Tabea Knobbe
email: tabea.knobbe@math.uni-giessen.de

Students with special educational needs verbalise their calculation strategies

Keywords: Calculation strategies, language impairment, special educational needs

Introduction

Children with a language development disorder may be given a special educational need (SEN) in Germany so that they are entitled to special support. The language development disorder can manifest itself in different ways and affect various levels of language, e.g., pronunciation, vocabulary, grammar, or communication and pragmatics.

Because language is an important learning medium and subject in mathematics education, the children need special additional support in order to be able to achieve the learning goals. Empirical results show a close relationship between mathematics performance and language skills, for example when children with special needs in the area of language perform worse in mathematics and have lower verbalisation skills than children without them [START_REF] Werner | Math, I don't get it": An exploratory study on verbalizing mathematical content by students with speech and language impairment, students with learning disability, and students without special educational needs[END_REF]. Verbalisation skills mean here "a specific form of linguistic expression in regards to verbal and nonverbal aspects" (Werner et al., 2019, p. 381). This verbalisation is needed, for example, in the Communication competency, which should be targeted as a process standard in mathematics instruction. Part of this competence is to be able to describe one's own approach and to discuss the tasks with others (NCTM, 2003).

A central element of primary mathematics instruction is to understand meanings of operations, to compute fluently and to use strategies for addition and subtraction tasks. This is part of the content standards in the section on Number and Operation (NCTM, 2003). In the number range up to 100, children can use calculation with whole numbers and notation to develop an understanding of arithmetic operations and number relationships. Individual steps or intermediate results can be notated, but the notation can be done variably [START_REF] Rathgeb-Schnierer | Developing flexibility in mental calculation[END_REF]. Main strategies for this informal calculation are, for example, the jump strategy, the split strategy, or modifying the task to an easier one [START_REF] Rathgeb-Schnierer | Developing flexibility in mental calculation[END_REF].

Research project

In order to present students with an explicit verbalisation occasion, they participate in a lesson in which they are to create a math audio podcast, meaning a voice recording, in which they explain their calculation strategy. The process of podcast creation is based on [START_REF] Schreiber | The use of artifacts and different representations by producing mathematical audio-podcasts[END_REF].

First, the children solve a calculation task on their own and write down their calculation path. Afterwards, they discuss the calculation with a partner and write a script together, which serves as a template for the following speech recording. In an editorial meeting the first podcast version is discussed with the teacher. This is followed by a revision of the script and the final podcast recording.

During the production of such an audio podcast, students have to act in different communicative situations. The calculation path is described in writing, orally negotiated with a partner, and recorded as prepared speech. Lessons become language sensitive by using language aids and mathematical tools, to help them solve the tasks and to support in communication and argumentation.

The students are filmed during the lessons. The interactions are then transcribed and evaluated with the help of interpretative classroom research [START_REF] Krummheuer | Grundlagen und Beispiele Interpretativer Unterrichtsforschung [Basics and examples of interpretative classroom research[END_REF]. Of particular interest are the phases in which the students are negotiating a lot, for example when they write their script together. The aim is to reconstruct how the students verbalise their calculation paths. The poster will provide an overview of the research design and present an example for a verbalization.

Research interest

In mathematics education research, the connection between language and mathematics learning is an intensively researched field (for example see TWG09 of CERME12, [START_REF] Ingram | Introduction to the work of TWG09: Mathematics and language[END_REF]. In this research project, the focus will be especially on students with a SEN language and how they verbalize calculation paths to tasks that suggest an informal strategy. Notations on their pathways are very welcome. The aim is to gain insight into the linguistic and mathematical learning of children with language impairments and to reconstruct connections between the linguistic expression and the mathematical understanding that emerges. Of particular interest is how the transition of tens is mastered, how the operation (addition or subtraction) is understood, and which means of expression are used.


Bea Kristinsdóttir
email: bjarnhek@hi.is

Silent video tasks in pre-service mathematics teacher training

Keywords: Classroom discussion, pre-service teacher education, silent video tasks

Teaching mathematics in multilingual diverse settings

Even though majority of pre-service teachers might be aware of diversity (backgrounds different from the dominant culture, e.g., cultural and linguistic), still, most of them do not transfer their awareness into diversity-aware teaching strategies or approaches -ways that "combine awareness of differences among pupils, appreciating those differences, and taking actions to promote equity in classrooms" [START_REF] Hummel | Diversity-aware teaching of mathematics: An explorative study on pre-service teachers' views[END_REF]. For example, in the case of second language learners in mathematics, one such teaching strategy involves building norms in the classroom such that students are expected to and supposed to explain their thinking and thus to actively engage in a joint production of mathematical explanations and solutions [START_REF] Barwell | Learning mathematics in a second language: Language positive and language neutral classrooms[END_REF]. In other words, teachers would be required to facilitate classroom discussion. It is very challenging for most teachers and teacher students to orchestrate productive mathematical discussions (Smith & Stein, 2018, p. 54 and p. 70). For professional training in orchestrating group discussions, [START_REF] Smith | 5 practices for orchestrating productive mathematics discussions[END_REF] have described five practices of anticipating, monitoring, selecting, sequencing, and connecting students' thoughts. Building on these practices, Kooloos (2022) collaborated with a group of Dutch in-service teachers, supporting them in developing their practice of teaching through mathematical whole-class discussions. To do this, teachers videorecorded their own practice and collectively reflected on their lessons with the researcher. This poster (tiny.cc/BK23) suggests that teacher educators might use silent video tasks to approach the task of supporting teacher students in their first steps toward orchestrating whole-class discussions in mathematics. It reports on how the variety of responses to a silent video task gave rise to discussion on diversity and the role of language and mathematical discussions in an international pre-service teacher training. It raises the question whether silent video tasks might be a useful tool in pre-service teacher courses for teacher educators who wish to raise awareness of linguistic diversity and to illustrate the importance and complexity of leading a group discussion in mathematics class.

Silent video tasks as a tool for training orchestration of mathematical discussions

A silent video task revolves around a short, animated video clip with no sound or text, focusing on a specific mathematical topic that is displayed for the whole class before a link is distributed to students for repeated viewing. In pairs, students prepare and record their description, explanation, and/or narration to the video and the teacher leads a group discussion based on students' voice-over recording responses [START_REF] Kristinsdóttir | Silent video tasks -their definition, development, and implementation[END_REF]. Such a task might be implemented in secondary school classrooms either before starting a new topic, to gather students' pre-ideas about the topic, or toward the end of working on a topic, e.g., with the aim to develop common understanding of the mathematical topic at hand. In a pre-service teacher-training program, within the frame of a 90-minute-long workshop (taught in English), participants (N=22) from Germany, Greece, Italy, and Portugal were introduced to silent video tasks and given the opportunity to experience a silent video task in the student role by recording a voice-over in English for a short (1 m 33 s) animated video. The selected video illustrates the property described by Viviani's Theorem that the altitude of an equilateral triangle equals the sum of the distances from any interior point to the sides of the triangle (see youtu.be/-nUDBmplu_Y). Immediately after handing in their voice-over recordings (11 in total), they were listened to in no particular order, collecting concepts and phrases on the blackboard in such a way that equivalent or related words were grouped together. For example, we noticed that some recordings mentioned perpendicular, other orthogonal, one vertical, and one with a 90°angle when referring to the line segments that showed the distance from the interior point to the sides of the triangle.

The diversity of responses seemed to catch students by surprise and their attention certainly was brought to the importance of language and discussion. Ways to link the recorded responses to each other and to reach common understanding of the mathematics displayed in the video were discussed. For example, we discussed our perceived transparency of words, how this type of task might offer opportunities for second language learners to enter the classroom discussion, and questions that we might pose, such as "Can you tell me in other words what this group focused on in their recording?"

Conclusion

One way to introduce the practice of facilitating mathematical discussions is to put teacher students in the role of pupils like what was described above. Pair discussions and task responses could be recorded and transcribed, and students asked to reflect on what questions they might have posed to pupils during classroom discussion. In a linguistically homogeneous group of teacher students, they could be provided with and asked to plan whole-class discussion based upon recordings or transcripts of responses from secondary school pupils (including multilingual pupils). This poster illustrates in text and simple comics drawings how and why silent video tasks might be a tool for raising teacher students' awareness of diversity regarding language and access to the classroom discussion.


Katri Luomaniemi
email: katri.luomaniemi@utu.fi

Andreas Gegenfurtner

Sanni Kankaanpää

Jake Mcmullen

Minna Hannula-Sormunen

A meta-analytic review on early mathematical interventions for multilingual children -Preliminary results

Keywords: Early mathematical skills, multilingualism, meta-analytic review

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

Introduction

Multilingualism is a common phenomenon in today's early childhood education and care (ECEC). However, there is little evidence of how to best support multilingual children's early mathematical development, a crucial predictor of later life outcomes. This is problematic because multilingual children risk falling behind their peers in mathematics (OECD, 2016). Multilingualism per se is not a risk factor in learning early mathematics. However, low proficiency in the language of instruction in school or ECEC settings seems to be associated with lower achievement in mathematicsespecially for children from low-income families [START_REF] Sarnecka | Early number knowledge in dual-language learners from low-SES Households[END_REF]. The current study aims to provide a better basis for developing ECEC mathematics support for multilingual children by systematically reviewing existing empirical evidence of mathematical intervention studies for multilingual children aged 3 to 5 years. For the purposes of this meta-analytic review, we use the term multilingual as it encompasses all the definitions of a child learning two or more languages in their everyday life [START_REF] Grosjean | Bilingualism: A short introduction[END_REF].

Methods

We conducted a meta-analytic review of early mathematical intervention studies within multilingual children following the PRISMA statement protocol [START_REF] Page | The PRISMA 2020 statement: An updated guideline for reporting systematic reviews[END_REF] to identify, select, and critically appraise available evidence on the subject. The pool of 452 studies was narrowed down by examining the titles and abstracts of the studies that would fit the following inclusion criteria: (a) the participants in the study were within the 3 to 5 years age range, (b) the treatment group received additional mathematical instruction above their typical ECEC experience, (c) the focus of the intervention was solely on mathematical skills, (d) the intervention was implemented in ECEC settings, (e) children's mathematical skills were assessed, (f) the analyses focused on multilingual children either exclusively or as a subgroup, and (g) the study was published in English. Finally, to be included in the quantitative analysis, the study had to provide adequate data for effect size calculation. First, we reviewed the included studies by summarizing the study features, mathematical content, and intervention characteristics. Second, we performed a meta-analysis to quantitatively combine data from sufficiently homogeneous studies to determine the effectiveness of these interventions. Third, we explored the effects of three moderator variables: the use of home language, culturally responsive instructional approach, and mathematical language on children's mathematical outcomes. We selected these moderators because previous research (e.g. Celedón-Pattichis, 2018) recommended these practices for enhancing multilingual children's early mathematical skills.

Preliminary results

In total, 24 studies published in 1999-2021 met the inclusion criteria for our systematic review. The preliminary analysis of the mathematical content showed: (1) all interventions included a numeracy component, (2) seven interventions focused solely on numeracy skills, (3) 17 interventions were more comprehensive covering at least two mathematical domains. In addition to numeracy, the most investigated mathematical domains were geometry, measurement, and patterns. Eight interventions included a mathematical language component. Next, we calculated the average weighted effect size of the eight studies that had adequate data to be included in meta-analysis. The average weighted effect size with one outlier removed for these numeracy interventions was small (g = 0.27, 95% CI = 0.18, 0.37). Finally, we will conduct (a) qualitative analysis by examining the use of home language, culturally responsive instructional approach, and mathematical language in the interventions and (b) quantitative analysis on how these recommended practices are associated with children's mathematical skills.

Implications

The results add to the evidence base regarding the effectiveness of early mathematical interventions for multilingual children. We will discuss how the moderator variables that produce the largest effect sizes could be utilized in the ECEC as a resource to plan evidence-based mathematical activities that support multilingual children's learning. This research provides an important opportunity to advance the understanding of how to reduce the gap between mono-and multilingual children's early mathematical skills already in ECEC and provide children with more equal access to education. (The original poster presented in CERME13 is accessible online at https://sites.utu.fi/mathlearn/wpcontent/uploads/sites/1083/2023/09/CERME13_Luomaniemi_Meta-analytic-review_poster.pdf.)


Ludwig Laukert
email: laukert@idmp.uni-hannover.de

Thomas Hausberger

Reinhard Hochmuth

Calculus at the school to university transition: early stages of a structuralist perspective in real analysis

Keywords: Real analysis, university mathematics education, school-university transition, anthropological theory of the didactic, pre-structuralist praxeologies

Building on our prior work which described the notion of structuralist praxeology for the theory of teaching and learning of Abstract Algebra, we focus in this article on the case of Real Analysis. The theoretical perspective of the Anthropological Theory of the Didactic by Chevallard helps us to analyse the data which is drawn from an introductory course in Real Analysis at university level. We identify structuralist aspects and show how they develop into a structuralist praxeology later in the curriculum. Further, we introduce the notion of pre-structuralist praxeology as a characteristic of the School to University transition.

Introduction

The teaching and learning of Calculus at the transition between School and University has been the subject of intensive research [START_REF] Bressoud | Teaching and Learning of Calculus[END_REF]. These studies focus on core calculus concepts such as real numbers, sequences, series, derivatives, and integrals with the concept of limit as a cornerstone. In such an epistemological perspective, the modern vision of Analysis is mainly that of the turn of rigour due to Cauchy with corresponding issue in the school-university transition the implementation of proofs around formal definitions and their consequences. Winsløw (2009) also pointed out a shift towards abstraction since students gradually learn to relate properties of specific functions to general and abstract properties of arbitrary functions, expressed in conceptual terms (e.g., the existence of a maximum due to continuity on a compact interval). Using the Anthropological Theory of the Didactic (ATD) as theoretical framework and the notion of praxeology [START_REF] Chevallard | Anthropological Theory of the Didactic (ATD)[END_REF] as key theoretical construct, he proposed a three-phase model to characterise the evolution of students' activity in the analysis path at university. In a first transition, techniques learned at school are justified and unified on a theoretical level (by means of properties and theorems) and lead to the learning of "complete" Calculus praxeologies, endowed with a consistent mathematical logos (in the ATD sense). Secondly, theoretical tasks that relate to the latter logos are introduced: in this second transition, students are required to engage with generic functions and their abstract properties to produce modest theoretical developments.

The rationale of the praxeologies resulting from the second transition is not further elucidated by Winsløw. In our view, it is related to another striking feature of modern analysis that is not due to Cauchy: the rise of abstract analysis, in the avenue opened up by abstract algebra, under the influence of mathematical structuralism as a new epistemology of mathematics. This trend, led by mathematicians such as the Bourbaki group in France, resulted in a refoundation of mathematical domains around key mathematical structures. As a consequence, Real Analysis evolved into a body of knowledge called Abstract Analysis and organized around such notions as topological spaces, metric spaces or Hilbert and Banach spaces [START_REF] Hausberger | From Real to Abstract Analysis: the development of structuralist praxeologies in the analysis path[END_REF].

The didactical consequences of mathematical structuralism are being investigated in a research program opened up by [START_REF] Hausberger | Structuralist praxeologies as a research program on the teaching and learning of abstract algebra[END_REF] in the paradigmatic context of Abstract Algebra. Theoretical ideas, expressed in the language of ATD (in particular, the key notion of structuralist praxeology), nevertheless apply to any context where structures are involved. The initiating paper [START_REF] Hausberger | From Real to Abstract Analysis: the development of structuralist praxeologies in the analysis path[END_REF] aims to show the relevance of the notion of structuralist praxeology to account for the knowledge taught in Analysis courses at the transition from the Bachelor to graduate studies where Abstract Analysis lies in the horizon.

In this paper, we rather focus on the transition from school to university with a view to identifying how structuralist thinking influences the teaching and learning of Real Analysis at early stages of the journey to Abstract Analysis. The research question is: What are the structuralist aspects in the teaching and learning of Real Analysis in the first year of Bachelor courses? We begin by a description of our theoretical ideas and methodology, then present a praxeological analysis of excerpts from first-year Analysis course material. We then discuss the identified structuralist aspects and their significance with respect to the didactic issues of the school-university transition. [START_REF] Hochmuth | Transitions to, across and beyond university[END_REF] highlight the following main features of ATD that justify its frequent use in research on transitions: the consideration of knowledge as living within institutions, the institutionalisation of knowledge seen as the result of complex didactic transposition processes, and finally the central 4T-model of praxeologies (task, technique, technology, theory) that allows researchers to build reference models of the knowledge to be taught for application to teachinglearning phenomena. As these theoretical constructs (presented in a concise manner in the mathematics education encyclopaedia; [START_REF] Chevallard | Anthropological Theory of the Didactic (ATD)[END_REF] are increasingly used in university mathematics education research, we will focus now on the additional tools, specific to the point of view of mathematical structuralism, that have been developed.

Theoretical framework

The starting point is the consideration of mathematical structuralism as a methodology, which consists of reasoning in terms of classes of objects, relations between these classes and stability properties for operations on structures [START_REF] Hausberger | Structuralist praxeologies as a research program on the teaching and learning of abstract algebra[END_REF]. The general view of structures thus allows particular properties of objects to be demonstrated by making them appear as consequences of more general facts (theorems about structures). Dually, generalisations are put to the test of objects, hence a dialectical relationship between objects and structures. In praxeological terms, the structuralist method consists in the passage from a praxeology P = [T/?/?/Өparticular] where it is unclear which technique to apply, to a structuralist praxeology Ps = [T g /τ/θ/Өstructure] where, modulo generalisation of the type of tasks (T g ), the theory of a given type of structure guides the mathematician in solving the problem.

As a paradigmatic illustration in the context of Abstract Algebra, [START_REF] Hausberger | Structuralist praxeologies as a research program on the teaching and learning of abstract algebra[END_REF] analysed a thread from an online forum. In order to prove that the ring of decimal numbers is a principal ideal domain PID (T: show that a given ring is a PID), a forum participant proposes to prove that any subring of Q is a PID (T g : show that any subring of a given PID is a PID), with the (erroneous) structuralist technique τ: show that any subring of a PID is again a PID. The underlying structuralist technology is the idea that striking features of rings are preserved by taking subrings, which unfortunately is not true for being a PID (Z[X]⊂Q[X] is a counterexample). The objects-structures dialectic thus gives rise to abstract tasks t=τ that are meant to provide new powerful technologies to solve tasks of type T (here, showing that the given ring is included in a PID). In other words, the objects-structures dialectic subsumes both a dialectic of the particular and the general, and of the concrete and the abstract, which are interrelated. Furthermore, [START_REF] Hausberger | Structuralist praxeologies as a research program on the teaching and learning of abstract algebra[END_REF] distinguishes several levels of structuralist praxeologies: at level 1, structures act as a vocabulary and appear mainly through definitions (e.g., a task of type T is solved by showing, by hand, that the definition is satisfied); at level 2, the technique mobilises general results about structures (in our example, one shows the existence of an Euclidean algorithm, which invokes in the logos of the praxeology the structuralist theorem that any Euclidean ring is a PID). [START_REF] Hausberger | Structuralist praxeologies as a research program on the teaching and learning of abstract algebra[END_REF] explained how analyses of structuralist praxeologies may shed light on transition issues in the development of structuralist thinking among learners. This program was carried out in a study of course materials in Abstract Algebra throughout the Bachelor in France and Switzerland [START_REF] Jovignot | Transitions in abstract algebra throughout the Bachelor: the concept of ideal in ring theory as a gateway to mathematical structuralism[END_REF]. The present study aims at investigating similar issues in the context of Real Analysis in Germany.

Methods

To identify structuralist aspects in mathematical analysis we chose an Analysis course at a mediumsized German university that was offered during the winter semester 2021/2022. This introductory course was designed for the first year of the bachelor's program of mathematics, of pre-service teachers of mathematics, of physics and of meteorology. In this context, the students had to turn in homework assignments on a weekly basis. The problems assigned for homework were discussed in a tutorial that took place after the homework assignment was turned in by the students. In these tutorials, a document presenting the standard solution of the problems was made available to the students and to us. In the tutorials, the students were presented an additional set of problems, whose contents correspond to the problems assigned for homework, and to the lecture. This additional material as well as the textbook used for the lecture, were made available to us. This course is usually attended by students in their first year of studies. Thus, it is situated in the transition from School to University. This study is part of a research project of the first author, that aims to provide course material to support students in the transition process. This course material is going to refer to problem sets that students encounter throughout their first year of studies of Analysis. Hence, we focus on the homework assignments and the problem sets from the tutorials as well as on the solutions that are suggested by the lecturer, to conduct our praxeological analysis.

Data Analysis and discussion

The problems that we analysed reveal different structuralist aspects. There are a few problems that allowed us to conclude that their purpose is to develop a structuralist praxeology. Among them there are some problems whose potential to develop a structuralist praxeology is not used in this course. We chose two examples to illustrate how the potential to develop a structuralist praxeology is used successfully in one case, and how it is not in the other. In addition, these two examples contain some justification for structuralist aspects (e.g., particular/general, and concrete/abstract dialectics) which we could identify in many problems of the set. For each problem, we first present the task and its solution as it was suggested by the lecturer. Then the discussion will follow.

Analysis and discussion of Problem 11

Two main tasks are assigned in this problem, labelled (a) and (b). Task 11 (a) is about proving the squeeze theorem (figure 1). Since the squeeze theorem is used in Analysis to show the convergence of sequences, the praxeology P1 that gives this piece of knowledge its raison d'être can be modelled as follows:

T1: Prove the convergence of a given real sequence and determine its limit; 𝛕 1 : Find upper and lower bounds which converge to the same known limit; 𝛝1: Squeeze theorem; 𝚹1: Real Analysis. Indeed, we could find a problem in the tutorial material that corresponds to this praxeology. It also relates to problems that could be found in German Schools (approximation of the circle by polygons, integrals as areas...). By contrast, task 11 (a) would rather not appear in Schools, due to its level of abstraction (it deals with generic sequences). In praxeological terms, it aims at justifying the technology 𝛝1 of the praxeology P1, in other words at completing the School praxeology. This is an example of the first transition identified by Winsløw.

The sequence is a variable of the type of tasks T1. Several other techniques may arise depending on the value taken by this variable. By contrast, the transition from an explicitly given sequence of real numbers to a generic sequence entails a switch in the technique and logos of the resulting abstract praxeology, if task 11 (a) falls within a type of tasks -which we will defend, of a different nature. Our proposed praxeological model is indeed the following: T2: Prove that any real sequence that fulfils certain conditions is convergent and determine the limit (in terms of the conditions); 𝛕2: Apply the definition of convergence; use the conditions and standard bounding techniques; 𝛝2: Definition of convergence; properties of real numbers; 𝚹2: Real Analysis.

The formal definition of convergence is now a key element of the technology of the abstract praxeology P2. In the point of view of structuralist praxeologies, we may note T2=T1 g (a generalisation) and call P2 a structuralist praxeology of level 1 since it invokes a conceptual definition. In fact, we prefer to call it pre-structuralist since the role played by real numbers is not elucidated at this level of generality. A proper structuralist praxeology would deal with structures on abstract spaces, for instance the ordered field structure endowed with the order topology.

It shall be noted in the written solution of task 11 (a) that it is sufficient to link the problem to the case where an=0 and (cn) is converging to 0 since such a result has already been proved in the tutorials (by means of the praxeology P2). As a result, properties of limits used: in structuralist terms, these reflect the compatibility of limits with respect to the ordered field structure, in other words they may be regarded as pre-structuralist theorems. The modified praxeology, denoted P2', may be called a prestructuralist abstract praxeology of level 2 and described as follows:

T2: Prove that any real sequence that fulfils certain conditions is convergent and determine the limit; 𝛕2': Apply properties of limits; use the conditions and standard bounding techniques; 𝛝2': Properties of limits; properties of real numbers; 𝚹2: Real Analysis.

It is the presence of task 11 (b), dedicated to the proof of the Cesàro lemma, that allows us to argue that the praxeology P2 is effectively developed in the Analysis course (Figure 2). Indeed, it is another instantiation of T2 and 𝛕2 is applied. In this case, applications of the Cesàro lemma have not been previously encountered by students. The rationale of the lemma will thus appear later, when it effectively contributes to expand the punctual praxeology P1 into a local praxeology (in the sense of ATD, obtained by unification of punctual praxeologies) by providing another technique and technology (a new criteria for convergence and tool to determine the limit).

Analysis and discussion of Problem 28

There are four tasks in this problem: except task 28 (b), they all deal with piecewise defined functions (by explicit formulas) whose continuity is to be proved or investigated. The most difficult (task (d)) is the popular example f(x) = 1/q for an irreducible rational x=p/q and f = 0 for irrationals. The density of Q and R\Q in R plays a role, which draws a slight connection to task 28 (b): This task is about proving a theorem that states that two continuous functions, which are defined on the same domain and are equal on a dense subset of that domain, are equal all over their domain. One would use this theorem, for instance, to conclude that there is only one continuous extension of 2 𝑥 to the real numbers if it were defined on the rational numbers only. Just as in problem 11, we first consider the rationale of the task and model the following praxeology: T3: Show that two explicitly given real functions f and g are equal; 𝛕3: Show that f and g are equal on a dense subset (under the assumption of continuity); 𝛝3: Theorem, that two real continuous functions equal on a dense subset are equal; 𝚹3: Definitions of continuity and density.

We would like to call this praxeology P3 a pre-structuralist concrete praxeology of level 2 since it is completely contextualised to explicit real functions and involves the pre-structuralist theorem 𝛝3. Making the underlying structures further explicit would lead to the following structuralist praxeology P4 =P3 g and proper structuralist theorem 𝛝4: T4: Show that two given continuous functions on a metric space E are equal; 𝛕4: Show that f and g are equal on a dense subset; 𝛝4: Theorem, that two continuous functions on a metric space equal on a dense subset are equal; 𝚹4: Definitions of continuity and density.

In fact, tasks of type T3 appear neither in the textbook nor in the problem sets of the homework assignments and tutorial. Our model thus reveals a lack of coherence in the problem set, which may hinder the development of pre-structuralist praxeologies. As a result, students would not perceive the rationale of the abstract task 28 (b) which remains isolated whereas University teachers are well aware of the structuralist dimension of the task fully revealed by P4.

Conclusion and outlook

Assignments in introductory Analysis courses constitute a bridge in the transition from School to University Mathematics. On the one hand, they take up topics and contents of School Mathematics, but treat them based on formal definitions. Assertions are formulated on this basis and proven using logical principles. Their transitional character is not only evident in the topics, such as the treatment of the derivative or integration in the sense of the Riemann integral (or related concepts), but also in the fact that they do not yet consider the most general mathematical structures possible, such as those from Topology or Functional Analysis. However, the assignments we have analysed allow us to identify a pre-structuralist character of the praxeologies developed, and to illustrate it by means of exemplary tasks.

One structuralist aspect we identified in the exemplary tasks (problem 11 (a)) are the particular/general and the concrete/abstract dialectics. This aspect is used to develop a prestructuralist praxeology. It is done by presenting an abstract task, preceded (in the tutorials) by a contextualised task that gives the abstract task its raison d'être. Therefore, a shift takes place from a particular object (explicitly given sequence) to a general object (general sequence) which also causes a transition in the technique and logos such that the technique is justified by a pre-structuralist technology (formal definition of convergence of sequences). This allows us to identify this praxeology (P2) as a pre-structuralist praxeology of level 1. It readily generalises to a proper structuralist praxeology if the field structure and topological structure on R are clarified. Whenever this kind of generalisation is missing in the praxeologies that contain structuralist aspects we have called them pre-structuralist. This points at the transitional character of these praxeologies and indicates that the development of structuralist praxeologies is a process. Further, we could find additional tasks (problem 11 (b)) that are of the same type and use the same techniques as in problem 11(a), introducing yet another concept (Cesàro means). This is an indicator that the corresponding praxeology (P2) is indeed developed.

Finally, we have shown that when abstract tasks (problem 28) are not provided with additional concrete tasks exemplifying the rationale of the abstract task, the particular/general and the concrete/abstract dialectic is not evoked, and the corresponding pre-structuralist concrete praxeology is not developed. Here, it is pre-structuralist in relation to metric spaces.

In fact, we have shown that our pre-structuralist praxeologies P2 and P3 both connect to school mathematics (P1 and the issue of extending functions from Q to R). We would need to interview the teachers to identify if they pointed out such connections. We nevertheless could consider that the assigned abstract tasks aim at justifying claims and techniques from high-school, or at unifying these techniques in a coherent technological discourse, to build local praxeologies (in the sense of ATD). This is what [START_REF] Bosch | Incompletitud de las organizaciones matemáticas locales en las instituciones escolares [Incompleteness of local mathematical organisations in school institutions[END_REF] call the phenomenon of completion of praxeologies in relation to the school to university transition. In other words, pre-structuralist praxeologies contribute to achieve such a completion, and they do so by initiating the first steps of the structuralist methodology described in our theoretical framework.

We don't claim that the aspects which we identified in our data are exhaustive. Further analyses of tasks might reveal the need to specify these aspects in more detail or to consider additional aspects.

Since we have focused primarily on the transition from School to University the next step is to examine the role of pre-structuralist praxeologies in the transition from the introductory Real Analysis to advanced Analysis. Together with the structuralist praxeologies which have been discussed for advanced Analysis (see [START_REF] Hausberger | From Real to Abstract Analysis: the development of structuralist praxeologies in the analysis path[END_REF] this would complete the structuralist view on the entire learning and teaching of Analysis.

Figure 1 :

 1 Figure 1: Task 11 (a) and its standard solution

Figure 2 :

 2 Figure 2: Task 11 (b) and its standard solution

Figure 3 :

 3 Figure 3: Task 28 (b) and its standard solution

Acknowledgment

The work is funded by the Ministry of Science and Culture of Lower Saxony, Germany, within the PhD program "LernMINT: Data-assisted teaching in the STEM subjects", by the Deutsch-Französische Hochschule (DFH), and by the German Academic Exchange Service (DAAD).