Michal Tabach
email: tabachm@post.tau.ac.il

Boris Koichu
email: boris.koichu@weizmann.ac.il

Who is right? Theoretical analysis of representational activities

Keywords: Experientially real representations, socially shared representation, task design, controversy

A theoretical discussion of possible connections between some non-conventional external representations (i.e., textual tasks of a particular format) and their corresponding internal representations (i.e., perceptions of these tasks by mathematics learners) is presented in this paper. Our goal is to analyze the interplay between external and internal representations in relation to non-conventional textual tasks of "who-is-right?" format. Such tasks involve a relatively long textual story introducing a situation that can be interpreted in (at least) two contradictory ways, which are explicitly given. The solvers of the task are required deciding which interpretation is correct and support their decision by an argument that would convince their peers. The presented theoretical analysis in terms of representations can serve as a tool for teachers and teacher educators in designing tasks specifically tailored to their students' needs.

Introduction

Ways in which mathematical ideas are represented are fundamental for how people learn and use these ideas [START_REF] Heinze | Flexible and adaptive use of strategies and representations in mathematics education[END_REF]. [START_REF] Goldin | A joint perspective on the idea of representation in learning and doing mathematics[END_REF] refer to representation as a configuration of some kind that corresponds to, or is referentially associated with, or symbolizes something else. According to the National Council of Teachers in Mathematics (NCTM, 2000), representation refers "to the act of capturing a mathematical concept or relationship in some form and to the form itself" (p. 4). In other words, representations refer to both processes and products of learning and doing mathematics. [START_REF] Goldin | A joint perspective on the idea of representation in learning and doing mathematics[END_REF] differentiate between internal and external representations. Internal representations allude to possible mental configurations of individuals. As such, they cannot be observed directly. Rather, they can be inferred based on theoretical analysis, and when appropriate based on behavioral clues, like gestures, speech and acts. In contrast, external representations allude to physically embodied, observable configurations such as texts, graphs, pictures, or mathematical symbols. These are in principle accessible to observation.

The theoretical discussion of possible connections between some non-conventional external representations (i.e., textual tasks of a particular format) and their corresponding internal representations (i.e., perceptions of these tasks by mathematics learners) in this paper corresponds with the Call for Papers of CERME11 TWG24 in the following way. We concern with creation, interpretation and reflection on external representations in learners' minds, with the purpose of depicting and communicating information, thinking about and developing mathematical ideas, and advancing understandings. Hence, we address the following question posed in the Call: "How can non-conventional representational activity contribute to mathematical thinking?" Specifically, our goal in this paper is to analyze the interplay between external and internal representations in relation to non-conventional textual tasks of "who-is-right?" format.

In the sections below, we present a "who-is-right?" task format, followed by analysis of examples in the contexts of algebra and geometry, with particular focus on conjectured internal representations. The conjectures, though theoretical, are supported by our experiences of enacting the presented tasks with various audiences. The concluding discussion is related to aspects of designing tasks of "who-is-right?" format, specifically to taking some conventional mathematical tasks and turning them into un-conventional ones.

The "who-is-right?" task format An external representation of "who-is-right?" format (henceforth, WIR task) involves a relatively long textual story introducing a situation that can be interpreted in (at least) two contradictory ways. Rather than asking students to interpret the situation, the contradictory interpretations are explicitly given in the voices of two or more virtual characters. As a rule, each interpretation involves argument based on attention to some aspects of the story; otherwise, the argument supporting the interpretation must be revealed by actual solvers of the task. The actual solvers of the task are required deciding which interpretation is correct and support their decision by a convincing argument.

Such tasks have several representational characteristics. First, the textually presented situation alludes to experientially real world of the potential solvers of the task [START_REF] Gravemeijer | Context problems in realistic mathematics education: A calculus course as an example[END_REF]. That is, the situation is rooted in solvers' perception of experientially real, everyday-life phenomena. Second, the task is presented in everyday language, which might be ambiguous once referring to mathematical objects or mathematical objects in disguise. Third, the task does not require solving a textbook-like mathematical problem, at least not explicitly. Instead, two contradictory claims about the situation are presented. The learners are asked to take a stance and defend it. In order to be able to do so, the learners need to build two internal representations of the situation by taking into account each of two contradictory claims. This activity has the potential to advance their ability to overcome the limitations of the experientially real (but subjective) representations, towards building a shared representation based on mathematically valid argument that can convince the other learners when non-mathematical argument fails to do so. Fourth, the mathematically valid answer to the question "who is right?" may be not only of the "one of the characters is right" format but also of non-conventional formats such as "both are right", or "both are wrong", or "it depends". Such options potentially add an element of surprise to the task. Fifth, WIR tasks can be sequenced so that dealing with the next task requires not only a comparison between two interpretations of the given situation but also comparison of a new situation with the previous one.

Several WIR tasks have been used in previous studies as research instruments. For example, researchers [START_REF] Buchbinder | The role of examples in establishing the validity of universal and existential mathematical statements[END_REF]Buchbinder & Zaslavsky, 2013;[START_REF] Healy | A study of proof conceptions in algebra[END_REF] have used such tasks for revealing the student conceptions of proof and examples. In particular, Buchbinder and Zaslavsky (2013) designed a WIR task based on a claim "For every natural is a prime number" plus two responses to this claim by virtual student-characters. The first student-character argued that the claim is correct because she checked its validity for the first 10 integers, and the second student-character claimed that the statement is false because is not prime for Then 12 high-school students were exposed to the task and asked to determine who was right and explain why. All students exhibited indicators of understanding that for a universal statement to be true it has to hold for all cases. At the same time, some of these students remained convinced that a statement can be 'proven' through examination of several confirming examples.

Another WIR task was used in a teaching experiment conducted by [START_REF] Koichu | Enhancing an intellectual need for defining and proving: A case of impossible objects[END_REF] with a group of pre-service teachers. The teachers were exposed to two pictures: a picture of the famous Penrose triangle and a picture representing a sculpture of the Penrose triangle taken from an angle that created an impression that the Penrose triangle can be made as a real 3D object. The students were required deciding whether the Penrose triangle indeed could be a 3D object or proof otherwise. This study resulted in a description of a particular set of instructional conditions in which mathematical defining and proving could be intellectually necessitated for the students.

In spite of the fact that in both tasks one of the included responses was intuitively more appealing than the other one, the tasks proved themselves as triggers for evoking uncertainty in their solvers, as well as useful tools for revealing the students' (mis)conceptions and reasoning. These tasks stimulated our interest in further developing WIR representational activities as research and teaching tools that would correspond to curriculum-determined mathematical topics. In the next section, we present two WIR activities, one in algebra and another in geometry, and discuss their common features as well as subtle differences inherited in their design.

Examples and analysis

The WIR tasks presented in this section are designed for the use with middle-school students. The forthcoming discussion of internal representations associated with these tasks is based on our experience of enacting and discussing the tasks with two groups of in-service teachers.

Example 1: Subtle matters related to percentages

Figure 1 presents the first task in the sequence of three tasks.

Towards the end of the summer holidays, suitcases are sold with a 30% discount. The final price of a suitcase includes VAT of 17%.

In calculation of the price, one can first take into account the discount and then calculate VAT or otherwise. It appears that a buyer and a tax collector have different opinions on this matter.

Advice from Buyer: I think that one needs first to calculate the price including VAT and then the discounted price. In this way, the discount would apply to a greater number and the final price would be lower.

Advice from Tax Collector: I think that one needs first to calculate the discounted price and then to add VAT. In this way, the sum of VAT would not be influenced by the discount and the final price would be higher.

1. What do you think? 2. How would you convince a peer who disagrees with you?

3. How can we decide who is right? We now outline a conjectured scenario of dealing with this task. The student reads the story and decides who is right (a buyer or a tax collector) based on the mainly emotionally loaded process of taking the perspective of a buyer or of an authority. It was our intention to make the buyer's and the tax collector's responses more or less equally plausible for the solver. The process of the task comprehending requires the students to create competing internal representations of the situation, but at the individual level, one of two internal representation eventually prevails, and the student convinces herself that "a buyer is right" or "a tax collector is right". At the collective level, the class splits (always worked in our teaching), so that each student can see that his or her opinion is not the only possible one. At this stage, many students cannot think of an argument in support of the opposite opinion, so the split is perceived as a surprise. When asked by the teacher to convince the opponents, the students attempt to explicate their internal representation and produce explanations in everyday language often accompanied by gestures. Then, the students are surprised even more when they find out that these explanations are incomprehensible or perceived as too vague for their counterparts in order to be convincing. A collectively shared need to find a way of getting to the common ground arises. At this stage, an option to employ mathematical apparatus emerges, usually as a spontaneous suggestion of one of the participants. This is probably because of the general context: after all, the discussion occurs at a mathematics lesson. Some meta-level questions arise with the help of the teacher. For example, what argument can be convincing? How can the buyer's or the tax collector's argument be expressed mathematically?

Next, the teacher suggests the students to attempt to mathematize the situation, in the hope to construct a shared (external) representation that would be convincing for all. The students begin using mathematical language, first in order to test the correctness of their initial claims; then in order to assess their claims as well their opponents' claims. In this process, the classroom map of agency changes, and the teacher does not act as a source of knowledge about the correct solution but as a facilitator of establishing the new discursive rules [START_REF] Sfard | The interplay of intimations and implementations: Generating new discourse with new symbolic tools[END_REF], and as a mediator of the discussion. In this way, the teacher stops being an authority whose role is to approve or disapprove the student solutions, but becomes an authority who can support or disregard (for example, by revoicing) types of arguments produced by the students. This situation promotes the individual student argumentative talk based on gradually established rules [START_REF] Hershkowitz | Knowledge shifts in a probability classroom: a case study coordinating two methodologies[END_REF].

Eventually, the surprising to many students resolution of the given situationthe buyer method and the tax collector method lead to the same final pricemay ease the emotional tension in class. This would be a good moment to present the next task of the sequence (Figure 2). Presenting this task second, one cannot but compare it with the first task. In this context, the individual may create an internal representation of the task by attending to the following parallels: decreasing the length of one side is the counterpart of the discount, and increasing the length of the other side is the counterpart of the VAT. This may overshadow for the students an important difference between the external representations of the tasks. It is now not about the multiplication operations performed sequentially, where the second operation is applied on the result of the first operation, hence the final answer does not depend on the order. In the second task, each of two multiplication operations is applied to the same initial number, and then the results of these two operations are multiplied to find the area of a new shape, hence the final result may change.

The area of any square equals to the area of a rectangle built from that square by increasing one side by 10% and decreasing the perpendicular side by 10%. Is it so?

Reading the third tasks may immediately raise a déjà-vu feeling of we have already solved this one, and, indeed, Ron's claim gives room for this feeling. Yet, reading Gal's argument may lead the learners to re-consideration of Ron's stance. We believe that for this task to be the third in the sequence, middle school students may turn to the search for (external) mathematical representations of the situation quicker than in the previous two tasks. And once more, a surprise would await them (after all, Gal is right!), which may evoke a hot discussion of how this task is different from the previous one. From our experience, elaboration on the differences between the external representations of all three tasks in tacit association with students' internal representations, which gradually emerge in the classroom discourse, is a non-trivial endeavor. This was true even for the in-service mathematics teachers, who were exposed to the sequence, and even for those of them who were familiar with a variation of the first task. All the teachers acknowledged the plausibility of the above scenarios and expressed the wish to use the sequence with their students.

Example 2: Dynamic loci of points

A task presented in Figure 4 follows the WIR format, but it differs from the previous example in (at least) three important aspects: (1) it concerns geometry; (2) it requires reconstruction of two different arguments rather than comprehending the given arguments, hence the questions to the solvers are not as in Example 1; (3) the answer to the task is of "it depends" format.

Einat and Sarah like touring and exploring. One day they found an old wooden box containing a note, written, apparently, by pirates. The note said: "There is no choice, we retreat. The treasure is too heavy and we are forced to hide it. The treasure is in 200 steps from a segment connecting the tallest eucalyptus tree and the tallest maple tree of this area, and in 300 steps from the midpoint of this segment." Einat and Sarah found the eucalyptus and maple trees, but then argue what to do next.

Einat: There is no chance to find the treasure. It just does not exist.

Sarah: Let's try. With small effort, we have a chance to find the treasure.

1. What might be the Sarah argument? 2. What might be the Einat argument? 3. Unver which conditions could Einat or Sarah be right? With no exception, the solvers begin from drawing a sketch (external representation), as the one presented in Figure 5. At first glance, the sketch supports Sarah's opinion, because the stadium-like shape (represents the locus of points at the distance of 200 steps from segment

) and the circle (represents the locus of points at the distance of 300 steps from the middle of ) intersect in exactly four points. Therefore, according to the sketch, the treasure must be located in one of these points, and checking only four points is a feasible task. Of note is that creating the above sketch has been proven as difficult for the solvers. With in-service teachers, the most interesting discussions revolved around two questions: How can the distance between a point and a segment be defined?" and "What is the locus of points?"

The most challenging part of the task is to re-construct the argument that may underlie the Einat's opinion. The tentative (but supported by practice) scenario of doing the task relies on the following assumption: Any external representation of the task situation by means of a drawing relies on an internal representation that includes an assumption about the length of line segment . Most of the learners intuitively assume that it is longer than 200 steps, as in Figure 5. However, if , the locus of points described in the situation would consist of exactly two points. Furthermore, if is less than 200 steps, the locus becomes an empty set, and in this case Einat is right. Since the story does not include information about the length of , but only the information that Einat and Sarah found the trees, each of them can be right. Hence, the answer to the task is "it depends".

The readers are invited to explore the situation by means of a dynamic GeoGebra applet at https://www.geogebra.org/m/n5mRrtMf (consider varying the position of point B at the applet).

Of note is that whether or not a dynamic external representation of the task by means of GeoGebra is used, dealing with the task requires the learner to create a dynamic internal representation of the situation. This is an additional point of difference: in Example 2, the dynamic representation might be continuous, and dealing with the change in Example 1 might require creation of a discrete series of static representations.

Discussion

On the face of it, turning "conventional" tasks into WIR tasks is simple. Instead of concluding a textual description of a situation by an open-ended question (e.g., "would the price change?" or "make a map and denote the treasure"), an additional layer of two contradictory arguments by virtual characters is added. Yet, we seek arguments that can be perceived by the learners as plausible, so that the class would split between favoring one argument over the other. Hence, designing worthy WIR tasks is not as simple as it may seem.

As mentioned in the introduction, any analysis in terms of internal representation cannot be conclusive just because these are not directly accessible. Accordingly, we do not assume that the above-presented scenarios are realistic or that we can realistically describe what is going in the minds of the solvers. Yet, the value of our analysis is in that it can serve as a design heuristic teachers and teacher educators, like in thought experimentation, when designing WIR tasks.

Our hypothetical analysis of the interplay between internal and external representations was mostly at the level of an individual solver, and thus is in line with the claim made by Rittle-Johnson and Star (2011): "comparison is a fundamental part of human cognition and a powerful learning mechanism" (p. 221). In addition, we approached the interplay between the individual and the social in implementation of the WIR tasks. In sum, we have argued that a WIR task can present students opportunities to engage with classroom-level argumentative talk based on an emerging need to resolve conflicting opinions expressed by individuals, and hence to be a step towards constructing some socially shared internal representations of the task.

Figure 1 :

 1 Figure 1: The first task (discount and tax)

Figure 4 :

 4 Figure 4: The treasure task

Figure 5 :

 5 Figure 5: A sketch for treasure task (E stands for the eucalyptus, M stands for the maple)

Acknowledgment

Our study is partially supported by the Israeli Scientific Foundation, grant no. 2699-17.

If the side length is 10 cm, the area is 100 cm 2 . The rectangle will have 9 and 11 as side lengths, and the area will be 99 cm 2 .

1. What do you think? 2. How would you convince your peer who disagrees with you? 3. How can we decide who is right? In this case, using the same 10% change for the adjusted sides of the square may add to the internal representation that leads to the (wrong) conclusion that "nothing changes". For these reasons, reading Josef's explanation may serve as an eye-opener, in line with the counterexample for in the task from the Buchbinder and Zaslavsky's (2013) study, as presented above. The calculations of Josef are easy to follow and validate. Hence, we expect that the discussion of the WIR question would rather be quick, and that the discussion of the question "how are the first and the second tasks different?" would be longer. Then, the discussion might develop, again, with the help of the teacher, in an epistemological direction. Namely, the class may discuss what can be concluded from just one counterexample and what argument can be made about whether the Josef example more convincing.

Our suggested sequence does not stop here. The third task is ready to be presented (Figure 3).

The perimeter of any squares equals to the perimeter of a rectangle built from that square by increasing one side by 10% and decreasing the perpendicular side by 10%. Is this correct?

We just solve this problem and saw that it would not be the same.

Gal: Yes.

Two sides became larger and two become smaller. The additions and cut-outs are of the same size; hence, the perimeter will not change.

1. What do you think? 2. How would you convince your peer who disagrees with you? 3. How can we decide who is right?


Marius Lie Winger
email: marius.l.winger@ntnu.no

Julie Gausen
email: juliegausen@gmail.com

Eivind Kaspersen
email: eivind.kaspersen@ntnu.no

Trygve Solstad
email: trygve.solstad@ntnu.no

Characterizing fraction addition competence of preservice teachers using Rasch analysis

Keywords: Fraction addition, competence, representations, Rasch analysis

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Research topic and theoretical framework

Norwegian preservice teachers (PSTs) produce lower scores on tests of prerequisite mathematical knowledge than other student groups, and fractions is a particularly difficult topic [START_REF] Bjerke | Når brøk ikke er tall-Eksempler på misoppfatninger knyttet til brøk som tallstørrelse[END_REF][START_REF] Ånestad | Et innblikk i laererstudenters forkunnskaper innen brøk[END_REF]. The low mathematical competence in turn poses a hindrance to acquiring didactical competence from courses in mathematics education.

Several theoretical frameworks for mathematical competence emphasize the role of representations and transformations between representations (e.g. Niss & Højgaard, 2011). If the target and source representation are the same, this transformation is called a treatment, whereas it is called a conversion if they are different [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF]. For example, a calculation in the symbolic representation is a treatment of the mathematical object, while graphing a function from a table is a conversion. Conversions between representations have been suggested to be a common source of incomprehension for students, where the direction of the conversion could substantially affect the perceived difficulty of a mathematical problem [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF]. As a step towards understanding and strengthening PSTs' competence with mathematical representations, we ask: What characterizes PSTs' competence in representations of fraction addition as they enter mathematics education, and how does their competence develop through teacher training?

In this poster, we report on the development of a Rasch measurement scale for answering these questions, and present quantitative and qualitative results from a group of PSTs at the beginning of their teacher training. By using Rasch analysis we will obtain an invariant measurement scale ensuring that the results can be compared confidently over a time period. Furthermore, this approach provides insight into the dimensionality of competence in representations of fraction addition.

Method

A Rasch instrument including 18 items on fraction addition was developed. The items concerned the transformation between symbolic and two kinds of diagrammatic (area and number line) representations of fraction addition. Data was collected from a group of 98 first-year PSTs who had not yet had instruction in fractions in their mathematics education courses. The group therefore reflects a wide range of experiences with fraction representations from classrooms across the country. The student answers were analysed qualitatively and scored according to pre-defined criteria developed in a pilot study.

Result and discussion

First-year PSTs were highly competent at solving fraction addition problems in the symbolic representation register. At the same time, there was large variability in the PSTs' competence in converting addition problems between the symbolic and diagrammatic representation registers, where conversion from symbolic to diagrammatic form was the most challenging.

Principal Component Analysis (PCA) of the instrument indicated two potential dimensions, or contrasts, in the data. The first contrast separated items according to which diagrammatic representation was involved (linear vs area model). The second contrast separated the items according to the direction of conversion between symbolic and diagrammatic representations, which could be considered a subdimension of the instrument. This can be interpreted as saying that competence in representation of fraction addition should be measured using more than one variable.

We conclude that competence in fraction addition was a multidimensional construct for the group of first-year PSTs. The PSTs' high level of competence in the fraction addition procedures did not necessarily transfer to representing fraction addition in a diagrammatic model, and producing diagrammatic representations was more difficult than interpreting diagrammatic representations, likely reflecting a difference in experience with the different kinds of conversion. These quantitative results were reflected in the qualitative analysis. The coding and scoring process showed that although many of the students that performed well on the area model tasks tended to apply partwhole interpretations to the number line. These results lend support to the theory that different aspects of competence in a single concept can be developed independently [START_REF] Usiskin | What does it mean to understand some mathematics?[END_REF].

The present study provides a foundation for investigating whether different representation registers of fraction addition, and transformations between these, become increasingly associated towards a coherent concept as PSTs progress through the teacher education program.


Valentina Postelnicu
email: valentina.postelnicu@tamucc.edu

Classroom communication: Defining and characterizing perpendicular lines in high school algebra

Keywords: Defining and characterizing perpendicular lines Classroom communication, perpendicular lines, high school algebra

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Classroom communication: Defining and characterizing perpendicular lines in high school algebra

Introduction

Research on students' difficulties with perpendicular lines and linear equations in the context of high school algebra is scarce [START_REF] Postelnicu | Didactic transposition in school algebra: The case of writing equations of parallel perpendicular lines[END_REF]. Some studies point to students' difficulties with graphical representations and connecting among various representations [START_REF] Arcavi | The role of visual representations in the teaching and learning of mathematics[END_REF][START_REF] Knuth | Student understanding of the Cartesian connection: An exploratory study[END_REF], and teachers' failure to identify the nature of their students' difficulties in geometric context [START_REF] Gal | Perpendicular lines -What is the problem? Pre-Service teachers' lack of on how to cope with difficulties[END_REF]. The purpose of this study is to gain insight into the students' difficulties with perpendicular lines and their equations in the context of high school algebra, as it is taught and learned in a public school in the United States. The focus is on the mathematical knowledge constructed during the classroom interaction [START_REF] Steinbring | The construction of new mathematical knowledge in classroom interaction -An epistemological perspective[END_REF], i.e., on the communication between the teacher and her students, the nature of their messages, and their different interpretations of those messages. [START_REF] Nührenbörger | Forms of mathematical interaction in different social settings: Examples from students', teachers' and teacher-students' communication about mathematics[END_REF] studied the different interpretations by the teacher and two students of a mathematical statement, and pointed to the dominance of the teacher's ideas during the observed interaction. In the study reported here, the relations between the teacher's language (her messages), the language of the learners (students' messages), and the language of mathematics (textbook's messages) [START_REF] Planas | Mathematics and language: Lessons and directions from two decades of research[END_REF]Planas & Schütte, 2018) are studied using [START_REF] Hall | The silent language[END_REF] theory of culture.

Theoretical framework: Hall's theory of culture

The teaching and learning of mathematics are cultural activities that take place in various institutions (e.g., the classroom from this study). The theoretical framework used to analyze the classroom culture abides by several principles [START_REF] Hall | The silent language[END_REF]: i) Culture is equivalent with communication. "Culture is communication and communication is culture" (Hall, 1959, p. 217). This principle of equivalence between classroom culture and classroom communication, allowed us to account for the teaching and learning of mathematics in the observed classroom by analyzing the communication between the teacher and her students. The idea of equivalence between doing mathematics and communicating mathematically is also found in [START_REF] Sfard | Thinking as communicating. Human development, the growth of discourses and mathematizing[END_REF].

ii) We communicate through messages. A message is constituted by sets communicated in patterns. Sets are constituted by isolates. When one communicates using a language, one uses words (sets) that are made of sounds (isolates). The words are communicated in some context using a specific syntax (pattern) "in order to give them meaning." (Hall, 1959, p. 124). Old words, communicated using a pattern, are used to give meaning to new words. When we learn a new language, the words (sets) are perceived first. It takes time to understand the pattern of communication in a new language and construct meaningful sentences. As well, it may take a long time to master the sounds (isolates) of the new language and pronounce them without the accent from the old language. At the academic level, the mathematical knowledge is communicated in statements like axioms, definitions, or theorems. These statements constitute the sets. Examples of such sets are the definition of perpendicular lines, or the theorem of algebraic characterization of two perpendicular lines. The isolates are the mathematical objects and their properties -like line, slope of a line, lines that intersect to form a right angle. In scholarly mathematics, the pattern of communication is the formal proof; it gives meaning to statements/theorems. Old statements, already accepted as true, are used to logically derive new statements, using the accepted rules of inference. Oftentimes, when validating a new statement, the steps of the argument are expressed in natural language and employ previously validated statements (e.g., see [START_REF] Postelnicu | Didactic transposition in school algebra: The case of writing equations of parallel perpendicular lines[END_REF] for an example of a proof of the theorem of algebraic characterization of two perpendicular lines described as a "Key Concept" in Table 1, below). The communication pattern changes with the cultural context/institution. In school mathematics, the pattern of communication may be a justification (e.g., see [START_REF] Wallace-Gomez | Slopes of parallel and perpendicular lines: A visual justification[END_REF] for a visual justification of the "Key Concept" from Table 1, below). In United States high schools, researchers point out that proofs are found mainly in geometry courses, and are almost non-existent in algebra courses [START_REF] Martinez | Integrating algebra and proof in high school: Students' work with multiple variables and a single parameter in a proof context[END_REF]. In the study reported here, the participants used an Algebra 1 1 textbook [START_REF] Larson | Algebra 1[END_REF] that contains mathematical statements (e.g., "key concepts") without proofs. The pattern of communication consists of stating the "key concepts" and providing examples of algorithms to execute tasks.

iii) We can only observe a message at one level of communication at a time. When a message is communicated from one cultural context to another, its meaning may change. As observed in this study, the message changes when the level of communication changes (from textbook to teacher, and from teacher to students). Consequently, it is imperative to account for the way the textbook message (with its pattern of communication, sets, and isolates) changes into the teacher's message, and then into the students' message.

Methods

1 In the United States, students take the first algebra course before the geometry course.

The study reported here is part of a larger study with an Algebra 1 high school teacher and her students from a public high school in the United States. The study lasted four weeks and included a teaching experiment with multiple episodes, with the author as a witness [START_REF] Steffe | Teaching experiment methodology: Underlying principles and essential elements[END_REF], classroom observations carried out by the author [START_REF] Erickson | Qualitative methods in research on teaching[END_REF], and the administration of various written tasks to students. Only 58 students out of 137 agreed to have their answers reported in this study. The focus is on the classroom communication, as observed by the author, the day after the teacher communicated to her students the definition of perpendicular lines and the "Key Concept" about the slopes of perpendicular lines (described in Table 1, below). The author observed all six classes taught by the teacher in one day, took observation notes, and collected copies after the students' written answers to the following task administered by the teacher, Task 1: "a) Sketch the graph of the line that is perpendicular to Before administering Task 1, the teacher assigned a warm-up task on parallel lines, similar to Task 1a 2 , and solved it on the whiteboard. She also included in her the post warm-up communication a review of the knowledge on perpendicular lines, necessary to execute Task 1a. Students' written answers to Task 1 were collected. Due to space constraints, I analyze only Task 1c. In my analysis I refer to the connections between Tasks 1a, 1b, and 1c.

Analysis and results

Table 1, below, represents the levels of communication taken from the Algebra 1 textbook [START_REF] Larson | Algebra 1[END_REF], the author's observation notes on the teacher's communication with her students, and the students' written answers to Task 1c. The analysis considers the messages at each level, as well as the changes in message from one level to another. The messages are decomposed in sets (titled in bold, for example Statements 1-4, at the student level), and the isolates are presented with bullets. The textbook messages are presented in the first column from Table 1, followed by the teacher's messages, preceded by "T:". The teacher's messages were subsets of the textbook messages, hence the presentation in the same column from Table 1. The Venn diagrams and the arrows illustrate the change in message from the teacher level to the student level. Table 2, following Table 1, presents examples from the students' written answers on Task 1c. Each of the Statements 1-4 is accompanied by Examples from the students' answers. For continuity of presentation, Table 2 restates in its first columns the student messages (sets with their isolates), exactly like in Table 1. The analysis follows each message (set and its isolates) horizontally in Tables 1 and2, respectively. As mentioned before, when one encounters a 2 Sketch the graph of the line that is parallel to

3 2 1   x y and goes through ) 1 , 2 (  .
new language, the words (sets) are perceived first, therefore the analysis was conducted at the level of sets.

It can be seen in Table 1 that in the context of Task 1c, three sets were communicated at the textbook and teacher level: the definition of perpendicular lines, the "key concept" stating the theorem of algebraic characterization of two perpendicular lines, and the first step of the algorithm for executing Task 1a, the step referring to identifying the slope of a line perpendicular to a given line. Each of those sets are analyzed at the textbook, teacher, and student level. It can be observed in Table 1 that the sets at the student level were constituted from the isolates at the textbook and teacher level.

Table 1: Representation of levels of communication

The first set in Table 1 (column 1) defines perpendicular lines. The textbook marks one of the right angles formed by two perpendicular lines in a graphical representation similar to the one from Table 1 and points to the "Key Concept" necessary to execute Task 1a (a 90 o rotation of a line generates a perpendicular line, and the two lines have slopes that are opposite reciprocals). At the textbook level, the definition contains isolates like "two lines in the same plane" and lines that "intersect to form a right angle" (Table 1, column 2). At the teacher's level, the isolate "two lines in the same plane" was omitted, and the right angle was sometimes referred to as a " 90 o angle." As noticed in Table 1 (columns 2 and 3) at the student level, the teacher's isolate referring to two lines that "intersect to form a right angle" changed into two sets, Statements 1 and 2, the first referring to intersecting lines and the second referring to intersecting lines that form a right angle. Consequently, more than half of the students (N=35) defined perpendicular lines as lines that intersect. The isolate "line" was not problematic for students with respect to their graphical representations (all students graphed lines), but the other isolates from Statement 1, "two lines" and "two intersecting lines" posed difficulties. The examples accompanying Statement 1 in Table 2 suggest the following difficulties: i) conceiving of the second line, given one line (Example 1, "a perpendicular line is a line that intersects with something"); ii) connecting between representations (parallel lines were represented in Example 2, but described as "lines that intersect, they both share the same point," and parallel lines that "do not ever touch" were described in Example 3, where intersecting lines were graphed); iii) conceiving of the intersecting point (the point of intersection was mentioned in three of the students' answers but was not marked in any of the students' answers, making questionable the meaning of "intersecting lines" in answers like in Example 4 where intersecting lines were described as "touching or crossing over each other").

Continuing with the Statement 2 in Table 2, only six students mentioned "lines that intersect to form a right (or 90 o ) angle, three of them did not have matching graphical representations (see Example 5, where the lines intersect but do not form right angles), and three of them had matching graphical representations like in Example 6. The analysis of the students' answers including Statement 2 indicates that while the isolate "two intersecting lines" did not pose difficulties, the isolate "right angle" was problematic. This issue raised the question if the difficulty was limited only to the measure of the angle or if it extended to the concept of angle itself.

To execute Task 1a, one needs the theorem of algebraic characterization of two perpendicular lines in a Cartesian system of coordinates (see the "Key Concept" in Table 1, column 1), and the algorithm to apply it (see also the Algorithm for Task 1a (Step 1)). At the teacher level, the theorem was stated "Perpendicular lines have slopes negative reciprocals," and sometimes "negative" was replaced by "opposite" to emphasize that the slopes of perpendicular lines have opposite signs. The isolate "nonvertical lines" was not mentioned by the teacher during classroom observations. As mentioned in Methods, the teacher reminded her students how to determine the slope of the line perpendicular to the given line (1/ 2) 3 yx  , by finding the opposite reciprocal of the slope of the given line: "You take the slope of the original line, a/b, that is 1/2, flip it, b/a, that is 2/1, and change its sign, -2/1."

The teacher matched the algebraic and graphical representations (see the last figure in Table 1, column 1). The students used the isolates from the teacher level (Table 1, column 2) to constitute their sets, Statements 3 and 4 (see Table 1, column 3). Statement 3 refers to a characteristic property used by students to define two perpendicular lines, while Statement 4 refers to the algorithm of finding the slope of a line perpendicular to a given line, y mx b  . None of the students who stated the property from Statement 3 used it correctly in the algorithm described in Statement 4. Although the students mentioned "slope" and "negative reciprocals," they failed to identify the slope of the given line or its negative reciprocal. Therefore, the students' answers were reported separately for Statements 3 and 4. 2). In Examples 7 and 8 perpendicular lines were described as "lines that have negative reciprocals," and in Example 7 "slope" was omitted. The isolate "perpendicular lines" remained problematic for students. Likewise, the isolate "slope" posed great difficulty, especially in the graphical representation. The isolate "negative reciprocals" or "opposite reciprocals" was used by 28 students (see Table 2, Statement 4 and Examples 9-10). However, isolates like "perpendicular line" and "given line" and its "slope" posed difficulties for students, as shown in their graphical representations (see Example 9 with parallel lines). In Example 10, the slopes of the two lines are those from the teacher's example used in the post warm-up communication, the student wrote "make everything opposite," although the y-intercept of the lines was not changed, and the graphical and algebraic representations did not match. Only 14 students determined correctly the slope of the perpendicular line in Task 1a, but they failed to find the correct y-intercept. Notwithstanding a minor sign error, two students found the equation of the perpendicular line from Task 1a, but their algebraic and graphical representations did not match.

Conclusions

The communication between the teacher and her students, analyzed at the level of sets, showed the following features:

1) The lessons observed in this study were "ritualistic" and abided by the following "routine" [START_REF] Lavie | Routines we live by: From ritual to exploration[END_REF]: like in the textbook, the teacher stated the "key concepts" and presented a "model" for executing a task, then asked the students to execute a similar task. The pattern of communication lacked justifications or proofs.

2) The teacher's messages were not understood by the students in the way the teacher intended. The teacher communicated using sets like those from the Algebra 1 textbook [START_REF] Larson | Algebra 1[END_REF], but only some of the isolates of those sets were received by students. The students communicated using sets that were constituted from the teacher's isolates, with a pattern lacking justification or proofs.

To overcome the effects of the "ritualization," [START_REF] Lavie | Routines we live by: From ritual to exploration[END_REF] recommend turning rituals into explorations. In our case, another way to think about teaching about perpendicularity would have been to look for patterns of isolates within each set. For example, in the case of the isolate "right angle," one such pattern might have been constituting all four right angles that are formed by two perpendicular lines, pointing to their vertex and sides (to constitute the isolate "angle"), describing them (their measure), giving counterexamples (acute and obtuse angles), and connecting the verbal and graphical representations. An example of an exploration to justify the "Key Concept" (see Table 1, column 1) is described by [START_REF] Wallace-Gomez | Slopes of parallel and perpendicular lines: A visual justification[END_REF].

  What is the equation of the new line you created? b) How did you determine this? c) A fellow student does not understand what perpendicular lines are. Explain to them in as much detail as possible what perpendicular lines are and how you find them."

  Two lines in the same plane are perpendicular if they intersect to form a right angle."(Larson et al., 2007, p. 320) T: " Perpendicular lines intersect and form a right ( 90 o ) angle." If two nonvertical lines in the same plane have slopes that are negative reciprocals, then the lines are perpendicular. If two nonvertical lines in the same plane are perpendicular, then the slopes are negative reciprocals."(Larson et al., 2007, p. 320) T: "Perpendicular lines have slopes negative (opposite) reciprocals."Algorithm for Task 1a (Step 1):Identify the slope m of the new line based on the "key concept" (perpendicular lines have slopes negative reciprocals).(Larson et al., 2007, p. 321) T: "You take the slope of the original line, a/b, that is ½, flip it, b/a, that is 2/1, and change its sign, -2/1." (Note: Given the line and the point from the warm-up activity, the teacher drew the line through (2, -1) with the slope -2/1). ), after Statement 1, represents the number of students who communicated that statement.

  "Perpendicular lines are lines that are side by side and do not ever touch each other." Example 4: "Perpendicular lines are lines that are straight but are touching or crossing over each other."  , then graph." Table 2: Representation of the communication at the student level Only one of the eight students who referred to Statement 3 in their answers had a graphical representation of perpendicular lines. The other answers had graphical representations of parallel lines and non-perpendicular, intersecting lines (see Examples 7 and 8, respectively, accompanying Statement 3 in Table


Susanne Prediger
email: prediger@math.uni-dortmund.de&angela.uribe@math.uni-dortmund.de

Ángela Uribe

Taha Kuzu

Disentangling students' personal repertoires for meaning-making: The case of newly arrived emergent multilingual students

Keywords: Multilingual repertoires, language, qualitative case study, newly arrived multilinguals

Empowering multilingual students for mathematics learning requires building on their multilingual repertoires. Although this claim has often been repeated, the students' repertoires and the interplay of their components have so far only been partly understood. The repertoire comprises different state languages but also different registers and representations serving as multimodal sources for meaning-making. This paper contributes to unpacking students' personal repertoires for meaningmaking of mathematical concepts (comparing fractions). A qualitative study of three newly arrived Syrian immigrants' learning processes identifies (re)sources that the study subjects have that are different from those of native residents that had been found in former studies. These results show that teaching-learning arrangements should connect the registers and representations in more flexible trajectories.

Relating registers and representations as a teaching approach for building upon students' personal repertoires

Dynamic, multimodal conceptualizations of multilingual repertoires resonate with teaching approaches that outline the didactical potentials of relating registers and representations for meaning making (see Figure 1 from [START_REF] Prediger | Purposefully relating multilingual registers[END_REF]similarly, Moschkovich, 2013). In many teaching approaches, teaching-learning arrangements start by activating students' informal experiences in concrete and graphical representations or familiar contexts and mobilizing their everyday languages, all of which are later systematically related to the target formal registers and representations. For multilingual students, this approach is extended to home languages [START_REF] Moschkovich | Principles and guidelines for equitable mathematics teaching practices and materials for English Language Learners[END_REF][START_REF] Planas | On the process of gaining language as a resource in mathematics education[END_REF]: Informal experiences and everyday language resources in home languages can be activated in multilingual mathematical discourses. Still, "meaning-making is relational" [START_REF] Barwell | From language as a resource to sources of meaning in multilingual mathematics classrooms[END_REF], that is, it takes place by relating languages, registers, and representations to each other across all languages of the learners.

Previously, based on this teaching approach, a bilingual German/Turkish teaching intervention on conceptual understanding of fractions was designed for multilingual seventh graders (native residents whose parents or grandparents born in Turkey). The study of this intervention provided quantitative evidence for the efficacy of the intervention despite the students' limited experience with the technical register in Turkish and qualitative insights into the functioning of relating languages, registers, and representations along the sketched learning trajectory from informal resources to formal learning content [START_REF] Schüler-Meyer | Is formal language proficiency in the home language required to profit from a bilingual teaching intervention in mathematics[END_REF].

In the current step of this research, the qualitative study has been extended to newly arrived immigrants from Syria. The analysis of Syrian textbooks and interviews with Syrian teachers gave indications that Syrian math classrooms prioritize the formal register and symbolic representation over the meaning-related register and the contextual and graphical representations. Thus, empirical research is required to investigate whether these different mathematical practices might influence the students' use of their multimodal repertoires.

Research gap: Disentangling the repertoires for newly arrived students from Syria

In order to adapt the main ideas of the teaching approach of relating registers and representations to multilingual repertoires of newly arrived students, it is therefore crucial to identify these personal repertoires and their activation empirically. Therefore, the learning-process study in this paper pursues the following research question: Which personal repertoires do newly arrived students activate for meaning-making in the subject of comparing fractions?

Methodological framework for the learning-process study

The research question was pursued in a learning-process study that was part of the larger mixedmethods project, MuM-Multi (Schüler-Meyer, Prediger, Kuzu, Wessel, & Redder, 2019), based on a teaching-learning arrangement developed by [START_REF] Prediger | Fostering German language learners' constructions of meanings for fractions[END_REF].

Methods of data gathering

Intervention. The project investigates a bilingual teaching intervention for developing multilingual students' conceptual understanding of fractions (especially organized for the research project). The intervention was conducted in small groups of 2-5 students each, and all groups were videotaped. The teaching material was provided in German and Arabic (or Turkish in the previous study), teachers spoke both languages.

Sample. The intervention addressed newly arrived Syrian students (n = 18, 12-15 years old) who are emergent German speakers (abbreviated as emergent multilinguals). Their prior formal education in Syria (5-8 years) took place in their Arabic. We present a case study of three students, Manal, Malik, and Zarah, who have learned German for 1.5-2.5 years and reached a German proficiency level of A2.2 in the Common European Framework of Reference for Languages. Among the three, Malik has the strongest mathematics achievements in the pre-test, and Zarah is slightly more proficient in German language than the others. The data was later compared to findings from the first intervention with native residents who spoke both German and Turkish (n = 41, 12-14 years old) who had grown up in Germany with no prior experience in activating their home language, Turkish, for institutionalized mathematics learning situations.

Selected task for the case study. While many episodes from the video data corpus were analyzed in order to identify the students' repertoires, the case study presented here is focused on the task in Figure 2. It stems from the second session of the intervention, when students had already worked on the part-whole concept. The task aims at consolidating the students' partwhole concept by addressing shares for different wholes (10 GB and 5GB) and at preparing the comparison of fractions. Subtask C follows the variation principle: Students usually solve it by either lengthening the bar in the graphical representation (the teacher then asks them to also relate this to the symbolic representation, e.g., articulating that 7/10 is less than 8/10 or that 7/10 must be added by 1/10 to reach 8/10) or starting in the symbolic representation and formally adding 1/10 (the teacher then asks them to also show what this means in the graphical representation). The relation of meaning-related, formal language registers and different representations is at the core of developing conceptual understanding.

Methods of qualitative data analysis

The transcripts of the selected videos were qualitatively analyzed in two steps with respect to students' conceptual development across languages, registers, and representations:

 In Step 1, teachers' questions and students' answers were analyzed turn by turn with respect to the uttered individual theorems-and concepts-in-action (Vergnaud, 2016) on the part-whole concept and the students' emergent ideas for comparing fractions.  In Step 2, each utterance was coded by the register and/or representation the speaker refers to and with a "+" if two registers/representations were related within the utterance or between two utterances. For example, "multiply," "denominator," and "times" are coded as terms from the technical register, "part of whole," "colored bar," and "equally long" as terms from the meaningrelated academic register [START_REF] Prediger | Fostering German language learners' constructions of meanings for fractions[END_REF]. Although some terms appear in different registers, their use is specific to the different registers, for example, "equally large" as a technical term vs. "equal" as an everyday term. As some English translations of terms do not exactly reflect the register character of the German or Arabic original, the original language was focused.

Insights into learning processes of newly arrived emergent multilinguals: Case study of Manal, Malik, and Zarah and equal-sized parts of the bars

In the episode in view from the first session, the newly arrived students Manal, Malik, and Zarah work with a German-and Arabic-speaking native resident bilingual pre-service teacher (TeaA) and a German-speaking teacher (TeaG) on the comparison of fractions (tasks printed in Figure 2). In earlier tasks, the fraction bars were introduced in the download context. Prior to seeing the printed part of the transcript, the students solved Task A (see Figure 2 for Zarah's solution), drew the colored parts of the bars and wrote down the fractions. They then discussed Task B with TeaA in Arabic: Which child has downloaded a larger part of the whole, Badr's share of 7/10 or Aziza's share of 4/5. Although the graphical representation was drawn, their discussion with TeaA in Task B mainly refers to the technical Arabic register and the symbolic representation and less to the download context and related everyday language. In contrast, neither the graphical representation nor any meaning-related term, such as "share," "part," or "whole," are explicitly addressed. TeaG takes over when students arrive at Task C. In Turn 45, the teacher asks how to modify 7/10 so that the colored part of the bar becomes equally long as the colored part in the 4/5 bar. He uses the everyday terms "make equal" and the technical term "equally large" while pointing at the graphical representation, intending to connect these three registers and representations. But his plans are not fulfilled: Zarah (in Turn 46) does not refer to the graphical representation but only to the verbalized symbolic representation. In Turn 47, the teacher acknowledges that her procedure for finding an equally large fraction for 4/5 is mathematically cor-rect. He continues asking her to explain also how to modify 7/10 so that it reaches 8/10, and makes explicit that he wants her to refer also to the graphical representation. This repeated hint leads Manal to refer to the expected representation: Manal uses the graphical representation as the source for the comparison of fractions for the first time in Turn 48. Although she visibly struggles while searching for adequate German terms ("pretty great") and does not complete the sentence in a grammatically correct form, her German is still sufficient to express her meaning-related ideas on the comparison, together with gestures. In contrast, Malik stays in the more familiar technical register: The teacher's repeated prompt to focus on the graphical representation finally leads to Zarah's idea to relate the graphical and the symbolic representations in Turn 58 ("change this"). Again, she uses a minimal set of words to address her meaning-related ideas with gestures, deictic means, and prosody, but without explicitly articulating meaning-related terms. Once she has expressed her meaningrelated ideas of modifying the part of the fraction bar, she can make explicit the meaning of "equally large" in the meaning-related register describing the graphical representation: "That becomes equally long" (in Turn 62). Here, the graphical representation is used to solve the task for the first time, not only as a diagram to read off numbers. In her solution, she considers both colored bars and their relation, as originally intended by the task. After this, the teacher asks for a justification: The Arabic discourse gives Manal the space to admit that she has not understood (Turn 80). The teacher makes sure that she constructs the meaning of the graphical representation (Turns 81, 83, and 85). Malik explains in the technical register what the graphical representation means (Turn 84), and then the teacher explains the graphical modification, also, remarkably, in technical terms ("add one"). During the whole discussion of Task C, the Arabic terms for "whole bar," "colored part of the bar," and "share" are not formulated by either the students or by the teacher.

In total, this episode shows that students and teachers activate different sources for meaningmaking: Whereas the German teacher intends to construct meanings for the symbolic 7/10 + 1/10 = 8/10 = 4/5 by working within the graphical representation and the meaning-related register, the students show a huge distance from the graphical representation, but a familiarity with the symbolic representation and a preference for the technical register. After longer negotiations, the teacher and the students jointly construct the meaning for the graphical representations by means of the context (not shown here) and by means of the symbolic representation. This huge difference in the students' sources of meaning-making and felt need for meaning-making for unfamiliar representations are more influential to the situation than the missing meaning-related vocabulary, which the students compensate for by gesturing and other multimodal sources.

Contrasting the learning processes of native resident multilingual students and newly arrived emergent multilinguals

To contrast the presented case of Manal, Malik, and Zarah with native resident multilingual students, we draw upon findings in earlier publications (e.g., [START_REF] Schüler-Meyer | Is formal language proficiency in the home language required to profit from a bilingual teaching intervention in mathematics[END_REF]) from native resident students who spoke both German and Turkish. The case study of Manal, Malik, and Zarah suggests drawing the emergent multilinguals' repertoires in a different way: not in hierarchical levels of successive abstraction from bottom to top, as shown in Figure 1, but in three columns, where the symbolic, graphical, and contextual representations serve as mediators between the home language and German language in each of the registers, as shown in Figure 3.

Although theoretically the languages and registers could be related in different combinations, Figure 3 shows the combinations that were empirically identified. Native resident multilingual students in previous studies have often appeared to feel alienated from symbolic representation and prefer the everyday register and contextual representations, from which they access the graphical representation and the meaning-related academic register in both languages and, in the end (with teacher's support), also the technical register and the symbolic representation. In their processes, travelling between both languages in bilingual connective modes was a major resource for their meaningmaking processes [START_REF] Schüler-Meyer | Is formal language proficiency in the home language required to profit from a bilingual teaching intervention in mathematics[END_REF]similarly in Moschkovich, 2013;[START_REF] Barwell | From language as a resource to sources of meaning in multilingual mathematics classrooms[END_REF]. In contrast, for the three newly arrived students in the current case study, connecting both languages in either of the registers rarely occurred, mainly because they strictly reserved German for the German-speaking teacher and Arabic among themselves and for the bilingual teacher. The fact that they adopted only a dual monolingual mode with rare exceptions might be traced back to the teachers' languages [START_REF] Planas | On the process of gaining language as a resource in mathematics education[END_REF], emphasize the language context as a crucial influence) and also to the fact that their German was not yet fluent, which shows why the term emergent bilingual applies. Furthermore, newly arrived students who are familiar with the symbolic representation can easily transfer the basic technical terms from their home language to German. With some teacher support, they can unpack the meaning of the symbolic elements by using implicit meaningrelated references to the graphical register. In our video material, all newly arrived students were acquainted to verbalizing the symbolic representations in German, and they activated this resource to construct meanings for a more unfamiliar representation.

Hence, for the investigated students, the different prioritization of representations in Syrian and German lessons appeared to have a direct impact on different uses of students' repertoires. Those who grew up in classrooms where the technical register and the symbolic representation were more valued developed other learning pathways through their individual resources for meaning making. These pathways have not yet been considered, in either the international theoretical discourse [START_REF] Barwell | From language as a resource to sources of meaning in multilingual mathematics classrooms[END_REF][START_REF] Planas | Language as resource: A key notion for understanding the complexity of mathematics learning[END_REF]Planas & Setati-Phakheng, 2018) or in classroom practices in Germany.

Discussion and outlook

As this case study has shown, students' personal multilingual repertoires can be very diverse. Whereas native resident students in Germany usually refer to contexts, graphical representations, and everyday registers as their sources for meaning-making [START_REF] Moschkovich | Principles and guidelines for equitable mathematics teaching practices and materials for English Language Learners[END_REF]Planas, 2013;[START_REF] Barwell | From language as a resource to sources of meaning in multilingual mathematics classrooms[END_REF], some of the newly arrived immigrants have strong resources in symbolic representations and thereby more quickly acquire the German technical register that their resident peers have to first get to know. In contrast, they can be unfamiliar with an emphasis on graphical representation, which is a typical shared practice in German classrooms, but not all over the world.

Although the case study is far too selective and limited in sample and scope to generalize these patterns, it already contributes to theorizing the dynamic conceptualizations of personal repertoires: repertoires are diverse and include registers and representations, beyond state languages [START_REF] Barwell | From language as a resource to sources of meaning in multilingual mathematics classrooms[END_REF][START_REF] Planas | Language as resource: A key notion for understanding the complexity of mathematics learning[END_REF]. We add that it should also take into account different mathematical practices in dealing with representations and registers. These theoretical extensions also have practical consequences for classrooms: Rather than planning only one learning trajectory that starts from the everyday register and concrete representations towards the formal and symbolic [START_REF] Prediger | Fostering German language learners' constructions of meanings for fractions[END_REF], more flexible learning trajectories should be planned. Here, we can add that for newly arrived students, it can even be beneficial to start from symbolic representations and technical registers and then continue towards graphical representations and meaning-related registers.

Figure 1 :

 1 Figure 1: Relating registers and representations

FindingFigure 2 :

 2 Figure 2: Fraction tasks in the case study and its preliminary solution by Zarah

Figure 3 :

 3 Figure 3: Contrasting the multilingual repertoires in parallel episodes

  have an idea how to do this by [symbolic] calculations. But then, they are not yet equal, I think, because there is 8/10 [points at the bar of 4/5] and 7/10 [points at the shorter bar]. Do you have an idea, perhaps, when you look at the picture? Manal, you look as if you have an idea.

	45 TeaG	Now, we want to think about, how can we make this equal [points at	T graphical repr. +
		the end of the colored parts of the bars]? How can we make 7/10 and 4/5	technical reg. +
		equally large? Do you have an idea?	everyday reg.
	46 Zarah Five times two and four times two.	Z symbolic repr.
	47 TeaG	Yeah, you T (explicit): not
			symb., but graph-
			ical repr.
	48 Manal Aziza is very, pretty large [points at the 4/5 bar of Aziza]	Mn graphical repr.

  to the bar. In Turn 56, he finally gestures to the graphical representation. The teacher tries once more to focus students' attention to the elongation of the colored part of the bar:

	49 TeaG	Exactly, Aziza is larger and what can we do so that Badr is equally large?	T graphical repr.
	50 Malik Simple, Simplify […] 10 gigabytes.	Mk context lang. +
	…		symb repr.
	55 TeaG	How can you do it in the picture? [hints to graphical representation] […]	T graphical repr.
	56 Malik The ten divided by two becomes five [points at the longer and shorter bars].	Mk technical register,
			hints to graph. repr.
	Although Malik receives several prompts to refer to the graphical representation (Turns 49, 51, 53,
	55), he talks in the technical register (Turns 50-56). His interpretation of "equally large" seems to
	be "of same denominator," without satisfying the teachers' (perhaps unusual) expectation of adding
	1/10 And then, five times two and four times two becomes eight, like that?	Z graphical + symb repr.
	59 TeaG	Hmm, exactly, but what you said first, this was a very good idea.	T strengthens
		What did you say?	graphical repr.
	60 Zarah We can here [imitates the coloring of the eighth piece]	Z graphical repr.
	61 TeaG	Do it, actually. Because, what happens if you color this piece?	T graphical repr.
	62 Zarah That becomes equally long [points at the ending points of the colored bars]	Z graphical repr.
	63 TeaG	Then, the shares are also equally large.	T graphical repr.
	64 Zarah Then I do this? [all students now color the eighth piece in the 8/10-bar]	Z graphical repr.

57 TeaG

[...]

Then

[…]

, but this would be until here and for Aziza this would be until here [hints again to the ending points of both bars of unequal length] What would we have to do so that both shares become equally large We have seen, 7/10 is less than 4/5. But do you have an idea how to modify 7/10? T graphical repr.

58 Zarah Well, we can change this [points at the eighth piece in the 7/10-bar].

  The teacher's "why" question in Turn 65 brings Zarah back to the technical register and the symbolic representations (Turn 66), and Malik joins her (Turn 67). Based on her experience with mathematical practices, justification in mathematics requires symbolic representation, as she formulates in another part of the transcript. However, this time, the symbolic procedure is related to the graphical shift from one bar to the next. Thus, Turns 58-67 indicate an activity of relating registers and representations. Some turns later, the Arabic-speaking teacher talks to the students again and they switch to Arabic. Manal admits to have not yet understood (grey letters mark Arabic parts of the transcript):

	79 TeaA	[…] You do not know how to write it?
	80 Manal No, I have not understood it.
	…
	83 TeaA	[to Malik] […] Can you explain to her [points at Manal] how you found	T graphical repr.,
		the result, why you have drawn one more?	meaning-rel. reg.
	84 Malik Because they are equally large.	Mk graphical repr.
	85 TeaA	First, this is longer and that is why she got the idea to add one.	T + technical reg.
	65 TeaG	Yes, exactly, and why are the shares equally large, then? [...]	T graphical repr.
	66 Zarah Five times two and four times two #	Z symbolic repr.
	67 Malik #Eight	Mk symbolic repr.

Acknowledgement. This paper was developed in the "MuM-Multi: Fostering Language in multilingual mathematics classrooms" project, funded from 2014-2020 by the German ministry BMBF (grant 01JM1403A to S. Prediger & A. Redder). We thank the colleagues for the interdisciplinary collaboration.


Frode Rønning
email: frode.ronning@ntnu.noheidi.stromskag@ntnu.no

Heidi Strømskag

Sense and reference of signifiers for elements of polygons

Keywords: Register, sense and reference, polygons, milieu

This paper is based mainly on a planning meeting involving teachers and researchers before classroom sessions in a Norwegian school where 7-8-year-old children are working with concepts connected to geometrical shapes. During the planning meeting different conjectures were made about possible confusion that might occur concerning the names of the parts (elements) of a polygon (edges and vertices). Comparing these conjectures to what actually happened in the classroom, it turns out that the group of experienced teachers and teacher educators were not able to foresee all the nuances in the children's sense and reference concerning concepts connected to polygons. The findings suggest that these concepts may be more challenging than previously assumed. The classroom sessions are designed using principles from Brousseau's Theory of didactical situations (TDS).

Introduction

The project Language Use and Development in the Mathematics Classroom (LaUDiM) is an intervention study carried out in collaboration between researchers at the Norwegian University of Science and Technology and two local primary schools in the period 2014-2018. A central part of the project is to design and implement teaching sequences. The design process takes place in close collaboration between researchers and teachers. The teachers are responsible for the implementation in their respective classes, with researchers present in the classroom. The core of the project is twofold: to study pupils' development and use of mathematical language to express their ideas, and their use of language in arguing and justification; as well as to study and develop teachers' mathematics teaching practices. The design of each teaching sequence is guided by principles from The theory of didactical situations, TDS [START_REF] Brousseau | The theory of didactical situations in mathematics: Didactique des mathématiques[END_REF]. A teaching sequence typically consists of a planning meeting between teachers and researchers followed by two or three classroom sessions. Between the classroom sessions there are reflection sessions to discuss what happened in the classroom and, if necessary, revise the plans. This paper is mainly based on data from the planning meeting preceding one particular sequence of teaching sessions where the main aim is pupils' development of a precise mathematical language for describing elements and properties of polygons. The actual classroom sessions were reported on in [START_REF] Rønning | Entering the mathematical register through evolution of the material milieu for classification of polygons[END_REF]. In the present paper, some of the findings from the first classroom session are mentioned in order to be able to reflect on what happened in the planning meeting. Also the reflection session after the first classroom session will be mentioned. The paper will shed light on the importance of the planning session and how different discourses influence the planning, as well as deepen the knowledge about the complexity regarding the, apparently simple, concepts involved. We will in particular compare teachers and researchers' conjectures about pupils' language use about polygons to what actually happened in the classroom.

Theoretical framework

In this paper we see a word as a sign, or signifier, and, in the language of Frege (1892) we connect the sign with its Sinn and its Bedeutung. The German words Sinn and Bedutung can be seen translated into English in different ways. We shall follow the translation in (Geach & Black, 1960), using the words sense and reference, respectively. Here, the reference is the object that the sign refers to and the sense comprises all thoughts and ideas connected to the sign. In his work, Frege gives examples to show that two different signs may have the same reference but different sense. One example used by Frege is the following, from Euclidian geometry. Let a, b and c be the medians of a Euclidian triangle. Let A be the intersection point between a and b, and B be the intersection point between b and c. Then A and B have different sense but from a theorem in Euclidian geometry the points A and B coincide, so A and B have the same reference (see e.g., Geach & Black, 1960, p. 57).

A distinction similar to Frege's sense vs. reference is made by Peirce in his definition of a sign: "A sign is a thing which serves to convey knowledge of some other thing, which it is said to stand for or represent. This thing is called the object of the sign; the idea in the mind that the sign excites, which is a mental sign of the same object, is called the interpretant of the sign" (Peirce, 1998, p. 13). Also [START_REF] Ogden | The meaning of meaning: A study of the influence of language upon thought and the science of symbolism[END_REF]Richards (1923/1948) describe a similar model when they talk about a symbol as being connected to a referent via a thought or reference. They state that "[b]etween the symbol and the referent there is no relevant relation other than the indirect one, which consists in its being used by someone to stand for a referent" (p. 11). They further state that "[a] true symbol = one which correctly records an adequate reference. … [W]hen it will cause a similar reference to occur in a suitable interpreter" (p. 102). A given word is therefore a true symbol if it causes the same reference in different interpreters. Successful communication in a mathematics classroom depends on symbols causing the same reference in the actors involved, usually teachers and pupils. [START_REF] Halliday | Language as social semiotic. The social interpretation of language and meaning[END_REF] defines the term register as "a set of meanings that is appropriate to a particular function of language, together with the words and structures which express these meanings " (p. 195). So, changing between registers can mean that the same word gets both a different sense and a different reference. The mathematical register is characterised by the property that words have very precise sense and reference. Sometimes the same word may be used both in the mathematical register and in the register of everyday language but with different sense and reference. This feature is language specific so that for a given word it can be part of both the mathematical and the everyday register in one language but translating into another language may lead to different words in different registers. When the same word is used in both the mathematical and the everyday register in a language, it is of particular importance to be aware of possible discrepancies in sense and reference because such a word may not be seen as a challenge in the learning process.

Design and research question

Each teaching sequence in the project starts with a planning meeting where teachers and researchers work together to plan the activities for two or, in this case three, classroom sessions. In this planning meeting TDS plays an important role. TDS provides a model for instructional design, where a didactical situation is designed with a problem that has the target piece of knowledge as an optimal solution [START_REF] Brousseau | The theory of didactical situations in mathematics: Didactique des mathématiques[END_REF]. The didactical situation consists of four phases-action, formulation, validation, and institutionalisation-designed with milieus so as to make the knowledge (necessary to solve the given problem) progress from informal knowledge to increasingly formal (mathematical) knowledge. More precisely, in action, the pupils use an implicit strategy to solve the problem given to them; in formulation, they need an explicit strategy because they are supposed to use the knowledge to make another pupil solve the problem; in validation, the knowledge needs to be mathematical knowledge in order to jusitfy that the solution is valid in general; and, in institutionalisation, the teacher decontextualises the knowledge reached to become scholarly knowledge. For a more detailed account, and exemplification, of the evolution of knowledge and the teacher's role in the different phases, see [START_REF] Strømskag | A methodology for instructional design in mathematics-with the generic and epistemic student at the centre[END_REF].

A teaching sequence was designed according to the teachers' and researchers' expectations regarding pupils' sense and reference of words connected to polygons (see [START_REF] Rønning | Entering the mathematical register through evolution of the material milieu for classification of polygons[END_REF]. However, the data collected during implementation of the sequence gave us insight into the pupils' actual sense and reference of the words in question. This design was instrumental for creating opportunities for revealing the challenges of the learning process. For this paper however, the concepts from Brousseau's theory will not play a significant role.

Between the first and the second classroom session follows a reflection session, where adjustments are made for the second session. Researchers present in the classroom interact with the pupils but are not directly involved in the teaching. The planning meeting and classroom activities are video recorded, and the reflection meeting is audio recorded. After completing a cycle of planning, reflection and classroom sessions, teachers and researchers meet to watch parts of the video recordings from the classroom. This is also video recorded, but is not used as data for this paper.

The topic of polygons turned out to be interesting for several reasons. It demonstrated how the phenomenon that words (signs) may have different reference in the mathematical and everyday registers constrained pupils' conceptual development and although parts of this phenomenon could be foreseen in the planning meeting, unexpected observations were made, showing that there are challenges involved in the apparently simple concepts involved when talking about polygons.

This paper is addressing the following research question: To what extent do teachers' and researchers' expectations regarding pupils' sense and reference of words connected to polygons fit with actual experiences from the classroom?

The main data are the transcribed video recordings from the planning meeting. In a line by line analysis of the transcription, we look for evidence of contributions from the different participants in the meeting. Then these contributions are compared to what actually happened in the classroom at one of the schools. Here we will refer to findings in [START_REF] Rønning | Entering the mathematical register through evolution of the material milieu for classification of polygons[END_REF].

The Norwegian language and the mathematical terms involved

The relevance of the work with the topic of polygons is connected to particular features of the Norwegian language. In Norwegian, polygons are named after the number of edges, using ordinary number words. That is, an "n-gon" is called an "n-edge", where n can be three, four, …. The Norwegian word for edge is kant. For n  5 this is in line with English, except that the English words are based in Greek (e.g. pentagon), and therefore they have no obvious meaning for young children. In the paper, we will use the Norwegian word kant (plural kanter) and whenever this word occurs, it is written in italics. In everyday language, the word is used in expressions like "falling off the kant of the cliff", "walking along the kant of the road", "sitting on the kant of the table". Edge is also a word in English everyday language but vertex is a mathematical term, not used in everyday language. The Norwegian language does not have a strictly mathematical term for vertex. The word hjørne is used (corner in English). In the sequel, we will use hjørne (plural hjørner), written in italics. Since the names for the figures are made up of everyday terms, it is generally believed that learning the names of and distinguishing between different polygons is an easy task. In German, the situation is very much like in Norwegian, except there the focus is not on the edges but on the vertices, i.e. a firkant (four edge) is referred to as a Viereck, meaning "four corner". Since the number of edges is equal to the number of vertices (V = E) it does not matter whether the edges or the corners are referred to, the number word connected to the figure will be the same.

Another feature is that the edges may be referred to by different words depending on the situation. For instance, when stating the formula for the area of a square, it is often said "side times side". Then the word "side" (or "sidekant") is seen as a variable denoting the length of the line segment making up the edge. The word side corresponds to the English word 'side'.

The planning meeting

A starting point for the discussion is that the participants (teachers and reseachers) present their experiences with the topic and also some evidence of confusion regarding language. Present at the planning meeting were the teachers Ruth and Pam, and the researchers Anne, Becky and Cathy.

Ruth:

We mostly talk about kanter so far. There is not so much talking about hjørner. Anne.:

Or they do not agree on what a kant really is. Is this the kant (strikes along the side of the table), or is this the kant (marks the transition between the table top and the side). Ruth:

We have discussed this a little in the staff room. Is it side or is it kant? Because when you talk about the area of a square, for instance, then it is like side times side, and why is it then called side when it is called trekant [triangle] and firkant

[quadrilateral] and how many kanter has a … ? But when you come to three dimensional figures, then the kant is in a way, when you see from one side, then it is one of the sides. Pam:

Sidekant I think it is called. Ruth:

Yes, because you also have the concept sideflate [face], and that is on a threedimensional figure. And that is the side, that is what we call the side. So, I think it is best that the line in a two-dimensional figure is called a kant.

Ruth has observed that the same object in a geometric figure may be given different names in different contexts. When talking about the line segments making up a polygon, the word kant is used but when the length of the line segment is referred to, e.g. for calculating the area of a square, the word side is used, as in the formula side times side. This indicates that the words kant and side have different sense, although they refer to the same element of the polygon, i.e. they have the same reference. It could be argued that in the formula side times side the reference is not the actual line segment itself, but a number representing the length of the line segment. In the beginning, Ruth also admits that they have not talked so much about hjørner, mostly about kanter. This may be because it is the word kant that appears in the names of the polygons. Ruth also refers to the threedimensional situation where the word sideflate is used for the face (F) but also sometimes just side, for short. She finds this confusing and suggests to stick to the word kant.

Anne:

A hjørne is where two kanter meet. Becky:

Yes, so then at least we agree. Ruth:

This is a much simpler definition. Anne:

Yes, but it isn't. When we discuss this with student teachers, they do not agree on what a kant is. Some envisage, yes, a trekant, it has three kanter, yes, what I would call kanter, and like you Pam, call sidekanter, but it also has three hjørner. So if you say trekant and think about hjørner, there is no mismatch. Pam:

And I heard a mother asking, when we had about hjørner and kanter on the working plan; hjørner AND kanter, but isn't that the same? That is the same, so why? Cathy:

I have not at all thought like this. I have thought like you, a hjørne is a hjørne. Pam:

Yes, and that is correct. But I think many have a wrong opinion about this.

Anne gives a precise mathematical definition of the term hjørne, and both Becky and Ruth seem to be happy about this, recognising it as a simple definition. However, Anne claims that it may not be that simple because she has evidence from student teachers not agreeing on what a kant actually is. They say kant and mean hjørne, and because of the relation V = E for polygons they get away with it. This is supported by Pam who has evidence from a mother of one of the pupils claiming that hjørne and kant are two words for the same thing. Cathy has not reflected on this being an issue. This conversation shows evidence that the word kant may be used by different people with different sense and different reference and also that some people think that the two words are two different signs with both the same sense and reference. The fact that the group discussing this is a group consisting of experienced teachers and teacher educators indicates that the topic may not be as simple as it may seem at a first glance.

There is agreement that one should stick to the concepts kant and hjørne, and that it is important to stick to one word in the beginning, and Ruth says that kant seems to be the most precise word.

The reflection session

The reflection session is based on experiences from the first classroom session and took place immediately after the classroom session. The classroom session is discussed in [START_REF] Rønning | Entering the mathematical register through evolution of the material milieu for classification of polygons[END_REF]. These are some of the observations made:

 Most pupils used the word kanter to refer to the vertices but there were also some who used it correctly, to refer to the edges.

 Some pupils refer to a vertex as a hjørne when they approach it from the inside and as a kant when they approach it from the outside.

 In a quadrilateral with one reflex angle and three acute angles, the vertex at the reflex angle is sometimes referred to as a hjørne and a vertex at an acute angle is referred to as a kant.

The classroom session revealed that for most of the pupils the sense of the word kant was something sharp. This was emphasised when they talked about kanter as "the pointed parts [spissene]". The word hjørne was not often in use but when used, it had the sense of a space, something one can stay in, corresponding to the expression "stand in the corner" [stå i hjørnet]. It can therefore be said that the pupils use two different words for the same reference but each word has its own sense which is connected to the size of the angle at the vertex, or whether they see the vertex from the inside or from the outside.

Pam was surprised that they did not use the words correctly. They mainly used the word kant with the reference expected of the word hjørne and paid very little attention to the actual kanter, i.e. the edges. Diana, a researcher who was not present at the planning meeting, suggests to use sidekant about what they call strek [line]. Then there is a discussion about how the words kant and hjørne could be connected to everyday settings. Diana had observed in the classroom that some pupils talked about a "soft kant", and said that that is a hjørne (with reference to a part of a local shopping centre), but a "real kant" is sharp. For 3D objects it makes sense to talk about an edge (kant) as being sharp. This may transfer to 2D objects. One of the tasks for the pupils was that they should write the name of a given geometrical figure and explain why it had this name. Based on the pupils' workings it seemed that they used the words in a correct way. They could draw e.g. a quadrilateral and write "this is a firkant because it has four kanter". Here the reference of the word firkant is the drawing of a quadrilateral, as it should be, and the sense of the word is that the figure has four kanter. This is also correct, and the written statement will not reveal the misconceptions.

In the reflection session, it is discussed how the activity could be developed to overcome the misconception. In the first classroom session, it turned out that using the abstract geometrical figures as reference context was inadequate to stimulate the correct concepts connected to the words kant and hjørne. Therefore, the teacher suggested to use a mini-pitch as a reference context. This was assumed to be a context from the pupils' everyday experience and at the same time a context with properties sufficiently similar to an abstract geometrical figure, in this case a rectangle. Then the idea was presented to take the pupils out into the mini-pitch in the school yard and give commands like "go to the hjørne", "walk along the kant" and similar expressions.

Discussion

The target knowledge of the teaching sequence was that the pupils should develop the scientific language for naming 2D shapes and become aware that these names are based on the number of edges (kanter) in the shape. To know the difference between edges (kanter) and corners/vertices (hjørner) will then also be part of the target knowledge.

The planning session revealed certain conjectures about the use of the words kant and hjørne. The different participants in the session had different expectations, based on different experiences. The teachers had experienced confusion based on previous work with pupils and also based on conversations with parents. Teacher educators in mathematics had experienced confusion among student teachers, whereas the general educators were not necessarily aware that these words could be a source of confusion. For most adults, at least adults with some mathematical background, it is so obvious what is a hjørne and what is a kant that they do not envisage any problems with this. And also, the connection V = E, ensures that the naming of geometrical figures will be consistent even if there is confusion about what is what. But when the sign is broken down into components, tre = three and kanter = edges, we see that the sense and reference are not as desired. Even if some confusion was expected, the classroom sessions brought new knowledge to the sense and reference of the words kant and hjørne. This means that even for such seemingly simple mathematical concepts as here, the group of experienced teachers and teacher educators could not fully forecast how the terms would be handled by the pupils. Using the terms from [START_REF] Ogden | The meaning of meaning: A study of the influence of language upon thought and the science of symbolism[END_REF]Richards (1923/1948) we may say that the words kant and hjørne did not function as true symbols because they did not cause the same reference in all persons involved.

Kant

The insight we got during the classroom sessions was used to redesign the adidactical situation with respect to the conventional sense and reference of the words kant and hjørne. The desired statuses of knowledge in formulation and validation, as described above, were decisive for the team in designing a material milieu and a game (for acting on the milieu) with adidactical potential regarding the target knowledge, so the pupils could know whether their responses were adequate or not. The new material milieu is shown in Figure 1. It consists of 12 tiles, where on one side of the tile was depicted a polygon where the edges had one colour and the vertices were marked with another colour. On the reverse of the tile was written descriptions like e.g. "firkant (quadrilateral) with blue sidekanter". This game was played in pairs of pupils both having the full set of tiles. One pupil reads the text and the other one is supposed to pick the correct shape. After picking he/she can turn the tile and read the text to see if the correct shape has been picked.

The conditions described in this paper indicate that there is a need to address the concepts hjørne and kant in Norwegian schools. The naming of figures by counting the kanter using ordinary Norwegian number words, like in firkant, may indicate that there is no need to go further into the topic. Also, the books that are used for the pupils seem to take the concepts for granted, there is never any indication of what is actually hjørne and what is kant. The discussion in the planning session also shows that there is varying consciousness about the topic among teachers and researchers. And those that are conscious about it are so because they have previous experiences indicating that there may be some confusion.

Distinguishing between edges and vertices can also be seen to be important for future learning, e.g.

about polyhedra, where the number of edges and the number of vertices are not the same.

Figure 1 :

 1 Figure 1: Material milieu for classification of polygons

Table 1 : Sense and reference of kant

 1

	Hjørne	Sense	Reference
	Mathematical discourse	A point where two kanter meet E.g. A, B, C and D in quadrilateral
			ABCD.
	Pupils' discourse	A spacious area	E.g. A, B, C and D in a convex
		A soft kant	quadrilateral ABCD, when seen
			from the inside.
			A non-convex vertex.

Table 2 : Sense and reference of hjørne

 2