfellow's productions, close activities regarding proof texts. Collective discussions were orchestrated by the teacher and in most cases concerned the comparison and the critical analysis of a few students' individual productions selected by the teacher. According to the aim of developing competencies related to the culture of theorems, the assessment method consisted of: the individual revision of individual work at the end of each of the three parts in which the pathway was divided, with careful identification and remediation of "what does not work" in each individual production, and an overall synthesis on the individual itinerary concerning difficulties met, still obscure points, reasons for mistakes, emotional problems, etc. This evaluation method was derived and adapted from a similar method currently adopted in several Genoa University courses for pre-service teacher education aimed at developing professional competencies of cultural analysis of the content to be taught (see [START_REF] Guala | Cultural analysis of mathematical content in teacher education: The case of elementary arithmetic theorems[END_REF], for more information on the assessment method and its motivations).

Further episodes

We have chosen three further episodes. Like that presented in the Introduction, they concern relations among circles and straight lines. They show different roles that may be plaid by transformational reasoning and continuity in the field of plane geometry, and how to legitimate them within the proposed theoretical framework, in particular through the continuity principle.

The rolling circle

Students are requested to find if it exists a position of a given circle, such that the circle is tangent to two sides of an angle, and to justify the answer. Some students imagine to roll the circle on one line towards the second line and they discover that "Yes, it exists, because there is a moment in which the rolling circle starts to touch the second line; in that moment it is tangent to both lines". This intuition of the rolling circle facilitates also the discovery that, in the found position of the circle, the center of the circle is the point of intersection of the two straight lines that are parallel to the sides of the angle at the distance of the ray of the circle. Students "see" the movement of the center of the circle in parallel with the first line, and (after the tangency position) with the second line.

By considering the generic case of the straight lines y=0 and y=x and of the rolling circle (x-K) 2 +y 2 =1, the continuity principle may be applied to legitimate the solution find by the students.

All this suggests a way both to find the center of the circle when it is tangent to both lines (a heuristic function for the solution of a construction problem), and to explain why in that position it is tangent to both lines (a proving function, once the continuity principle is adopted)in a perspective of cognitive unity of theorems.

Comparing the length of the chords of a circle

The diameter of a circle is its longest chord, and the length of the chord increases when its middle point approaches the center of the circle. This is part of proposition 15 of Book 3 of the Elements:

Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the centre is always greater than the more remote.

Euclid proof is rather complex-it needs the proof of 6 intermediate statements. A student produces a reasoning that may be reported this way: he fixes a point on the circle and considers a chord on that side of the circle "which is opposite to it", with corresponding angles at the center of the circle. He imagines to move the chord towards the center and he observes that the nearer the chord is to the center, the bigger is the corresponding angle -till when the chord becomes the diameter by collapsing on the two aligned rays. This is not yet a proof of the theorem, but the student's line of reasoning might be integrated with elements that allow to prove that the length of the chord increases when the chord approaches the diameter. Indeed, the triangles obtained by joining the extremities of the chord with the center of the circle have two sides of equal length (the ray of the circle) and the width of the angle between them increases, thus also the length of the side opposite to the center increases. This may suggest to exploit the triangular inequality -the chord is shorter than the sum of the rays-to prove that the diameter (two rays long) is the longest chord; and it may suggests also to use Pythagoras' theorem in order to prove that the length of the chords increases when the distance between the chord and the center of the circle decreases, or to find the length 2√1-K 2 of the chord intersected by y=K on the generic circle x 2 +y 2 =1: the length of the chord attains its maximum value 2 when K=0, i.e. when the chord becomes a diameter.

Tangency between a straight line and a circle.

By composing two statements of Euclid's Elements, we get the following statement: Given a circle centered in C and a straight line a intersecting the circle in the point T, the straight line a is tangent to the circle (i.e. T is their only common point) if and only if the straight line is perpendicular to CT. While trying to prove the "if" part of the statement, students were suggested by the teacher to reason by contradiction; they were also invited to consider a second point of intersection T'. Some students reacted to the discovery of a contradiction (an isosceles triangle with two rectangular angles) by imagining to make CT' collapse on CT by rotating it around C. This movement might be exploited to expand their reasoning through the consideration of the isosceles triangle TCT' and its height CH. The identity CH = √(CT 2 -TH 2 ) (Pythagoras theorem) allows to prove (by continuity) that TT' is 0 (i.e. there is only one point of intersectionwhich means tangency) if and only if the height CH of the triangle collapse on CT, i.e. if and only if CT is perpendicular to the straight line a.

In this case transformational reasoning produced by some students might play a double heuristic role: first, to suggest the necessity of the condition of perpendicularity for the tangency while proving its sufficiency; second, to suggest the way to get a proof (by the continuity principle) of the "if and only if" statementstill, cognitive unity might allow to arrange a valid, simple proof.

Discussion

Transformational reasoning combined with the continuity principle may play three roles for the approach to the culture of theorems and, more generally, for the development of students' mathematical rationalities: first, it may help solving a construction problem and simplifying its proof, in comparison with the Euclid's proof (like in the episode presented in the introduction). Second, it may play a heuristic function by suggesting a method of proving in Euclidean geometry, or eventually a method of proving by relying on the continuity principle (like in the last two episodes). Third, it may allow to compare different ways of proving the same theorem in synthetic geometry, thus contributing to the development of the culture of theorems and at the same time to an initial understanding of the fact that a theorem may be tackled with different strategies and according to different criteria of truth and of validity of proving methods. Concerning the second and the third role, we may observe how the adoption of a principle of continuity would imply changes in the rationality of the discursive activity of proving, in comparison with Euclidean rationality. On the epistemic side: a new criterion of truth is introduced. On the teleological side: for a construction problem, the existence of the solution may be got through transformational reasoning; for the proof of a theorem, transformational reasoning in several cases (see our episodes) guarantees the possibility of the cognitive unity between the exploration phase and the proving phase. On the communicative side, new verbs and expressions are necessary to account for the specificities of transformational reasoning, in comparison with the language of Euclidean geometry. In the perspective of rationality, the principle of continuity might allow to compare (for some theorems) methods of proof corresponding to Euclid's ones with methods of proof which depend not only on Euclids' axioms and theorems, but also on the additional principle, thus contributing to the culture of theorems and prepare students to move to other theoriesin particular, Analytic geometry -and help to "cross the borders" between the two domains (and the inherent rationalities: see [START_REF] Boero | Crossing the borders between mathematical domains[END_REF]. We observe how, in the problem situations of our episodes, the principle of continuity may be easily related to the algebraic modelization of the situations at stake. For instance, in the case of the last episode we may consider the circle of center (0,0) and ray 1, whose equation is x 2 +y 2 =1, and its intersections with the straight line of equation y=kx+1, which meets the circle in the point (0,1). The algebraic treatment of the system of the two equations offers an immediate answer to the problem of tangency by discussing the solutions of the second degree equation: (1+k 2 )x 2 +2kx=0 derived from the system. The equation has the solutions (x-coordinates of the intersection points) x=0, x=k/(1+k 2 ). They are coincident (i.e. tangency, in terms of analytic geometry) for k=0, i.e., when the straight line of equation y=kx+1 becomes perpendicular to the ray of the circle with extremities (0,0) and (0,1). Conversely, including the principle of continuity in synthetic geometry might allow to give sense to the algebraic expressions that are used in the algebraic modeling process of geometry problems, particularly as regards the role of the variable(s) which not only are signs to be dealt with according to the syntactic rules of the algebraic language, but also represent dynamic phenomena (in our case, geometric transformations).

In this section we have used the expressions "may", "might allow" several times to outline possible directions for studies, intended to develop the culture of theorems in the classroom on the reflective and on the operational sides, once the continuity principle (possibly, after further elaboration) is assumed to legitimate an extension of the scope of synthetic geometry and the inherent rationality.

Figure 1 PO:

 1 Figure 1

Figure 2 [

 2 Figure 2 [I. 26], therefore DE is equal to DF. For the same reason DG is also equal to DF. Therefore the three straight lines DE, DF, DG are equal to one another; therefore the circle described with centre D and distance one of the straight lines DE, DF, DG will pass also through the remaining points, and will touch the straight lines AB, BC, CA, because the angles at the points E, F, G are right.

Figure 3

 3 Figure 3


Fiene Bredow
email: bredow@uni-bremen.de

The role of the teacher in the development of structure-based argumentations

Keywords: Argumentation, teacher influence, symbolic language, algebra

Motivation

It is a desideratum that mathematical argumentation should be integrated into today's mathematics teaching. In reality, opportunities for learning mathematical argumentation are infrequent. Algebraic language is often understood only as a series of signs that can be transformed according to certain rules and the same applies to algebraic argumentation [START_REF] Pedemonte | Argumentation and algebraic proof[END_REF]. Such argumentations present an epistemological challenge for students because they are detached from concrete examples. The integration of mathematical argumentation into teaching and the appreciation of different types of arguments is therefore of crucial relevance. The role of the teacher in this needs to be clarified.

The aim of my PhD project is to investigate the role of the teacher in the development of structurebased argumentations in classroom practice. What kind of support can teachers in mathematics education give their students in order to guide their argumentation from concrete examples to general structures? How does this support influence the students' argumentation and their conception of algebraic language? For this purpose, a learning environment at the transition from arithmetic to algebra was designed and implemented in three eighth-grade classes in Germany. The main ideas of my teaching design and learning environment will be presented in this poster. [START_REF] Sfard | On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin[END_REF] distinguishes between an operational and a structural conception of mathematical objects. In an operational conception, mathematical objects are perceived as a process. In a structural conception mathematical objects are conceived as static constructs, objects. Reification is necessary to conceive mathematical objects structurally. There are different types of mathematical arguments. Structure-based arguments use properties of the involved objects instead of only calculation-oriented transformations of algebraic expressions without relation to the content. [START_REF] Mason | Expressing Generality and Roots of Algebra[END_REF] describes that, as a first step, learners should investigate examples and their structure in order to be able to develop general arguments using symbolic language. This idea guided the construction of the learning environment in my project. Connections between the conception of algebraic language and the development of structure-based argumentations are examined as well.

Theoretical framework

All in all, the teacher has an important role in mathematical argumentation. [START_REF] Conner | Teacher support for collective argumentation: A framework for examining how teachers support students' engagement in mathematical activities[END_REF] describe three different types of support for collective argumentation by teachers: "Direct contributions to arguments", "Asking questions" and "Other supportive actions" (e.g., evaluating, repeating). In addition to that, it is interesting how teachers enact written tasks in classrooms. This can lead students to an interpretation of tasks that can provide rich opportunities for argumentation or create obstacles. Whether and how this support can affect the arguments of pupils has been researched little so far and is a focus of this study.

First, tasks for eighth-grade classes and teacher prompts are theoretically constructed, which are supposed to stimulate a structure orientation and provide opportunities for mathematical argumentation. Then, a learning environment is designed and empirically enacted. Two interviews with teachers are conducted to get insights into the teachers' understanding of mathematical argumentation and into their experiences with the designed learning environment. Then, from transcripts of the lessons, argumentation processes and students' conceptions are reconstructed and analysed with a focus on the prompts of the teacher that promote a structure orientation. All this will inform a revision of the learning environment. The following research questions guide my study: How can teachers constructively support their students in developing structure-based argumentations? How do a teaching design and a learning environment for the development of structure-based argumentations in mathematics lessons look?

Structure of the learning environment

An intervention of four lessons (90 min each) was designed and performed. All tasks provide opportunities for mathematical argumentation. The teaching environment and tasks are designed in a way that allows students to build on concrete examples, examine the structure of these examples, and finally develop structure-based arguments. All tasks support this strategic approach to argumentation: 1. Observe; 2. Assume; 3. Analyse and check; 4. Justify. Learners are not simply asked to do calculations in the tasks, but to observe and reflect the mathematical structures of the numbers to support structure orientation (see [START_REF] Wittmann | Objekte-Operationen-Wirkungen: Das operative Prinzip in der Mathematikdidaktik[END_REF][START_REF] Mason | Expressing Generality and Roots of Algebra[END_REF]. In the first two lessons, the students get to know four different types of arguments (explanatory arguments in form of a dot pattern; generic examples; algebraic and narrative arguments) and have to solve conjectureand-proving tasks. In the third and fourth lesson, the students learn to argue in a new format and more sophisticated arguments are demanded in context of "arithmogons": the sum of two corners is a side [START_REF] Wittmann | Objekte-Operationen-Wirkungen: Das operative Prinzip in der Mathematikdidaktik[END_REF].

All in all, this learning environment should provide students opportunities to learn mathematical argumentation and to establish a structure orientation. How teachers enact this learning environment, support their students and which impact this support has will be analysed in my study.


Games as a means of motivating more students to participate in argumentation

Jenny Cramer

University of Bremen, Germany; cramerj@math.uni-bremen.de

Mathematical argumentation and proof require students to engage in complex thought processes in which they explore logical connections, solve problems, and learn to trust their own assertions. While in most mathematics classrooms there are some students who show an aptitude for trusting their own reasoning and making logical inferences, this does not generally hold true for the majority of students. In the context of playing games, however, many students overcome such obstacles and become motivated to use logical reasoning and argumentation. In this paper I examine the potential of exploiting logical structures in games as a means of fostering the motivation of students to engage in inner-mathematical argumentation. A three-step-approach leading from a board game into argumentation in calculus is introduced and indications towards the usefulness of the approach for fostering students' motivation are presented.

Argumentation, logical games, calculus, motivation.

Argumentationa mathematical activity for the few?

Mathematics is a discipline based on logical thinking and inferences, with argumentation and proof at its heart. Proving activities in mathematics provide opportunities to autonomously discover mathematical knowledge and reach a deeper understanding (de [START_REF] De Villiers | The role and function of proof in mathematics[END_REF]) and can therefore promote an experience of empowerment to students. They may also teach us how to reason logically outside of mathematics [START_REF] Grabiner | Why proof? A historian's perspective[END_REF]. The mathematics classroom should thus aim at engaging all students in "suitable mathematical activities of argumentation and proof" (Boero, 2011, p. 1). This demand is even more pressing in the light of existing gaps in the accessibility of high level mathematical activities for students from different backgrounds (Boaler, 2016, p. 173). Argumentation and proof threaten to act as social filters that may increase performance gaps between students of different backgrounds if they are only accessible to few students [START_REF] Knipping | The social dimension of argumentation and proof in mathematics classrooms[END_REF].

Being important mathematical activities, argumentation and proof have been receiving increasing attention in mathematics education and in school curricula for decades. However, in spite of efforts made to promote argumentation in class, only one in three students reported in the PISA study of 2012 that they like to engage in complex problem solving activities (OECD, 2013, p. 67f), and about 30% of students reported a feeling of helplessness when doing mathematics problems (OECD, 2013, p. 88f). While problem solving is not synonymous to arguing and proving, such tasks almost always rely heavily on logical reasoning, and the results may be taken as an indication for students' attitudes towards argumentation activities: few students appear to enjoy such activities, and a significant number of students is reluctant to engage in them at all. This is surprising, as more than half of the same students questioned for PISA 2012 stated that they sought explanations for things and could easily link facts together (OECD, 2013, p. 67f), and both of these skills are crucial for argumentation.

Mathematical argumentation is a discursive practice, into which students need to be introduced in order to be able to participate [START_REF] Boaler | Identity, agency, and knowing in mathematics worlds[END_REF]. Logical games and puzzles may provide a suitable way of overcoming obstacles to participation, as they require forms of logical, deductive reasoning similar to the reasoning used in mathematical argumentation and proof. Clear rules, balanced starting positions for all players, and the limited scope of knowledge required for argumentation in a game context may help to facilitate students' participation in discourse as they provide circumstances which may support fulfilling discourse ethical requirements [START_REF] Habermas | Moral consciousness and communicative action[END_REF]. These potential positive aspects of games have been discussed before [START_REF] Cramer | Using Habermas to explain why logical games foster argumentation[END_REF].

Another positive aspect of games lies in their potential to appeal to the intrinsic motivation of students. High motivation in games can be explained by the needs for competency, autonomy and relatedness postulated by [START_REF] Deci | Die Selbstbestimmungstheorie der Motivation und ihre Bedeutung für die Pädagogik[END_REF] self-determination theory. The motivational potential of games has been well documented for video games [START_REF] Rigby | Glued to games: How video games draw us in and hold us spellbound[END_REF], and similar effects may be expected for board games. The need for competency describes an inner desire to master challenges and new situations. Games provide such challenges to their players. A fulfillment of the need for autonomy is given when we perceive our actions as self-guided. While this is generally difficult to achieve in a school setting, a playful approach may help to fulfill this need. Lastly, the need for relatedness describes a need for meaningful social interaction. Board games are characterized by interaction and could prove even more helpful in this area than video games.

From game to graph: A three step approach

In this paper, a three-step approach to fostering the motivation of students to engage in argumentation is presented. It includes several rounds of the logical game Uluru, puzzles in the logical game environment, and puzzles in the area of calculus that aim at finding the graph of a function based on certain requirements. The research was conducted in a regular mathematics class with 15-16 year-old students in their tenth year of education (E-Phase) at a German high school (Gymnasium) over the course of five double lessons on four school days. Results are based on students' responses to three questionnaires at different points of the study. Figure 1 shows a timeline of the intervention. Step 1: Playing Uluru

In the first step of the intervention, the game Uluru (by Lauge Luchau, published by Kosmos) was presented to the students. In Uluru's scenario, animals in Australia transform into dream birds of different colors (white, yellow, orange, pink, red, green, blue and black) at night and fly towards Uluru (Ayers Rock). There are exactly eight spots around the Uluru on which birds may be placed. Each player receives bird tokens in the eight different colors and a game board. At the beginning of each round, wishes are randomly generated from a set of cards to guide the placement of the dream birds. The wishes correspond either to positions around the Uluru (e.g. "on the short edge") or to positions relative to birds of other colors (e.g. "next to the green bird"). Situations can arise in which not all wishes can be fulfilled. All players simultaneously try to place their birds according to the wishes within a set time limit. When the time has run out, points are given to the players according to the number of birds correctly placed. The game is played in several rounds.

The types of arguments generated in Uluru are very similar to typical argumentation patterns in mathematics. When evaluating the positions of the birds that each player has found, it is easy to tell whether the requirements given by the wish for a certain bird have been fulfilled or not, and the only possible results of this evaluation are true or false. Furthermore, the game naturally leads to questions characteristically mathematical, such as: "(Why) is this the only possible constellation?", "Is it possible to fulfill all wishes?", or "Why can't all conditions be met at the same time?". Thus, the game naturally provokes argumentation between players at the end of each round, especially in situations in which no player found a solution yielding eight points or when different solutions occur.

Step 2: Solving Uluru puzzles

For the second step of the intervention, I created eight different puzzles in which different wish constellations were depicted. Students were given the task to figure out the maximum of achievable points for each situation. Two of the puzzles could be solved in exactly one way, two could be solved in different ways, three puzzles had a maximum of seven possible points, and in one puzzle six points were the maximum. The students were asked to record and justify their solutions on a protocol sheet. They were allowed to use the game board and the bird tokens to find a solution. In the example puzzle shown in Figure 2, three wishes (white, pink and green) refer to positions around the Uluru and five wishes (yellow, orange, red, blue and black) refer to birds of other colors. This puzzle has a unique solution. Due to the wishes of the red snake and the black emu, red and yellow need to be placed on adjacent spots and the black bird must be placed opposite the red bird. The blue bat's wish means that black and blue need to be on adjacent spots. The orange kangaroo's bird must be placed opposite the blue bird and thus on the same side as yellow and red. There is only one side of the Uluru that has three spots (cf. Figure 3). The wishes of the yellow dingo (shared corner with pink) and the pink lizard determine the corner in which these two dream birds must be placed.

Combined with the already discussed wishes of the other animals, there remains only one spot of the places defined by the wish of the white echidna. The green bird token takes the remaining spot.

The students were allowed to work on these puzzles with a partner. They handed in their protocols at the end of the lesson. The examination of students' answers showed that while they were almost always capable of finding the best possible solution, many students had problems with justifying their answers. Before resuming work on Uluru puzzles in the next lesson, a plenary phase was initiated in which proof by exhaustion of cases and proof by contradiction were discussed in the context of Uluru puzzles.

Step 3: f(u)-luru puzzles

In a third step, the concept of f(u)-luru puzzles was introduced. In these puzzles, the animals of Australia wished for properties of the graph of a function. The puzzles followed the game design. To develop the graph, the students received a laminated coordinate system in a design inspired by the game board, as well as removable foil pens. For this puzzle, a maximum of 7 points can be achieved. The wish of green combined with the wishes of orange, white and black provide additional points and another minimum for . The blue and red wishes define the y-intercept as a local maximum or minimum. As the black and green wishes define two minima, pink's wish determines that the yintercept must be a maximum. This contradicts yellow's wish . Thus, not all conditions can be fulfilled simultaneously. As seven is the second largest number of achievable points, this is an ideal solution to this puzzle. Figure 4 shows the solution. To help students determine possible solutions and formulate arguments, they received an overview which served to clarify and simplify the possible wishes in the f(u)-luru puzzles and to make the mathematical task more playful. Figure 5 shows an excerpt of the overview. The students also received protocol sheets in a design analogous to the protocol sheets for the Uluru puzzles to document their solutions. Like in the Uluru puzzles, the students were again allowed to work in pairs. The puzzles did not ask the students to find an algebraic solution (no functional equations were expected from the students).

From game to graph: A three step approach

To assess motivation, the Intrinsic Motivation Inventory (IMI, McAuley et al., 1989) was used. It is a suitable tool to measure motivation based on the needs postulated by self-determination theory. The questionnaire consists of 18 items and covers the dimensions interest-enjoyment (int-enj), perceived competence (perc-comp), effort-importance (eff-imp) and tension-pressure (tens-pres). For each item, agreement or disagreement was measured on a 7-point Likert-scale. Assessments took place at three points during the intervention. All examined tasks target solving a problem involving a variety of conditions that require reasoning and justifying. The students answered to the questionnaires using a pseudonym, allowing them to freely utter criticism (the researcher being their teacher).

Questionnaire 1 (Q1) was given to the students before the intervention to act as a point of comparison for later responses (cf. Field & Hole, 2003, p. 68). It was coupled to a task from the math textbook in which the graph of a function was to be created from certain requirements. This task had briefly been discussed in class in a lesson before the intervention. Q1's purpose was to facilitate capturing students' attitudes towards tasks in the regular mathematics classroom involving reasoning.

The second questionnaire (Q2) was given after the students had played the game Uluru and after they had been working on Uluru-puzzles for approximately one hour. I decided to forego an assessment of motivation after playing so as not to overstrain students with too many questionnaires.

The students responded to the third questionnaire (Q3) after approximately one hour of f(u)-lurupuzzles. This last questionnaire also covered some free text questions and items focusing on an evaluation of the intervention by the students, which were not statistically validated and can thus only give an indication concerning students' attitudes towards the intervention.

The setting of the evaluation is as a repeated measures design (Field & Hole, 2003, p. 183ff) with interdependent data, which necessitates an analysis of variance [START_REF] Rasch | Kapitel 7: Varianzanalyse mit Messwiederholung[END_REF]. Nineteen students participated in all three questionnaires (n=19). Shapiro-Wilk tests allow the assumption of normally distributed results for all dimensions except tension-pressure (p<0.05) in Q2. Levene's test allows to assume homogeneity of variances except in the dimension of perceived competence.

Results

The needs dimension of interest-enjoyment is a measure of perceived intrinsic motivation. Results for this dimension are thus very interesting, as they refer directly to how students perceived the tasks. A non-significant Mauchly-test and the clearly significant analysis of variance justify the use of t-tests. Bonferroni-corrected values show significant (

) differences between Q1 and Q2 and between Q2 and Q3. The boxplot in Figure 6 shows a positive shift of means from the first to the second assessment 5.51; =0.83. The third mean 3=4.39; =1.06 lies between the others. Interest and enjoyment decreased in f(u)-luru puzzles compared to the Uluru puzzles. However, the histograms of questionnaires 1 and 3 (Figures 7 and8) also show a considerable decrease of very negative evaluations (scale range 1-2). These results can be interpreted as a positive tendency that students preferred the f(u)-luru puzzles over classic mathematical tasks. However, this tendency needs to be treated with care as working arrangements were different for the mathematical task and for the f(u)-luru puzzles.

For the dimension of perceived competence, homogeneity of variances cannot be assumed. A closer look at the data shows that this can be explained by the high mean and small standard deviation in Q2. A comparison of means of questionnaires Q1 , Q2

0.77, and Q3 3=4.4; =1.75 shows an increase in perceived competence for the Uluru-puzzles, followed by a decreased for f(u)-luru; the pairwise differences between Q1 and Q2 and Q2 and Q3 are significant (p<0.05). The means for perceived competence did not increase significantly between Q1 and Q3. However, as in the dimension of interest-enjoyment, the distributions of students for the lower scores are instructive. In the classic mathematical task covered in Q1, eight students perceived themselves as scarcely competent (scale range 1-2), whereas only two students gave this response in Q3 for the f(u)-luru puzzles. Taking into consideration that the latter were comparatively harder, this can be taken as an indication that students who otherwise trust little in their abilities might have benefited from the intervention.

Results for effort-importance show significant differences ( ) between Q1 and the means of Q2 and Q3 ( Mauchly-test , ANOVA

). The students appear to have put greater effort into both, solving Uluru-puzzles and f(u)-luru puzzles. These results could point to an increase in students' readiness to get deeply involved in a given task.

Results do not allow to assume normally distributed results for the dimension of tension-pressure in Q2. However, this dimension was rated low in all assessments ( . The students apparently did not perceive themselves under pressure during the intervention.

The students were asked in Q3 to comment on whether they had liked f(u)-luru puzzles better than classic problems in mathematics. Some notable answers:

I liked the f(u)-luru puzzles more, because is was possible to visualize and understand better.

Yes, because it was something different. Not so "dry", but another method to do math problems.

I liked it better because for some reason I was more motivated.

The majority of students responded positively. Only one student replied: "I personally liked the Uluru puzzles better, because you didn't need to show mathematical understanding like in the f(u)luru puzzles. The f(u)-luru puzzles were just as tiresome as normal mathematical problems." While this answer shows that the intervention did not manage to motivate all students equally to engage in mathematics, it underlines the positive effects of the second step of the intervention.

In addition to the IMI-questions, Q3 also covered items aimed at a more direct evaluation of the intervention, which were rated on the same 7-point Likert scale. The students' answers to these questions show a positive evaluation. Table 3 shows the items and the means.

Item mean

I had fun playing Uluru (the game) 6.65 I had fun solving Uluru puzzles 6.15 I had fun solving f(u)-luru puzzles 4.9 I was motivated by the game Uluru to make an effort in Uluru puzzles 5.55 I was motivated by Uluru puzzles to make an effort in f(u)-luru puzzles 4.53

Discussion

The three-step intervention described in this paper covered only a brief timespan of five double lessons, so only small effects may be expected. Furthermore, different working arrangements at different stages of the intervention must be considered when examining the results. However, the results presented here may be taken as an indication that game-based approaches may have motivational benefits for students. The quantitative evaluation of students' answers yields a significant and noticeable increase of motivation for the Uluru-puzzles. While results for the f(u)luru puzzles are not as unequivocally positive, the distribution of student responses compared to their assessment of the mathematical task in the dimensions of interest-enjoyment and perceived competence allows for the tentative assumption that the f(u)-luru puzzles are more likely to get a larger number of students involved in mathematical thinking than classic problems. Further research is needed, both to determine the effect of the different conditions in this study (e.g. working arrangements, difficulty of the tasks in Q1 and Q3) and to look at potential long-term effects.

While this paper shows potential motivational benefits of game-based approaches, this study did not include an evaluation of students' argumentation quality in the Uluru puzzles or in the f(u)-luru puzzles. Therefore, it is unclear whether the intervention managed to actually improve mathematical argumentation skills. This question needs to be tackled in future research. For an evaluation in this regard, a consideration of Hintikka logic can prove helpful [START_REF] Soldano | Learning with the logic of inquiry: game-activities inside Dynamic Geometry Environments[END_REF].

Interventions like the one presented in this paper can be first steps towards a better integration of more students into argumentation discourse in the classroom. Observations in class and student responses to Q3 show that students worked on all given tasks in a highly concentrated way, which naturally included arguments and justifications. Clearly positive reactions of students to the game and the game-based puzzles show that most students like to engage in logical thinking. Exploiting logical games and puzzles as a stepping stone for more students to get involved in mathematical argument seems a promising path towards more equally distributed student participation and might provide a means of tackling existing gaps in students' access to argumentation discourse.

Fig 1 :

 1 Fig 1: Temporal overview of the study

Fig 2 :

 2 Fig 2: An example for an Uluru-puzzle (colors annotated)

Fig

  Fig 3: Unique solution

Fig

  Fig 6: Boxplot int-enj for Q1-Q3 Fig 7: Histogram int-enj Q1 Fig 8: Histogram int-enj Q3

Table 1 Conditions in the f(u)-luru-"lightning puzzle" (simplified design)

 1 Table1gives an example for conditions of a puzzle.

	white	pink	yellow	orange	red	green	blue	black
		exactly 1				symmetrical	minimum
		maximum				to y-axis	for

Table 3 Evaluation items in questionnaire 3 and their means

 3


Kerstin Hein
email: kerstin.hein@math.tu-dortmund.de

The interplay of logical relations and their linguistic forms in proofs written in natural language

Keywords: Formal proof, logical relations, Toulmin model, linguistic analysis

The Toulmin model and the systemic functional grammar are combined to analyse logical relations and their linguistic forms in students' written proofs for identifying obstacles and possibilities to foster the understanding of proofs. The qualitative analysis of 63 students' products reveals a parallelism between syntactical and content-related explications and condensations. In particular, the use of conjunctions seems to support more options 1.) to make explicit logical relations between premise and warrant or conclusion, 2.) to combine several steps of a proof, and 3.) to recycle conclusions as new premises. The logical relation from the warrant to the conclusion is often only made explicit using causal prepositions as linguistic condensed forms of relations.

Introduction

There is still a lack of knowledge about the concrete language demands for specific topics, although the importance of identifying academic language and the epistemic role of language has been explored (e.g. [START_REF] Schleppegrell | The language of schooling: a functional linguistics perspective[END_REF]. For these reasons, a linguistic analysis of the classroom discourse to support teaching-learning processes should be pursued for several subjects and topics, in particular mathematics [START_REF] Schleppegrell | The Linguistic Challenges of Mathematics Teaching and Learning: A Research Review[END_REF]. Especially, for the challenging topic of proof and proofing an analysis of students reasoning is important (e.g. Mariotti, Durand-Guerrier, & Stylianides, 2018, p. 80), in particular, the analysis of logical rules and their linguistic forms, which cannot be translated from formal language directly (Durand-Guerrier, 2004, p. 2). Within the logical structures, the "implicit logical relationships" (Schleppegrell, 2007, p. 141) in their linguistic forms are one of the major challenges in proof. Because of the high density of academic language, it is demanding for the students to be aware of and to understand the academic language. Therefore, it is important to unpack the meanings in more explicit language [START_REF] O'halloran | Classroom Discourse in Mathematics: A Multisemiotic Analysis[END_REF]. The suggestion for teachers is to be explicit and to hold syntactic control within their own language during teaching logic [START_REF] Durand-Guerrier | Examining the Role of Logic in Teaching Proof[END_REF].

As a first step, a linguistic analysis is needed for analysing the language demands of logical relations. The structural and linguistic analysis of students' products in this study follows the above mentioned, general suggestions to identify language demands in logical structures, here for logical relations. It pursues three research questions: 1) How can logical relations and their linguistic forms be analysed? 2) Which pattern can be seen in the interplay between the logical relations and their linguistic forms? 3) Which linguistic forms of logical relations can be used to make logical relations explicit? The first two sections present the theoretical background and methodology of the analysis. The outcome of the qualitative analysis of the written proofs is presented afterwards illustrated by case studies of two texts.

Theoretical background: Logical relations and their linguistic forms

In line with [START_REF] Mesnil | A reference for studying the teaching of logic teaching in high school in France: A complex request for teachers[END_REF], the interaction between logic and language and the importance of explicitness of the language in teaching and learning proof is assumed. This study focuses on the logical relations as one challenging part of the language of proof [START_REF] Schleppegrell | The Linguistic Challenges of Mathematics Teaching and Learning: A Research Review[END_REF]. For this reason, the study aims at identifying the linguistic forms of logical relations.

Logical elements

For the analysis of proofs, the Toulmin model ( 1958) is often applied, although it was developed to describe everyday argumentations. Within mathematics education the structural analysis with the Toulmin model was also applied to analyse mathematical classrooms [START_REF] Krummheuer | The Ethnography of Argumentation[END_REF], the structure gap between argumentation and proofs [START_REF] Pedemonte | How can the relationship between argumentation and proof be analysed[END_REF], and for proofs with several steps (e.g. [START_REF] Knipping | Reconstructing Argumentation Structures[END_REF], even if there are limits for the analysis of proofs and proofing (also discussed in Mariotti et al., 2018, p. 78). In the structure model of Toulmin, the function of the logical elements as premise etc., not their relations, is crucial. This becomes more important when it is applied for the logical structure analysis of proof. This article refers to the short versions of Toulmin's model and considers premise, warrant and conclusion as the relevant logical elements within deductive steps. These logical elements are connected more or less implicitly by logical relations, which need to be unpacked, here.

Logical relations

In this article, logical relations are understood as relations between logical elements. In deductive proofs, the relations between premise, warrant and conclusion are crucial (as described by Duval 1991, p. 235), but also within a warrant with the logical form of implication or equivalence [START_REF] Selden | Unpacking the logic of mathematical statements[END_REF]. In particular, the logical relation from premise to warrant is crucial for the verification of the premises [START_REF] Duval | Structure du Raisonnement Deductif et Apprentissage de la Demonstration[END_REF]. These logical relations are often implicit in the language, creating an obstacle for students [START_REF] Schleppegrell | The Linguistic Challenges of Mathematics Teaching and Learning: A Research Review[END_REF] as they have to be unpacked in order to understand them [START_REF] Selden | Unpacking the logic of mathematical statements[END_REF].

Different linguistic forms

Logical relations are often expressed in natural language by logical connectives such as "because", "due to" or "if…, then…." [START_REF] Clarkson | Researching the language for explanations in mathematics teaching and learning[END_REF]. These logical connectives can be classified as causalconditional principal markers by the systemic functional grammar of [START_REF] Halliday | Introduction to functional grammar[END_REF], describing language from a functional perspective. They have different linguistic forms such as causal conjunctions ("because"), conditional conjunctions ("if... then….") or causal prepositions ("due to"). However, there is still a research lack of the analysis of the combination of logical relations and their linguistic forms (e.g. [START_REF] Durand-Guerrier | Examining the Role of Logic in Teaching Proof[END_REF][START_REF] Schleppegrell | The Linguistic Challenges of Mathematics Teaching and Learning: A Research Review[END_REF].

Methodology for the analysis of written proofs

Data collection

The students' texts of written proofs have been generated within a teaching-learning arrangement on deductive reasoning in grade 8-12 all with angle sets (Hein & Prediger, 2017;Prediger & Hein, 2017). In the teaching-learning arrangement conjunctions were used to express logical relations. The students were also asked to reason why a theorem can be applied (premises are met). In this If there is a triangle, then these three angles in total measure 180 degree. (Line k which is parallel to the side AC can be used as an auxiliary line). paper, the learning process and the effects of the teaching-learning arrangement are not analysed. Instead, it focuses only on the written products of the teaching-learning arrangement in order to investigate the interplay between logical relations and their linguistic forms. The data corpus consists of 63 written texts from 48 students (20 in grade 8, 6 in grade 9, 4 in grade 10, 18 in grade 12). The result section presents the results of two cases with texts of the twelfth graders Linus and Petra.

Methods for qualitative data analysis

The qualitative analysis of logical relations and their linguistic forms in the written products combines two analysis models:

(1) Toulmin model: The Toulmin model in its short version ( 1958) is applied for the logical structure analysis to identify the addressed logical elements (premise, warrant, conclusion) [START_REF] Pedemonte | How can the relationship between argumentation and proof be analysed[END_REF] and to disentangle the several steps of the proof [START_REF] Knipping | Reconstructing Argumentation Structures[END_REF]. ( 2) Systemic functional grammar: The linguistic analysis of the logical connectors as language means for logical relations draws upon systemic functional grammar [START_REF] Halliday | Introduction to functional grammar[END_REF], which can also be used to identify linguistic challenges in mathematics education [START_REF] Schleppegrell | The Linguistic Challenges of Mathematics Teaching and Learning: A Research Review[END_REF]. The analysis approach systematically identifies lexis used for logically connecting sentences or elements within sentences and classifies their syntactical forms as conjunction (con) respectively prepositions (pre). For these, the English functional grammar (in which the products are presented here) and the German functional grammar (the original language of the products) resonate with each other. For example, causal conjunctions can be described in English and German with conjunctions and prepositions and function in a similar way. Here, first the existence of a logical relation as links between logical elements and then their grammatical form (conjunctions such as "therefore" or preposition such as "according to") are identified.

Empirical insights into the cases of Petra's and Linus' written proofs

Petra's and Linus' texts were chosen for illustrating the results because their texts contain typical linguistic forms for the logical relationscondensed and non-condensedwhich were also found in many other written proofs about angle sets. The students work on the proof for the sum of angles in a triangle (Figure 2). Before starting to write individually, they discuss the proof, identify which theorems have to be applied and draw the sketch printed in Figure 3 to which both texts refer.

Analysis of activated mathematical statements: Both students refer to the same mathematical statements, the alternate interior theorem (Step 1 and 2), supplementary angle theorem (Step 3), calculating with angles theorem (Step 4), but both texts show substantial differences with respect to making the logical relations and their linguistic forms explicit. Presentation of the structural and linguistic analysis: The logical elements are illustrated with colours (green: premises (P); blue: warrant (W), orange: conclusion (C)), both in the text and the graphical representations. Only partly explicated logical elements are illustrated with * in the text and with dashed boxes in the graphics.

The logical relations (R) (conjunctions (con), prepositions (pre)) are illustrated in black and by arrows in the graphical presentations (continuous arrow for conjunctions, dashed arrow for prepositions).

Linus' text with causal prepositions for the logical relations

Step The detailed explicated premises (without parallelism) are not verbally connected with the warrant.

With "According to" the warrant (alternate interior angle theorem) is connected with the conclusion. "According to" is a preposition within one sentence.

3

According to (R/pre) the supplementary angle theorem (W), the angles β, π and δ sum to 180 degree (C).

(German Original: Gemäß des Nebenwinkelarguments sind die Winkel β, π und δ zusammen gleich 180 Grad groß.) "According to": analysis analogue to Step 1+2 4 π can be substituted by α and δ can be substituted by γ (P*: Conclusion of step 3 as new premises is missing.).

According to (R/pre) the calculating argument (W), the interior angles α, β and γ have a measure of 180 degree (C).

(German Original: Für π kann α eingesetzt werden und für δ γ eingesetzt werden. Gemäß des Rechenarguments sind die Innenwinkel α, β und γ gleich 180 Grad groß.)

Linguistically, it is not marked as previous conclusion of Step 1+2 (implicit recycling). Conclusion from Step 3 is not explicitly used. "According to" connects the theorem with the conclusion with a preposition.

Table 1: Analysis of Linus' text

In Linus' first sentences, the premises for the Steps 1, 2 and 3 are almost explicated. In further sentences, the warrants and the conclusions for Step 1, Step 4 is not made explicit. However, logical relations between these elements are only explicated from the warrant to the conclusions, whereas the logical relations from the premises to the warrants or to the conclusions are not explicated. In each occurrence of logical connectives, Linus activates prepositions such as "according to". With "By the fact that…", the conclusion from the premise is drawn. "By" is a preposition. The premises are condensed to an object with "as a fact" which is explained in a sub-clause. "According to" (preposition) is used at the end of the sentences to add the logical relation from the warrant to the conclusion. 4 So, if (R/ con) β+π+δ= 180 degree and α=π and γ=δ (P), then (R/ con), according to (R/ prep) the calculating argument (W), β+γ+α= 180 degree (C).

(German Original: Wenn also β+π+δ= 180 Grad und α=π und γ=δ, dann sind nach dem Rechenargument β+γ+α= 180 Grad.)

With "If…, then" (conditional conjunction) the warrant is applied to the case. With "so" there is a reference to the previous conclusion, which is used as new premise (recycling). For "According to…", see Step 3.

Table 2: Analysis of Petra's text

In her text, Petra addresses the premises, the warrant with the name of the theorems and the conclusions as Linus. For making the logical relations explicit, she connects the premises with the warrants or the conclusions with causal conjunctions such as "because" and "therefore". In one case, Petra connects the premise with the conclusion by a preposition (Step 3). Petra also uses prepositions such as "due to" by adding the expression "due to the ...-argument" as last part of the sentences after connecting the premise to the conclusion (Step 3 and 4).

Comparison of the case studies of Linus and Petra

In both texts, almost all contents of the elements of the short Toulmin Model are explicated, even Linus only mentions the premises partly at the beginning, respectively not completely in Step 4. The conclusions as new premises in Linus' example are not explicated in their function, in particular, the logical relations from the premises to the warrant or conclusion. The linguistic analysis shows that causal conjunctions such as "because" and "therefore" of logical relations are only used in Petra's text, where also the logical relations from the premises to the warrant respectively conclusions and the recycling of previous conclusions are expressed. These findings are in line with those of other written products (see Prediger & Hein, 2017). According to [START_REF] Halliday | Introduction to functional grammar[END_REF] conjunctions are non-condensed forms for relations as in everyday language (called by him as coherent). In this case, these conjunctions were also used to make explicit the logical relations from the premises to the warrant or from the premises to the conclusion. With causal prepositions such as "according to", "due to" or "by", virtually nothing but the logical relations between the warrant and the conclusion are made explicit. This was also found in other texts (Prediger & Hein, 2017). Only in Step 3 of Petra's text, the relation from the premises to the conclusion is made explicit with a preposition ("by the fact that…"). Here, the content of the premises is expressed by a sub-clause. One reason for virtually explicating nothing but the logical relation from the warrant to the conclusion (exception: Step 3 in Petra's text) may be that prepositions need nominalizations and here only the warrant is condensed to a nominalization by its name and can be easily integrated by a preposition ("due to the …-theorem"). adding the phrase with the preposition ("…. according to the …-argument.") at the end of the sentences (such as in Step 3 and Step 4 of Petra's text), the logical relations from the premises are also made explicit in the same sentence. In other cases, in sentences with prepositions only the logical relation from the warrant to the conclusion is expressed. Prepositions are linguistically condensed forms of logical relations (called metaphorical by [START_REF] Halliday | Introduction to functional grammar[END_REF]. This kind of linguistic phenomena is one of the most important characteristics of academic language to increase lexical density. However, these condensed forms (e.g. prepositions for logical relations) are challenging for students [START_REF] Martin | Mentoring semogenesis: 'genre-based' literacy pedagogy[END_REF] and have to be unpacked first into non-condensed forms before students can understand their meaning (O'Halloran, 1998, p. 382). In the context of this case study, mainly the logical relations between warrant and conclusion are articulated by condensed forms of prepositions instead of conjunctions. The qualitative analysis shows that the prepositions provide not only challenges to understand the logical relations, they also seem to hinder the explication of logical relations, from the premises to the warrant or to the conclusion or within a theorem. Prepositions also seem to be obstacles for combining several logical elements and steps.

Conclusion

The analysis of students' products reveals three main findings: 1) The Toulmin model has not only limits for analysing proofs in general (Mariotti et al., 2018, p. 78), but in particular, for capturing the logical relations. 2) The linguistic analysis here suggests that in non-condensed forms some aspects are more often expressed than in condensed forms. This finding can be used to be more explicit on language while teaching proof as [START_REF] Mesnil | A reference for studying the teaching of logic teaching in high school in France: A complex request for teachers[END_REF] has recommended. 3.) For these reasons, it might be useful to first offer non-condensed linguistic forms (conjunctions), before condensed forms (prepositions) are used in the classroom. This approach resonates with [START_REF] Martin | Mentoring semogenesis: 'genre-based' literacy pedagogy[END_REF] observations regarding the challenging condensed forms in academic language.

Of course, this case study has significant methodological limits as only 63 texts were analysed, that were produced within the specific setting and content. It has not yet been taken into account how the setting and the content influence the students' articulations. Future research is required to overcome these limits. Furthermore, the texts are written in German and not in English, so every interpretation can only be made in the original.

  Sketch of Petra & Linus within their pair work

Figure 2 :

 2 Figure 2: Mathematical statement to be proven by Petra and Linus Figure 3: Students' joint sketch

Figure 1 :

 1 Figure 1: Analysis tool with Toulmin for several steps and both linguistic forms for logical relations

  Translated (and original) text Structural and linguistic analysis 1+2 The triangle ABC has the angles α:= CAB; β:= ABC, γ:= BCA. Additionally, a line passes through point B. Additionally, the angle π is located at line k and at and δ is located at line k and the side . (p*: Parallelism of k to BC is missing.) According to (R/pre) the alternate interior angle theorem (W), the angles α and π have the same measure and δ and γ have the same measure. (C) (German Original: Das Dreieck ABC hat die Winkel α:= CAB, β:= ABC, γ:= BCA. Zusätzlich geht durch den Punkt B eine Gerade. Zusätzlich liegt der Winkel π an der Geraden k und an und δ liegt an der Geraden k und an der Seite . Gemäß des Wechselwinkelarguments sind die Winkel α und π gleich groß und δ und γ gleichgroß.)

Figure 4 :

 4 Figure 4: Summary of the analysis of Linus' text

3

  with α, β and γ is given. The line k is parallel to line k and crosses point B. To identify the measures of the angles α and γ (crossed out), at the point B to the angle β the angles π and δ are defined (P). Because (R/con) the line k and the side AC are parallel and are crossed by the side AB (P), the angle theorem (W) can be applied. Therefore, (R/con) α = π (C). (German Original: Es ist ein Dreieck mit α, β und γ gegeben. Die Gerade k ist parallel zu der Geraden k und schneidet den Punkt B. Um die Größe der Winkel α und γ (durchgestrichen) zu erfahren, definiert man an dem Punkt B zu dem Winkel β die Winkel π und δ. Da die Gerade k und die Seite AC parallel sind und durch die Seite AB geschnitten wird, kann man das Winkelargument anwenden. Daraus folgt, dass α = π.) "Because… can be applied" connects the premise with the theorem. "Because" is a causal conjunction (within one sentence). "Therefore" connect the warrant (here angle theorem) with the conclusion. "Therefore" is a causal conjunction and connects two sentences. 2 Additionally, the side BC crosses the parallel line k to the side AC (P), therefore (R/con) γ = δ (C) reasoned by (R/prep) the alternate interior angles theorem (W). (German Original: Außerdem schneidet die Seite BC die parallele Gerade k zu der Seite AC, deshalb ist γ = δ begründet durch das Wechselwinkel-argument.) With "therefore" (causal conjunction), the conclusion is drawn. "Reasoned by" is used at the end of the sentence to add the logical relation from the warrant to the conclusion. By the fact that (R/prep) Ω and δ are supplementary angles (Ω = β+π) (P), β, π and δ =180°(C), according to (R/prep) the supplementary angle argument (W). (German Original: Dadurch dass Ω und δ Nebenwinkel sind Ω = β+π, sind β, π und δ =180°, laut dem Nebenwinkelargument.)

Figure 5 :

 5 Figure 5: Summary of the analysis of Petra's text

Acknowledgment

I would like to thank my supervisor Susanne Prediger for the great support and intense cooperation. The doctoral thesis on which the article is based was carried out with the financial support of the Foundation of German Business.