Stephan Michael Günster
email: stephan.guenster@uni-wuerzburg.de

Utilizing dynamic representations to foster functional thinking

Keywords: Functional thinking, dynamic representation, variation, tablet. Theoretical Framework

This article proposes a theoretical framework concerned with fostering functional thinking in Grade 8 students in relation to dynamic representations. To explore how functional thinking can be promoted through dynamic representations, tasks were varied systematically for the implementation in the classroomas evidenced by two exemplar problems presented in this paper. Students' perceptions of engaging with these tasks were ascertained through interviews conducted with Grade 8 students using tablet technology as part of an evaluation of the study. The focus of this evaluation was the effectiveness of the designed tasks for the development of functional thinking and students' skills when dealing with interactive dynamic representations.

Introduction

The development of the concept of function is an important part of a mathematical education. Working with representations of functions and how to translate between different forms are core skills in this area of learning [START_REF] Duval | A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics[END_REF][START_REF] Höfer | Das Haus des funktionalen Denkens: Entwicklung und Erprobung eines Modells für die Planung und Analyse methodischer und didaktischer Konzepte zur Förderung des funktionalen Denkens: Franzbecker[END_REF]. While there is considerable research concerning the understanding of functions and especially the associated difficulties (e.g. [START_REF] Nitsch | Diagnose von Lernschwierigkeiten im Bereich funktionaler Zusammenhänge: Eine Studie zu typischen Fehlermustern bei Darstellungswechseln[END_REF], the increasing availability of digital tools provides the possibility for working with interactive, dynamic and multiple representations. The way students use these representations when developing an understanding of functions has been the focus of substantial research, for example, studies that investigated how students use dynamic representations in connection with different dragging modalities in geometry [START_REF] Baccaglini-Frank | Generating conjectures in dynamic geometry: The maintaining dragging model[END_REF] or when students explore functional dependence through the use of a dynamic algebra and geometry environment [START_REF] Lisarelli | Students' use of movement in the exploration of dynamic functions[END_REF]. While the benefit of dynamic representations to students is not yet resolved, there is evidence that in certain learning arrangements they are advantageous compared to static representations for the development of functional thinking [START_REF] Rolfes | Funktionales Denken: Empirische Ergebnisse zum Einfluss von statischen und dynamischen Repräsentationen[END_REF] and that computersimulations are more beneficial than material-based experiments [START_REF] Scheuring | Computer-Simulationen oder gegenständliche Materialien -Was fördert funktionales Denken besser?[END_REF].

However, even though interactive, dynamic and multiple digital representations are available, the main goal in mathematics teaching and learning is to develop appropriate mental representations [START_REF] Weigand | Looking back and ahead-didactical implications for the use of digital technologies in the next decade[END_REF]. In this paper, a theoretical framework is proposed as the foundation to design tasks aimed at promoting students' functional thinking through the use of dynamic representations. To this end tasks are designed to support moving from visual to mental representations and foster functional thinking. Insight into this process is provided via the implementation of sample tasks within a Grade 8 classroom. This paper presents the initial findings of this study exemplified through the analysis of two interview protocols.

Functional Thinking

The teaching and learning of functional thinking have been widely discussed. Usually three characteristic aspects are distinguished (e.g. [START_REF] Doorman | Tool use and the development of the function concept: From repeated calculations to functional thinking[END_REF][START_REF] Dubinsky | The nature of the process conception of function. The Concept of Function: Aspects of Epistemology and Pedagogy[END_REF][START_REF] Vollrath | Funktionales Denken[END_REF]. In the framework utilized in this study, Vollrath's notion is used:

-Assignment aspect: a function creates a relation between two variables. For a given input, an output is calculated. The level of understanding can be determined by how relations are recognized and worked with in different forms of representation. -Co-variation aspect: a function describes how changes of the independent affect the dependent variable. Typical activities for this aspect are to plan, execute or analyze variations of the independent variable and the resulting covariations. -Object aspect: a function can be seen as a whole and therefore be dealt with as a mathematical object. This means that attributes can be used to describe a function as a whole (e.g. high points, slope) but can also be derived from it. Furthermore, one can treat functions like mathematical objects in their own right that can be operated on (e.g. add or substitute).

All aspects can be visualized and analyzed if typical representations are seen under a special perspective. However, some representations are more suitable for certain aspects than others. For example, graphs offer the opportunity to display a wide range of pairs of values. This is helpful to view the function as a whole whereas a table only shows a limited number of values. A learner should therefore be able to not only work with but also choose a suitable representation flexibly [START_REF] Acevedo Nistal | What counts as a flexible representational choice? An evaluation of students' representational choices to solve linear function problems[END_REF].

The operative principle to develop functional thinking

According to Dubinsky et al. understanding a concept starts with an action as "an action is a repeatable mental or physical manipulation of objects" (Dubinsky & Harel, 1992, p. 85). As actions are repeated and reflected, they may be interiorized as mental processes and later encapsulated into new objects to which then actions can be applied (for details see Arnon et al., 2014, pp. 17-26).

Those mental processes are also called operations. Operations can be characterized as reversible, associative and that they can be composed (Piaget, 1967, 47 ff.). However, actions as well as operations cannot be viewed on their own but must be considered as they act on an object. The changes caused to the object by these actions and operations and its properties and relations are to be evaluated. As Wittmann comments "to comprehend objects means to investigate how they were constructed and how they behave when operations are applied on them" (transl. by author from Wittmann, 1985, p. 9). In order to use an investigative approach in the learning process, one should adapt tasks by varying the involved objects, operations or the relations between them to find interesting effects or invariants. Such an approach is referred to as the operative principle [START_REF] Wittmann | Objekte-Operationen-Wirkungen: Das operative Prinzip in der Mathematikdidaktik[END_REF].

The operative principle is useful for developing functional thinking utilizing interactive dynamic representations via digital tools for various reasons. Actions can be performed directly on the given objects, for example, by dragging points or using sliders and the changes can be viewed accordingly. Thereby one can investigate invariances and variations. Relations between different variables can be studied by varying one variable and examining the changes caused to a dependent one. This can be viewed from all aspects of functional thinking.

To use the operative principle in the classroom, tasks need to be developed which guide the learner to view the problem's different aspects by varying relevant ones. Usually there is a starting condition provided to which operations must be applied to get a target configuration. It seems, therefore, constructive to begin by varying one of those attributes.

In this article we try to answer the following research questions:

1. What is the form of a theoretical framework that describes the relationship between the aspects of functional thinking and the operative principle? 2. How can interactive and dynamic tasks be developed and implemented in order to develop a relationship between aspects of functional thinking and the operative principle? 3. How do students utilize dynamic representations in the context of the developed tasks and how do they reflect and elaborate on their functional thinking?

The Function-operation-matrix (FOM)

In figure 1 these three aspects of tasks and the three aspects of functional thinking are arranged in a gridthe function-operation-matrix (FOM). To create tasks the problem definition has to first be analyzed regarding which aspect of functional thinking is of interest and its setting (e.g. what is given, what is the target). The FOM is meant to serve as a guide for developing meaningful tasks by selecting cells and varying the problem accordingly.

Sample tasks

To illustrate this concept two sample sequences of tasks are presented. One typical change of representation which is practiced repeatedly in the beginning of learning about functions is from symbolic formula to a corresponding graph. A coordinate system and a formula are provided, and the learner is required to sketch the graph. One possibility could be to vary the starting condition: Instead of the coordinate system a line is provided and the task is to adjust the coordinate system in such a way that the line represents the given formula (Herget, 2017, p. 9). Using a GeoGebra learning arrangement the coordinate system can be stretched, shrunk, rotated about the origin or shifted as a whole by dragging the blue points (see figure 2a). Afterwards the result can be checked by comparing the resulting function equation with the given one. The task is described as follows: "A line is given. Adjust the coordinate system in such a way that the line corresponds to the function ." This task would be categorized in the FOM within the cell starting conditionobject, as the learner has to operate with the function as a whole and use its attributes to find the proper configuration. In varying this task further, operations can be restricted, which means moving to the cell operationsobject [START_REF] Günster | Die Bedeutung des operativen Prinzips für die Entwicklung funktionalen Denkens im Tablet-unterstützten Unterricht[END_REF]. For this variation the wording of the task is: "Revisit task 1, with the following restriction. Do not use a) rotation b) shrinking/stretching of the axis."

A task concerning proportional functions reads as follows: "There is a square ABCD with a side length of 2 cm given. How much does the perimeter of the square change when the side becomes 1 cm longer?" In the FOM, this variation would be located at starting conditionco-variation because the change of the dependent variable, the perimeter, under the influence of the starting condition, the length of the side, should be evaluated. As a second subtask one could ask by how much the side must be lengthened to get a perimeter which is 6 cm longer, therefore varying the task according to the cell target configurationco-variation. Subsequently, it can be discussed whether this depends on the initial length of the side of the square or how the problem changes if, rather than a square, a triangle or hexagon is treated. The task then reads as: "For comparison consider an equilateral triangle and regular hexagon. For which figure do you need to change the length of the side the most, to obtain the same change of the perimeter. Explain your answer!" This means extending the task with respect to the cell target configurationobject. A dynamic representation is a powerful tool in this case, as the variation of the side can be realized and the relation between the length of the side and the perimeter viewed as a whole by using the trace facility (see figure 2b).

Methods of the empirical investigation

The interviews presented in the following section were conducted as part of a study at four German Gymnasien (grammar schools) where students are offered the choice of joining a tablet class. Two of the schools made use of iPads, one utilized Android tablets and one used Microsoft Surface devices. In total, five grade 8 classes without and five with tablets were includeda total of n = 216 participants. Teachers were given access to tasks designed according to the FOM described above to use them freely in regular class.

Students were administered a pen-and-paper test at the beginning and at the end of the school year to examine their knowledge regarding functional thinking. Additionally, selected students of different skill levels were interviewed in pairs after half and the full year. Questionnaires evaluated the general usage in school as well as at home through teachers, parents and students. As the study was conducted in the school year 2017/18, only preliminary results are available at this point.

In this paper parts of two interviews are discussed which were run after half a year of working with the tablet in the classroom. The first two students were both female and rated as mathematically capable by the teacher. For the second interview, student 3 was female and student 4 was male while both were estimated to be average students. The students already studied linear functions in class. Both interviews featured first some general questions about the usage of the tablets in class and at home, a few pen-and-paper tasks and, most importantly, two tasks to be solved using GeoGebra for which the screen was recorded using screen capture. The first problem was the one already described above as a sample task to adjust a coordinate system in such a way that a given line corresponds to the function equation .

The interviews are analyzed and categorized via qualitative content analysis regarding three guiding questions: how did the students adopt their practiced routines in this scenario? In what way did students use the given dynamic representation? Which aspects of functional thinking did students show? While for the first two questions categories are extracted inductively from the transcripts, the aspects of functional thinkingassignment, covariation and objectare set deductively from the theoretical framework.

Findings

The presentation of the initial findings is structured according to the three guiding analysis questions.

Application of routines

First, both student groups adjusted the coordinate system in order to get the correct y-axis intercept. This is in line with their practiced procedure as it is always the first step to draw in the y-axis intercept. For the first group it was possible to just rotate the coordinate system as they rearranged it while testing a number of options, the second one used the shift-option by dragging the origin. This was followed by attempts to try to match the slope.

Again, the students tried to follow their routine implementing the slope by calculating and and thereby finding a second point the line must pass through. They then adjusted the coordinate system accordingly (see figure 3.1). In their first attempt, they made the careless mistake of selecting instead of since the slope is but were able to correct it on their own while at the same time explaining their thinking processes. The second group determined the slope by gradually turning and shifting it to make the line pass through to two calculated points. However, they made the common mistake of mixing up and (see figure 3.3) [START_REF] Nitsch | Diagnose von Lernschwierigkeiten im Bereich funktionaler Zusammenhänge: Eine Studie zu typischen Fehlermustern bei Darstellungswechseln[END_REF]. Checking their result with the function equation and a little help from the interviewer, the students were able correct the error.

In summary, three possible steps can be identified in students' solution processes: draw in the y-axis intercept; identify and and calculate the slope from the y-axis intercept; or calculate two points and make the line run through those.

Use of dynamic representation

In implementing these steps using the given dynamic representation, students used similar strategies. For the y-intercept they used whichever tool was available to move the point to the intended position making no difference between them. To make the line run through two calculated points however, they gradually shifted and the coordinate system, e.g.:

Student 2:

First, I looked for the y-axis intercept and then I tried to use the rotate turn … with the slope 2, to pass through this point. And then the y-intercept moved again and again, but then one just has to readjust.

As for implementing and it seemed to the students more appropriate to use the shrink and stretch feature. The second group even adjusted the slope with the line running through the origin and then shifted the coordinate system to match the y-intercept (see figure 3.4.)

Aspects of functional thinking

The students show only the assignmentthrough calculating pointsor at times the co-variation aspect of functional thinking. For example, when asked about the changes they caused by varying the x-axis, they answer:

Interviewer: What changed, when you were varying axis? Student 3:

The boxes got larger, when one was at 1 the boxes got like longer.

Interviewer: When we are talking about linear functions, did the y-axis intercept change? Did the slope change?

Student 3: No, because it's still 1 to the right and 2 up.

Interviewer: And regarding this line, which is here…what changes when one varies the x-axis? Does the y-axis intercept change?

Students are shrinking and stretching the x-axis. The students are able to describe the dynamic representation as they used it, stating that the boxes got larger as well as that it seemed like the coordinate system was getting more precise because of this, since visually it resembles zooming in. However, they have difficulties relating it to the line and the changes which arise through it. They are therefore not able to view the line as a function as a whole.

Conclusion

This paper describes and illustrates a frameworkthe function-operation-matrix (FOM)utilised for the design of tasks used to foster functional thinking via dynamic representations. The FOM combines the operative principle and three aspects of functional thinking. The goal behind this is to develop appropriate tasks for the development of mental representations of functional thinking.

Interactive dynamic representations are capitalised upon by visualizing situations and associated mathematical connections and to change these representations through performing actions related to the different aspects of functional thinking.

Two sample tasks demonstrate how the FOM can be used to develop tasks related to different cells within the framework. One can also vary a single problem regarding multiple cells of the FOM. This offers the opportunity to inspect different aspects of the problem and of functional thinking. Thereby, it can also serve as an intuitive way to prepare the focus on the object aspect of functional thinking.

The analysis of the students' solution processes shows that the students had few problems using the given dynamic representations. They adopted their practiced routines for drawing the graph of a linear function according to the available tools. However, they seem to lack the ability to view the changes they made to the representation in respect to the function as a wholethe object aspect of functional thinking. Nevertheless, the task offered a situation in which this aspect could be discussed based on the interactions of the students with the graphical representation. Future research will include additional analysis of other interviews and concerning other tasks in order to confirm the preliminary finds presented in this paper.

Figure 1 :

 1 Figure 1: Function-operation-matrix (FOM) as a guide to create meaningful tasks using dynamic representations for the development of functional thinking

Figure

  Figure 2: (a) possible solution ( ) by shifting the coordinate system and stretching the xaxis (b) studying the relation between the length of the side and the perimeter of regular polygons

Figure 3 :

 3 Figure 3: Screenshots taken during the solving process to adapt the coordinate system so that the line corresponds to the function equation


Sue Hough
email: s.hough@mmu.ac.uk

Steve Gough
email: s.j.gough@mmu.ac.uk

Yvette Solomon
email: y.solomon@mmu.ac.uk

Connecting the everyday with the formal: the role of bar models in developing low attainers' mathematical understanding

Keywords: Bar model, realistic mathematics education, singapore bar, low attainers

Use of the bar model has gained momentum in England in recent years through the introduction of Singapore maths. Yet bar models originating from the Dutch approach known as Realistic Mathematics Education (RME), such as the fraction bar, the percentage bar and the double number line, have been available since the late 1990s. In this paper, we discuss the use of the bar in an intervention with low-attaining students in which we employed the RME approach. RME bases understanding in the everyday, where the role of the bar is to sustain modelling across multiple contexts, building on students' informal models. We argue that this context-driven 'bottom up' use of the bar is crucial in supporting progress towards formal mathematics, highlighting important issues to consider in the use of bar modelling, particularly with low attaining students. We suggest a consequent need for caution in use of the Singapore bar as a potential 'top down' model.

Background: different versions of the bar model

The Singapore bar

The success of the Pacific Rim countries in international tests such as PISA (OECD, 2016) has led to the promotion of approaches from Shanghai and Singapore in England, with 'Singapore maths' gaining considerable popularity. Drawing on the work of a number of theorists, but in particular on Bruner's enactive, iconic and symbolic modes of representation [START_REF] Bruner | Toward a theory of instruction[END_REF], Singapore maths' Concrete, Pictorial, Abstract (CPA) framework focuses on the shift from concrete to abstract via pictorial representations, and features bar modelling as a means of analyzing and solving arithmetic and algebraic word problems. Children in Singapore are taught the method in first or second grade, beginning with the introduction of pictorial representations of quantities using familiar objects such as teddy bears before moving to rectangles in the bar model [START_REF] Ng | The model method: Singapore children's tool for representing and solving algebraic word problems[END_REF]. Students are taught to recognize a problem type (part-whole, comparison, before-after) and then apply the bar model procedure to find the solution [START_REF] Ban Har | Bar Modelling A Problem-Solving Tool[END_REF]. Figure 1 shows the basic comparison bar model. Taken up by the influential National Centre for Excellence in Teaching Mathematics (NCETM) in England, bar modelling 'is introduced within the context of part/whole relationships… It exposes the relationships within the structure of the mathematics, which are used to find the unknown elements and thus supports the development of algebraic thinking' (Griffin, nd). Presented in this way, bar modelling focuses on applying a model once the problem structure has been analysed, or as part of the process of its analysis, with the teacher supplying the model and its associated method.

The RME use of the bar

In RME, bar modelsincluding the fraction bar, the percentage bar, the double number line and the ratio tablealso take the role of bridging the gap between the learner's informal understanding of 'reality' on the one hand, and the understanding of more formal systems on the other. However, the connection between reality and formal mathematics differs quite fundamentally, in that models are generated from students' understandings of the context, rather than being offered by the teacher. Mathematical activity emerges from students' informal models, in contrast to the Singapore use of the bar model which focuses from the outset on the structure of the mathematical problem and the relationships between the numbers involved. This emergent characteristic of RME models means that context plays a crucial role in the process of formalization, supporting extended discussion of models which are then generated as mathematizations of reality rather than being presented as tools for solving particular types of problems1 .

RME materials expose students to multiple contextual situations which can be represented by a 'model of' each particular situation. Designers deliberately choose contexts with potential for developing mathematical thinking; for example, asked to depict how to share a subway sandwich, students may begin by making 'realistic' drawings, showing round ends and fillings, but in time will move to a rectangular bar 'model of' a sandwich with the ends squared off. Producing a drawing to show how to share the sandwich fairly, marking cuts on the rectangular representation and labelling the pieces with fractions, leads to a bar model picture -a fraction bar. Other contexts such as shading in a rectangular cinema layout to represent the number of seats sold would lead to a percentage bar type of bar model. For some contexts such as marking bottle stops on a race route, it may be more appropriate to draw a line showing distance on one side and bottle stop positions on the other. This is sometimes described as a 'double number line' but would still be classed as a type of bar model, where the bar has been flattened to look like a line. Overall, models in RME "should 'behave' in a natural, self-evident way. They should fit with the students' informal strategiesas if they could have been invented by them…" (Van den Heuvel-Panhuizen, 2003, p. 14).

Conceptualising progress and the role of the bar model

How models are used in teaching has implications for the conceptualisation of progress. Whereas the Singapore model is based on Bruner's enactive-iconic-symbolic framework as a progression heuristic, in RME, progress is indicated when students start to see the similarities between 'models of' situations, and are consequently able to generalise the use of these models and apply them to other problems. At this point they are able to make a shift towards using a bar to represent a situation which is not obviously 'bar like', using a bar as a 'model for' solving a problem. Thus progress is defined in terms of formalisation of models (Van den [START_REF] Van Den Heuvel-Panhuizen | The didactical use of models in realistic mathematics education: an example from a longitudinal trajectory on percentage[END_REF], and in particular the progression from 'model of' to 'model for' [START_REF] Streefland | Wiskunde als activiteit en de realiteit als bron [Mathematics as an activity and reality as a source[END_REF]. So, for example, students may begin to see how they can use a bar to represent survey data, and to develop connections between segmented bars, pie charts and fractions, ultimately using a bar to compare and add fractions. This shift is also described as vertical mathematisation [START_REF] Treffers | Three dimensions: A model of goal and theory description in mathematics instruction: The Wiskobas project[END_REF], where students recognise the mathematical similarity between different problems and are able to choose an appropriate model to solve a problem. Models, in this view, are far more than given strategies for solving problems; they are central to building the understanding needed to deconstruct a problem.

In particular, models in RME depend on a 'bottom-up' process in contrast to a 'top down' approach which offers the model ready-made, to be overlaid onto the problem.

The contrasts between the use of models in RME and the Singapore method raise some questions regarding the current popularity of the Singapore bar -in particular, "what is being modelled, and by who?" The Singapore bar appears to be introduced by the teacher as a ready-made model for a mathematical problem rather than being generated by the learner from a context that will support the development of a situated model of that has meaning for them, and on which they can build to generate a model for which they genuinely own. In this paper, we explore how a group of lowattaining students worked with the bar in an RME-based intervention which aimed to support them to progress beyond poorly learned algorithms towards a deeper understanding of mathematics which is based on their own informal models. Thus, we ask the following research questions:

1. How do students use the bar within an RME context? 2. To what extent do students make progress in terms of moving towards 'model for'?

We consider the implications for the widespread adoption of Singapore bar modelling in the English context of pressure to move quickly towards formal mathematics and algorithmic approaches.

Methodology

In England, a large percentage (around 30%) of students fail to gain an acceptable pass grade in mathematics in national examinations (GCSE) at the age of 16 each year, and resit success rates are poor (DFE, 2017). Resit courses tend to be short (around 8 months) and focus on revising the same methods which students have already failed to learn and remember, hence the larger study of which this is a part investigated the impact of materials based on RME in four GCSE resit classes across three different sites in the North-West of England, during 2014-15. The research team developed and delivered two short modules focusing on number (12 hours) and algebra (9 hours), employing a quasi-experimental design. For reasons of space, we focus on the number module in this paper. Number teaching materials were designed to draw on contexts that supported students in producing situation-dependant bar-type representations and solutions, thus enabling the natural emergence of the bar as a 'model of'. For example, fair sharing problems required students to share out candy strips, pizzas and ribbon, all of which were chosen for their rectangular and hence bar-like properties. Lessons were delivered by three members of the research team, all experienced RME teachers. They followed the RME practice of emphasis on discussion of the context, moving from students' informal (and life-like) representations of the context (e.g. buying ribbon, mapping water stops on a fun run, sharing a subway sandwich) to discussion of their various strategies for problem solution, and finally to bar models produced by the students.

One of the questions for this intervention is whether students can recognise the potential of the bar model and associated strategies as 'models for' tackling problems where use of the bar is not suggested by situation-specific imagery (see Van den Heuvel-Panhuizen, 2003, pp 17-29). All students took a short test prior to and at the end of each module. The number test problems were chosen to reflect a range of topics in proportional reasoning, and covered ratio, proportion, finding a percentage of an amount, finding a fraction of an amount, comparing two rates and a reverse percentage calculation. Each question made reference to a context, but unlike in the teaching materials, the contexts were not suggestive of a bar-like representation. Questions were designed to reflect typical GCSE questions on the target subject matter, but students were additionally asked to explain their answers in order to reveal differences in levels of conceptual understanding.

Seventy-five students participated in the intervention classes and 72 in control classes. In the main study, independent evaluators found small but significant gains for the intervention group on the number post-test (F 1,93 =4.55, p=0.035, Cohen's d = 0.26). They also found a significant correlation between students' improvement from pre-to post-test and the extent to which they used an RME approach (use of the bar or ratio table to solve the problem) (r = .258, n = 86, p = .016) (see Hough et al, 2017, Appendix 8.5). Here, we investigate these findings further through qualitative analysis of the number pre-and post-test scripts in order to identify how students used the bar and its contribution to their learning. Our analysis involved comparing pre-test and post-test solutions by categorising responses given in terms of:

(1) use of the bar or ratio table versus standard methods/trial and error;

(2) correct labelling/division of bar parts, and scaling accuracy;

(3) the bar strategies employed, for instance partitioning/ adding; (4) error types: inappropriate halving, incorrect adding, incorrect bar representation; and

(5) evidence of progress.

Findings

The extent to which students could vertically mathematise their use of the bar

In post-test scripts, 73 % of intervention students chose to draw a bar-type model for at least one of the questions. In contrast, methods used in their pre-test scripts referred to operations denoted purely by numbers and symbols. Control group students were highly unlikely to draw a pictorial representation in order to solve a problem. These overall statistics suggest that the intervention led many students to recognise the potential of the bar model as a model for tackling a variety of problems, and in that sense to vertically mathematize.

However, it is not enough for students to simply apply the bar model method to the problem. One of the factors which distinguishes RME models is whether the learner perceives the model to be 'emergent' or 'imposed'. Emergent models are described by [START_REF] Gravemeijer | Emergent models as an instructional design heuristic[END_REF] as one of the key design heuristics of RME: models are grounded in reality, they make sense to the learner and even when the model progresses to a more formal version, the learner can always fall back on the original meaning in order to make sensethey are 'bottom up ' (p. 146). But although the designer may intend engagement with the model to be 'bottom up', this does not necessarily guarantee the model will be experienced in this way. Hence, there may be cases where a learner's experience of working with RME-based models results in a classical 'top down' engagement whereby the model is imposed by the teacher in the traditional sense of demonstrating formal procedures. In the next two sections, we look at examples of students using the RME bar model from both a 'bottom up' and 'top down' perspective.

The bar model as a 'bottom up' sense making strategy / model

For many post-16 GCSE resit students, their experience of learning mathematics involves being shown formal procedures which they can make little sense of, and which they are rarely able to accurately reproduce. The pre-test scripts contained numerous mis-representations of formal procedures, for example writing 17% of £3300 as ×100, calculating of £600 by replacing with 0.58, and incorrect additive strategies for two proportionally related quantities. However, , in an answer typical of 33% of scripts, she has drawn a bar split into 7 parts and labelled as a continuum from 0 to £140, and she has used shading to distinguish the portions on £140 allocated to Pat and Julie. Here, the student has created a bar model representation which correctly depicts the required sharing processes.

analysis
In the next example, Student B demonstrates knowledge of the standard formal procedure used to find a fraction of an amount in her pre-test, but is unable to go any further, commenting that she is does not know how to calculate the division 600 ÷ 8 (Figure 3, left). In the post-test, and in common with 20% of scripts, she was able to work out an alternative strategy, targeting the values of , of £600 and combining these to find (Figure 3, right). Her successful modelling of the problem on a bar enabled her to employ informal approaches to see the relationships between the numbers. The third example concerns a question requiring students to find the original price of a car, when the current price of £6820 is 20% less than the original. Although none of the control students were able to answer this question in either the pre-or the post-test, 18% of intervention students used a bar to model and solve the problem (Figure 4). Reverse percentage problems of this type are notoriously difficult for students to access, not least because the standard method involves setting up an equation in one unknown. Representing the information on a bar enables further sense-making in terms of providing opportunities for filling in what else the student knows, and enabling them to build up to the required percentage amount.

The bar model as yet another 'top down' procedure/model

We have seen examples above where students used the bar as a model for, enabling them to represent the question and link it to mathematical strategies which they could make sense of and therefore use effectively. However, in the next two examples, we see students able to operate with the bar model, but their subsequent strategies suggest that they are unable to connect the model with meaningful mathematical activity. For example, the student in Figure 5 (left) applied several cycles of halving and combining chunks to fill in as far as the th way point on his bar, but given that the bar is split into 7 pieces, these calculations served no purpose in finding one part. Some studentsaround 10% -always marked the half-and quarter-way points of their bars, even when the divisor was an odd amount, suggesting a top down use of the strategy. Figure 5 (right) shows a bar model being used to solve a question which asks how long a photocopier takes to produce 30 copies if it takes 18 seconds to produce 12 copies. The student has used a bar to model the situation for 24 copies, marking up the bar in one second segments, but has then simply extended the bar by 6 segments to reach 30 copies, omitting to realise that the model shows that one copy takes 1.5 seconds. The error of adding one second for every copy implies that drawing the bar has not provided this student with genuine insight into the underlying structure of this problem. Rather, the bar model has been used as a 'top down' model, here, applied without real linkage to the context.

Discussion

These findings show that the bar has the potential to enable progress. Drawing a bar-type model enabled students to represent problems in helpful ways, allowing some to engage with informal sense-making strategies. The model enabled them to fill in other quantities in order to provide a more complete and detailed representation of the problem. Many students applied an RME method to several post-test questions, suggesting that they were beginning to recognise the unifying potential of the models as a strategy for answering questions across a range of topics in number, and that they were beginning to vertically mathematise and see how to apply their context-specific models to a range of problems.

However, the transition to using a bar model is not unproblematic, and it does not provide an instant solution. Indeed, the issue of inappropriate halving suggests that students can move towards treating the bar as yet another algorithmic method. The intervention was short and as such represented an over-simplification of the RME process, but this served to expose the fragility of these particular students with regard to number sense and their ability to make connections. The complexity of their response to the bar, and the importance of its use for sense-making and not just method raises questions for the Singapore approach to bar-modelling. The significance of this issue is highlighted by [START_REF] Ng | The model method: Singapore children's tool for representing and solving algebraic word problems[END_REF] analysis of Singapore 5 th -graders' solutions to a variety of problems, in which high achievers scored well, but mid-level achievers' strategies often showed erroneous use of the model. Incorrect solutions involved the omission or misrepresentation of crucial information, changing unit generators mid-solution, failure to keep the overall goal in mind, or lack of necessary conceptual knowledge (of fractions, for example) -children were using the method algorithmically, failing to monitor what they were doing. Arguing for the importance of discussion in the classroom about different strategies for solution and the development of meta-cognition in problem representation, [START_REF] Ng | The model method: Singapore children's tool for representing and solving algebraic word problems[END_REF] note that the bar model must be used 'as a problem-solving heuristic that requires children to reflect on how they would accurately represent the information presented in word problems …. This art of representation has to be taught, but it is then the children's responsibility how they choose to use this heuristic effectively' (pp.311-2). Ng and Lee's warning underlines the potential danger in the Singapore bar's reliance on formal conventions in terms of which bar model is required to solve a problem and how the bars should be labelled. From an RME perspective, we would ask what are the contextual mediums that enable struggling students to gain access to these conventions. The model offers learners a way to represent algebraic word problems, but how do students lacking number sense negotiate meaning once the bars can no longer be drawn to scale, or when, as Ng and Lee (2009, p. 308) note, they are challenged by the concept of fraction and represent the relationship between rectangles erroneously? We suggest that these considerations need to be taken into account in the adoption of Singapore bar modelling, where students, not the teacher, need to own the model.

Figure 1 :

 1 Figure 1: Comparison bar model with arithmetic form on the left, algebraic form on the right (from Ng & Lee, pp. 287-8)

  of the post-test scripts revealed several examples of sense-making through use of an RME bar model, as in the following three examples.

Figure 2 :

 2 Figure 2: Student A pre-test (left) and post-test (right)Student A's pre-test solution (Figure2, left) may be interpreted as an example of a half-remembered procedure -multiply something by 2 and something by 5. In the post-test (Figure2, right), in an answer typical of 33% of scripts, she has drawn a bar split into 7 parts and labelled as a continuum from 0 to £140, and she has used shading to distinguish the portions on £140 allocated to Pat and Julie. Here, the student has created a bar model representation which correctly depicts the required sharing processes.

Figure 3 :

 3 Figure 3: Student B pre-test (left) and post-test (right)

Figure 4 :

 4 Figure 4: Representing reverse percentage on the bar

Figure 5 :

 5 Figure 5: Top down use of the halving strategy (left) and Using the bar in top-down fashion (right)

It is important to note that in RME 'real' means imaginable (Van den Heuvel-Panhuizen,

2003, pp. 9-10).


Kirstin Erath
email: kirstin.erath@math.tu-dortmund.de

Explorative study on language means for talking about enlarging figures in group work

Keywords: Language means, language repertoire of the learner, macro scaffolding, design research

This paper presents a first step for designing language support for Grade 9 students working collaboratively on the topic of similarity, more precisely an adaptation of Brousseau's tangram task. The qualitative study identifies (topic specific) language means which students use for talking about their ideas on how to enlarge figures, combining geometrical and arithmetical ideas. Furthermore, a textbook analysis shows the target language that students should use and understand at the end of the teaching unit. The identified difference between both repertoires of identified language means calls for developing language-responsive teaching-learning arrangements in the future in which both language repertoires can be bridged systematically by macro-scaffolding trajectories.

Background: Need for supporting language means for rich discursive practices

Students' participation is pointed out as crucial for learning mathematics by many researchers (e.g., [START_REF] Sfard | Thinking as communicating[END_REF] and is seen as "a process of enculturation into mathematical practices, including discursive practices (e.g., ways of explaining, proving, or defining mathematical concepts)" (Barwell, 2014, p. 332). Qualitative studies identify sequences of explaining and arguing as especially important for students' conceptual learning, in whole class discussions as well as in small group work (e.g., [START_REF] Mercer | Children's talk and the development of reasoning in the classroom[END_REF][START_REF] Erath | Talking about conceptual knowledge. Case study on challenges for students with low language proficiency[END_REF]. However, especially for students with low language proficiency, participating in discursive practices that allow talking about conceptual knowledge is challenging [START_REF] Erath | Discourse competence as important part of academic language proficiency in mathematics classrooms[END_REF]. Since language acts as a medium of thinking (Vygotsky, 1987) and following Interactional Discourse Analysis [START_REF] Erath | Discourse competence as important part of academic language proficiency in mathematics classrooms[END_REF], this calls for supporting students in accomplishing demanding discursive practices in order to provide access to mathematical learning opportunities. Whereas in whole-class discussions, overcoming language-related obstacles can be spontaneously supported by the teacher, group work requires a prepared support when the teacher is absent. One dimension of prepared support refers to the establishment of norms about the ways students talk to each other, such as the "ground rules" proposed by Mercer (e.g., Mercer et al., 1999), emphasizing activities such as sharing relevant information and discussing alternatives. Students with low language proficiency often need additional support for accomplishing these norms, when lexical means or syntactical structures are required for explaining and arguing in the mathematical topic. Thus, referring to Gibbons ' (2002) approach of macro scaffolding, these language means can be provided in prepared ways.

As [START_REF] Prediger | School academic language demands for understanding functional relationships. A design research project on the role of language in reading and learning[END_REF] emphasize, the identification of mathematically relevant lexical and syntactical means for a specific mathematical topic is an empirical task; it can be conducted by analyzing textbooks and teaching-learning processes. Following the research approach of empirically identifying relevant language means, this study investigates a textbook and empirical video data for identifying the language means students initially use, the target language means, and for exploring possibilities of bridging between them.

Research context: Exploring similarity in the Tangram Task

Especially teaching-learning arrangements that activate students' individual ideas and allow processes of solving problems and discovering mathematics have the potential to elicit vivid discussions about mathematics (e.g., [START_REF] Brousseau | Theory of didactical situations in mathematics[END_REF] and make demanding discursive practices necessary. For identifying students' initially used language means, the presented study focuses on the first task in a teaching unit on similarity using an adaption for four students of Brousseau's (1997, p. 177ff.) tangram task (see Figure 1; translated from German by the author). The task follows a more dynamic perspective on similarity since it leads to an understanding of similarity as the result of the process of enlarging figures true to scale: Hölzl (2018) terms approaches to similarity as dynamic if they relate to transformation geometry and as static if they refer to Euklid and define similar figures by identical angles and aspect ratios. Thus, this paper empirically investigates students' initial language means in group work related to a task on enlarging figures true to scale and compares this repertoire to the target language means as identified in a widely used textbook by following these two research questions: RQ1: Which language means can be identified in students' talk and in textbooks? RQ2: Do some of these means have the potential to bridge between the identified repertoires?

Methods: Qualitative analysis of students' interactions solving the tangram task

The presented analysis is part of the larger project MAGENTA that is conducted in the tradition of design research [START_REF] Gravemeijer | Design research from a learning design perspective[END_REF]. It aims at developing a language-responsive teachinglearning arrangement for the topic of similarity in German Grade 9 classrooms as well as building local theories on the students' learning processes.

Methods of data collection

This paper reports research from the second cycle of collecting video data. The design experiments were conducted by two preservice mathematics teachers for their master thesis. Altogether, 6 groups of two or three students each (at the end of Grade 8 or 9; aged 14 to 16; all from the same lower secondary school in an urban quarter in the Ruhr area) worked for 3 lessons on similarity. For all students, the design experiment was the first lesson on the mathematical concept of similarity. Following the idea of establishing ground rules, the importance of explaining and arguing and the way of working together was discussed with the students. For this paper, video data and written documents of the first task of the teaching-learning arrangement (Figure 1) were analyzed for investigating students' initial language means by making collections of the original expressions that were later on group by the used language means. The textbook analysis was conducted along the most sold textbook for comprehensive schools in North Rhine-Westphalia "mathe live" focusing on the pages introducing the topic of similarity until the summary (Böer et al., 2008, p. 20f.).

Methods of data analysis

First, all sequences were collected in which students were working independently on Task 1, until students proceeded with another task or until the teacher started to intervene. Second, these sequences were transcribed and enriched by students' written products. The third step followed inductive category formation [START_REF] Mayring | Qualitative content analysis. Theoretical background and procedures[END_REF] and coded all utterances by generalized versions of the language means that were used (i.e. language means using variables instead of numbers) in order to gain a repertoire of students' lexical and syntactical means. For the presented analysis, only categories related to talking about changing the sides of figures are considered due to space restrictions. The textbook analysis collected all language means in tasks related to enlarging figures and in the summary.

Empirical results: language means for talking about enlarging figures

First, students' initially used language means and the target language means in textbooks are identified. Afterwards, the idea of bridging between these identified repertoires is discussed.

Students' language means for talking about enlarging figures

Task 1 proved highly suited for eliciting discursive group work, especially if students had different strategies for enlarging their parts, like in the group of Julian, Lara, and Nitika. Julian is the only student in the data of the presented research that utters the mathematically sound idea of multiplying each side with a constant factor that he calculates by using the rule of three (in German "Dreisatz" is a term for the rule of proportion) and that he furthermore connects with the idea of scale: "I don't know if this is correct now, but this 4 cm (…) this is like such a scale thing, right? So if 4 cm are 7 cm and 4 cm it is asked fort wo, the half, and the half of 7 is 3.5. Do you know what I mean?" (F1_V1_T1_G1, line 53). All other students with multiplicative ideas think of enlarging a figure as doubling the lengths of the sides (which is only correct in special cases). Most students have additive ideas for enlarging figures. Hamsa, Jussuf, and Younis serve as an example for students talking about additive ideas, especially representing students struggling with expressing their mathematical ideas. The extract (F1_F2_V1_T1_G6; translated from German by the author) starts after the boys discover that their enlarged pieces do not fit together as a square: Like all groups, the boys first describe quite broadly that somehow the size of the figures or some sides of them are too small or large (lines 73 to 78). Afterwards, Hamsa explains Jussuf how to additively enlarge the sides of the figure using the language means "if… (then)…" (line 81), "x cm become y cm" (lines 81 and 83), and "making larger by x" (line 81). The mathematically inadequate idea of enlarging the sides by adding 3 cm to all or some sides occurs in nearly all groups of the study. An overview on students' language means for talking about enlarging figures is printed in Table 1, sorted by the related mathematical ideas.

Language means

Examples from students' talk Students' language means for talking about additive ideas to make x cm Did you also make everything 3 cm, Leonie? (to calculate) x plus y / plus x Yes, I did everything plus 3. / So then you calculated 6 plus 3? x become y That are 6 cm here. They become 9. x longer / larger / higher / more / … Yes, I though 3 cm longer. / Everything 3 cm larger. / Um, maybe it's just always 3 cm higher […] to be 3 cm more? x more / longer than You have to be 3 cm more than true to scale. / You have to write 3 cm more than normal. x on top There was written that there are already 4 cm that you should enlarge on 7 cm in that I just calculate 3 cm on top. to make larger by look, if 4 cm become 7 cm, you have to make larger by 3 to enlarge on There was written that there are already 4 cm that you should enlarge on 7 cm in that I just calculate 3 cm on top. difference between x and y I think 10.5, because 9 is only calculated plus 3. And that's just the difference between 4 and 7. Students' language means for talking about the idea of doubling to take / make the double / to double And eh it says 4 cm. And then I just took the double. Thus 8. And then this is enlarged. And quasi doubled. Students' language means for talking about multiplicative ideas x corresponds to y / x are y So, 4.5 are 7.875. So, this side is 4 cm long eh 4.5. But to begin with let's put away the 0.5. So, if we then calculate 0.5 corresponds to 0.875. x times y Just take the 6 cm and then 1.75 times 6. Simply just calculate times. to calculate with the rule of three Ehm, I did calculate with the rule of three for 1 cm 1.75 times 6, because it was 6 cm, that makes 10.5 on x cm I have the rule of three, thus on 1 cm. To 3 cm because my side, well two were 3 cm, I think at least […] from x to … From 100 to .. And then from 1 times 3 and 1.75 times 3 are 5.25.

Table 1: Students' language means for talking about enlarging figures identified in 6 groups

Like Hamsa in line 81, students with additive ideas often initially use language means related to visual impressions of larger, more, higher, huger and related verbs. Furthermore, they often describe processes (instead of e.g. talking about characteristics or relations). Some groups also use the phrase "x cm become y cm" (like Hamsa in lines 81 and 83) that is also offered in the text of the task.

Another interesting inside refers to the line "x more/longer than" in Table 1 (not printed in the transcript but from this group): Jussuf's utterance "You have to be 3 cm more than true to scale" (F1_F2_V1_T1_G6, line 249) combines two ideas in one sentence: the inadequate idea of additively enlarging the sides and the concept of scale which usually refers to the mathematically sound multiplicative idea of scaling with a magnification factor. Here, Jussuf seems to use "true to scale" for referring to the length in the original figure and/or for expressing that the shape is conserved. The phrase "true to scale" is taken up by the group and later on even used in their written general instruction for enlarging figures (Figure 2). But, since it is not discussed regarding its mathematical meaning, the term stays vague and is still used in connection with additive ideas. The analysis of further language means related to multiplicative ideas reveals two important points.

First, talking about the rule of three makes it necessary to talk about corresponding numbers, for example by the means "x corresponds to y" or "x are y" (see Table 1). These language means are relatable to a static perspective on similarity since they do not only describe a process but also relations. For example, the static perspective also talks about 'constant length ratios of corresponding sides'. The expression "x corresponds to y" could also work with additive ideas but is not observed in this way in the analyzed data. Instead, only the expression "x becomes y" is identified which is very close to the formulation of the task and also has another meaning than "x corresponds to y". Particularly, "x becomes y" (as a lot of the language means related to additive ideas) describes a process and not a relation and is thus more related to a dynamic perspective on similarity. Second, calculating with the rule of three makes it necessary to talk about calculating "on 1". Mathematically this parallels the idea of calculating the magnification factor which is introduced in most German textbooks as the constant result of dividing the lengths of corresponding sides (see next Section).

Language means identified in a textbook

The analyzed textbook starts with a page headed "Enlarging and Reducing" and introduces the new topic by referring to the dynamic perspective on similarity: several tasks ask students to draw enlarged/reduced pictures without giving any instruction. Task 5 is the only one talking about the process of reducing by "merging several boxes" (Böer et al., 2008, p. 20; see Figure 3 on the left You have to be 3 cm longer than the scale. Then you draw it on the sheet. When you is all true to scale you cut it out thoroughly. At the end you put together the figures.

side). The second page has the caption "Similarity" and starts with the questions "Which data does the drawer need to draw an enlarged version of the house? How do the line segments change, how do the angles change during enlarging?" next to a small and a larger drawing of a house, directly followed by the summary printed on the right side of Figure 3. Whereas the first question refers to a dynamic perspective on similarity, the second question is unclear but might be intended as a transition to the static perspective. In the summary, the first sentence refers to a dynamic perspective without talking about the corresponding processes, the remaining text to a static perspective. On the one hand, it was to be expected that the book does not offer a general instruction for enlarging figures on the first pages, since it follows the tradition of taking up students' ideas. Thus, except for "merging boxes", there are no language means for the dynamic perspective identified. On the other hand, the summary focuses on the static perspective on similarity with a strong emphasis on the concept of scale (most textbooks use the term 'magnification factor' which is here only mentioned as marginalia). The more mathematical perspective of this lack of transition between the perspectives will be discussed elsewhere. The analysis of language means shows, that "enlarging true to scale", "being similar to each other", "length ration" and "scale" (not as the concept scale but as the name for the quotient of length of the line segment in the picture and in the original) are language means that are of importance but not used in the tasks or texts before. The expression "corresponding" is used in this example only in connection to the angles. Other textbooks use this mean more prominent and especially for corresponding sides. A short review of more textbooks shows that starting with a dynamic perspective on similarity is quite common. After these first tasks of investigating how to enlarge figures, there is often a cut towards the summaries which are written from a static perspective on similarity, i.e. about "length ratios", "magnification factors", etc. which coincides with a change in the used language means from describing processes towards talking about relations. The comparison of students' initially used language means and the target language identified in textbooks shows that there is a gap: On the one side students talk about processes and visual impressions, especially if they refer to additive ideas. On the other side, textbooks initiate thinking and talking about this dynamic perspective on similarity (without offering related target language) but focus often exclusively on the static perspective in the printed summaries which is connected to talking about relations. Thus, even though using different tasks, the challenge of leading over from a more dynamic to a more static perspective on similarity can be identified in the textbook itself as well as by comparing the task from the project and the textbook.

Bridges from students' language means to the language means of textbooks

Of course, it is not possible to compare students' oral language in the process of understanding with the written language of textbooks that display the already understood version of mathematics. Nevertheless, students need to be able to relate their language and the corresponding mathematical ideas from the phase of discovering mathematics with the counterpart from the textbook that is often the basis for all other lesson phases (e.g., also as an idea in [START_REF] Gibbons | Scaffolding language, scaffolding learning[END_REF]. Exploring possible bridges between students' initial language means and textbooks, three points appeared central: First, the textbook analysis revealed a cut between the use of the dynamic perspective in the first tasks and the use of the static perspective in the summaries which is paralleled in the used language means. Since the textbook does not offer language means for talking about the process of enlarging, it can be assumed that students in classrooms speak quite similar to the students in this study. This emphasizes the need of developing tasks that explicitly work on this transition of mathematical ideas along with supporting the corresponding language means in small groups or teacher guided whole-class discussion. Second, the notion of scale is used by Jussuf and Julian as well as in the presented textbook. But, this notion seems problematic since it is tied to a specific mathematical concept that needs to be understood in order to be used properly. Third and most important, the identification of language means used for talking about similarity in a dynamic perspective shows that students with additive ideas often use language means related to the visual change of sides and their language is very tied to describing processes. This also holds for the most cases in which students talk about their idea of doubling. In contrast, if students talk about the multiplicative idea, they tend not to use language means related to visual change. Especially Julian uses the language mean "x corresponds to y" which is close to "corresponding angles/sides" used in textbooks and also displays a more static perspective. Thus, the language means around "corresponding" can be identified as possible bridges between students' language and the language in textbooks since it can be used to describe parts of the process of enlarging as well as to talk about relations between features of figures. This could also hold for the expression "calculate on 1" which refers to the rule of three used in the process of enlarging but can be also interpreted as talking about the magnification factor which describes a relation.

Conclusion and outlook

The study is limited in the small size of student groups that were observed. Thus, it is planned to extend the analysis to more groups and to enrich the data with more textbooks. Nevertheless, the presented research points to important directions: On the one hand, the study shows that students mainly talk about processes and refer to visually observable change, often staying a bit diffuse and not using mathematical wording (like adding, plus, etc.). On the other hand, textbooks often talk about similarity by means of characteristics and relations. This gap between students' initial language means and the target language parallels the difference between talking about similarity from a dynamic or static perspective. This highlights the need for developing a task that explicitly picks out this transition as a central theme and supports students by offering language related support. This especially holds for students with low language proficiency that are still acquiring language and discourse practices that allow talking about characteristics and relations of figures. The analysis points to two language means that have the potential to bridge between students initial language means and the target language: "x corresponds to y" and "calculate on 1" can be used for talking about similarity from a dynamic perspective and static perspective. Nevertheless, there is a need of further research for developing language-responsive teaching-learning arrangements in the future in which both language repertoires can be bridged systematically by macro-scaffolding trajectories as well as for teacher moves for supporting the group work and following whole class discussions.

Figure 1 :

 1 Figure 1: Task 1 of the teaching-learning arrangement of enlarging figures

Figure 2 :

 2 Figure 2: General instruction on enlarging figures from Hamsa, Jussuf, and Younis

Figure 3 :

 3 Figure 3: Extracts from the textbook (Böer et al., 2008, p. 20f.), translated by the author

Drawings in squared grid can be reduced by merging several small boxes. Merge four boxes each for reducing in your notebook. Is an original figure enlarged or reduced true to scale, the picture figure and the original figure are similar to each other. The length ratio between the line segments of the picture and original figure is determined by the scale k. […] Each angle of the picture figure is just as big as the corresponding angle of the original figure.

Acknowledgment

The Deutsche Telekom Stiftung funds 'MAGENTA-Explaining mathematics, discovering geometry, and participating in small group processes. Design research study on supporting mathematical explanations of language learners in grade 9' (Senior Fellowship Fachdidaktik MINT to K. Erath).


Vinay Kathotia
email: v.kathotia@mmu.ac.uk

Elizabeth Kimber
email: elizabeth.kimber@bishopg.ac.uk

Diagrammatic representations for mathematical problem solving

Keywords: Representations, diagrams, lived-in spaces

Introduction

 [START_REF] Noble | Experiencing change: The mathematics of change in multiple environments[END_REF]provide multiple references regarding the "strong support in the mathematics education community for the view that students should encounter mathematical concepts in multiple mathematical environments" (Noble et al., 2001, p. 85) and be able to connect these. This poster explores, via a case study, some of the related challenges for the restricted setting of using diagrammatic or graphical representations for arithmetic and algebraic problems. It uses a framing of 'lived-in spaces' [START_REF] Nemirovsky | Body motion and graphing[END_REF] and the underlying research question is, "how can we support the use of diagrammatic or graphical representations becoming a 'lived-in space' for users?"

The case study is structured around a particular problem, set out as follows. There are three circular cardboard discs. A number is written on the top of each disc: (6), ( 7), (8). There is also a number (not necessarily the same) written on the reverse side of each disc. Throwing the discs in the air, and then adding the numbers on the faces, I have produced the following eight totals: 15, 16, 17, 18, 20, 21, 22, 23. Can you work out what numbers are written on the reverse side of each disc? (Association of Teachers of Mathematics (ATM), 1977).

In at least two instances, onea professional development workshop with a group of 40 secondary mathematics teachers in England, and twoan online discussion group of mathematics educators, none of the initial shared approaches used a graphical representation of the problem, even when unknown variables were denoted by x, y and z, and could have suggested 3-D Cartesian space. This is striking as a graphical analysis of the problem can help bring to the fore much of the underlying structure. To clarify this, in our case study, two mathematicians, one who had solved the problem graphically and one who had solved it non-graphically, worked on it together for an hour. This was captured and analysed using multimodal microanalysis as by [START_REF] Nemirovsky | Diagram-Use and the emergence of mathematical objects[END_REF].

Theoretical Background

Those working on the above problem had access to graphical representations but what seemed absent is the creation of a graphical space [START_REF] Nemirovsky | Body motion and graphing[END_REF])a 'common place' where symbols and their referents are made accessible and sensible. [START_REF] Nemirovsky | Body motion and graphing[END_REF] develop three themes: tool perspectives, fusion, and graphical spaces to analyze students' use of a computer-based motion detector in the context of graphing. These three themes are not dependent on the technological nature of the tool and, we propose, apply equally to our setting, with graphical representation being the tool. Indeed, as [START_REF] Nemirovsky | Body motion and graphing[END_REF] observe, "Tool perspectives look at development of graphical space through simultaneously exploring the qualities of the tool and relation between actions and symbols. Fusion is about the blending of action and symbol in discourse within the graphical space" (Nemirovsky et al., 1998, p. 124). The growing familiarisation with a graphical space as it is populated with experiences and actions, which make it a space for purposeful and creative activity, is encapsulated by the notion of lived-in space. [START_REF] Noble | Experiencing change: The mathematics of change in multiple environments[END_REF] propose that "the mathematics that students learn from working in a given environment emerges from their process of making that environment into a lived-in space for themselves" (Noble et al., 2001, p. 86). Our work aims to draw out how the use of graphical representations can become a lived-in space.

Methodology and results

This case study uses a conversation between two mathematicians to investigate what actions and experiences contribute to fluid and effective approaches to solve the problem and how this can enable the fostering of a related lived-in space. Audio and video data were captured and analysed using constructs from the above referenced papers. The findings have two aspects: (i) an analysis of graphical and non-graphical solutions of the problem and (ii) observations on how the relevant lived-in space can be fostered, and related conclusions. We briefly outline the graphical approach:

1. The chosen number on a disc is independent of the other discs so choices can be modelled in 3-D space, one dimension for each disc. A choice of numbers, e.g., (6,7,8) gives a point in this space.

2. Flipping a disc results in a fixed addition or subtractionrepresentable by a translation vector.

3. It follows that there are eight choices of 3-tuples and these correspond to vertices of a cuboid.

4. The resulting constraints on the possible sums enable all possible solutions to be determined.

In relation to lived-in spaces, moving to-and-fro between different representations, translating expressions that are clear or articulable in one realm to the other, considering their affordances and constraints, and reflecting on these, provides opportunities for learners to make the graphical space a more familiar lived-in space. This also enables learners to experience and exercise graphical representation in ways which move it from being a representational tool to a more expressive tool.


Giovannina Albano
email: galbano@unisa.it

Cristina Coppola
email: ccoppola@unisa.it

Pier Luigi Ferrari
email: pierluigi.ferrari@uniupo.it

High school teachers' evaluation of argumentative texts in mathematics

Keywords: Argumentation, language, evaluation, teachers, mathematics education

The goal of this paper is to start an investigation on how high school teachers evaluate students' argumentative texts. In particular, as an argumentation requires dealing with some mathematical content, to produce a verbal text, possibly equipped with diagrams and formulas, and to make clear the links between premises and conclusion, we are interested in finding out whether teachers focus on all of the three aspects or on one or two of them only. We have gathered the comments of 12 high school teachers on five argumentations written by university science students in order to justify their answers to a problem involving formulas and graphs.

Introduction

The importance of argumentative practices in mathematics teaching at all levels has been increasingly highlighted in the last years. In the transition from high school to university these practices are especially important. This holds also for the mathematics classes for science undergraduates, as it is often necessary to teach some basic concepts to large numbers of first-year university students with assorted levels of competence and motivation, in a short span of time. In this context, it might happen that some students try to learn some content by heart or to solve problems through strategies not based on some interpretation of the meanings involved. The practice of asking for argumentations to explain and justify their solution procedures when dealing with mathematical problems has proved effective both to assess students' competence and to help them to adjust their learning processes. Still, a number of students seem not to be acquainted with argumentative practices, regard them as abnormal and cannot tell the difference between a correct and justified procedure and a correct but unjustified one. These remarks induce us to guess that there are some problems about the development of argumentative skills in high school. So we decided to start a study about how high school teachers interpret and assess argumentative text produced by students, in order to see which aspects are, in their opinion, relevant. Throughout the paper we use some terms (such as 'proof', 'argumentation', 'argument', 'explanation', 'justification') in a broad sense, without referring to some distinctions proposed in literature, such as, for example, Duval's distinction between proof and argumentation (2007), or Johnson's distinction between 'argumentation' and 'explanation' (2000), or the distinctions between 'argumentation ' and 'argument' proposed in literature (e.g., Johnson, 2000, p. 105;Bermejo-Luque, 2011, p. 68). We regard the proof of a theorem, as well as the justification of a resolution procedure of a mathematical problem, as texts that, in some way, describe and make explicit instances of the semantic relation of logical consequence1 . We shall try to make this point clear in the a priori analysis.

Theoretical framework

The evaluation of an argumentation involves at least three main aspects: the analysis of the ideas related to the semantic domain of the discourse, in our case mathematics; the linguistic analysis of the text produced and the analysis of how the links between the premises and the conclusion are made explicit. These three aspects are to some extent autonomous, since an argument might happen to be appropriate related to one or two of them and not to the other(s). As remarked by many authors, such as Bermejo-Luque ( 2011), an argument is, first of all, a piece of text. So language is to be regarded as a relevant factor in the analysis of arguments. We adopt a functional-linguistic perspective according to [START_REF] Halliday | An introduction to functional grammar[END_REF] and [START_REF] Hasan | Language, society and consciousness -The Collected Works of Ruqaiya Hasan[END_REF]. This perspective fits well with frameworks assuming that language plays a major role in the development of thought, and also with a sociocultural, non-platonistic view of mathematics. In particular, we assume Hasan's description of the central role of verbal language (2005) and Halliday's definition of the three metafunctions of language, ideational, interpersonal and textual (1985). In short, the ideational metafunction is related to the representational meaning, i.e. what the text is about. The interpersonal metafunction is concerned with interactional meaning, i.e. what the text is doing as a verbal exchange between people. The textual metafunction regards the organization of the message, i.e., how the text relates to the surrounding discourse, and to the context of situation in which it is produced. We believe that in a context of communication like a classroom, all of these components are involved and so are to be taken into account. In particular, the textual metafunction is closely related to argumentation, since explaining the links between premises and conclusion requires a well-organized text. The debate on argumentation theory is widely complex and involves a number of stances. In our opinion both the specific features of the semantic domain of mathematics and the fact that an argumentation is essentially a text are to be taken into account.

Goals

The main goal of this study is to begin to understand how high school teachers of different subjects evaluate the written argumentations produced by students, and in particular what weight they assign to each of the three aspects mentioned above, and how. This is important in order to reflect on the transition from a model of learning based on the acquisition of content only to a model aimed at developing the competencies required to take part in a discourse on such content. Arguing on a subject requires a different and somewhat deeper understanding of the subject itself.

More specifically, our main research question is: do the teachers involved also take into account the logical structure of the arguments they are evaluating (i.e., how the parts of the argumentative text are linked each other)?

Methodology

We asked a group of 12 high school teachers of different subjects (9 Mathematics, 2 Philosophy, 1 Science) to evaluate the argumentations proposed by 5 first year science undergraduate students to justify their answers to a problem administered during a tutorial session. The teachers were involved in a long-term in-service training course focused on transversal competences. We asked them to write down their evaluation of the arguments and any comments they felt appropriate. In particular we asked them to state if they regarded the arguments acceptable for students at the end of high school, with regard to mathematical correctness, linguistic adequacy and explicitation of the links between the parts of the argument. Afterwards we interviewed them. The aim of the interviews was not to elicit further information but to be sure of our interpretation of what they meant in their written evaluations, as it was possible that teachers of different subjects might use some words with different meanings. One example of such a word is 'formal' ['formale'], which can be associated to a range of different meanings, such as 'independent from content' or 'conformed to a given pattern' or even 'related to structure rather than to function'. The problem under discussion is shown below.

Let f be the function defined on real numbers by: 2 () 2

x fx e   . Among the following graphs one is associated to f. Mark it and explain in detail.

A B C D

A good translation from one language to another, which is usually based on grammar, often succeeds in rendering the ideational meaning of a text. With interpersonal and textual meanings there are sometimes problems. In particular, the textual organization of Italian is quite different from English. For example, in the text of Student 1 some subjects are missing. In English this is both a grammatical and a textual error, whereas in Italian it is textually inappropriate but not grammatically wrong. In the translation, if we put the subjects, as we have been taught at school, we hide the textual inadequacy of the text, if we do not, we add a grammatical error (which affects the ideational meaning) to a text which was just textually inappropriate. So we have added the Italian original version.

Student 1: I exclude the graphs A and B because we have calculated f'(0)=2/9 and 2/9 is the slope of the tangent line to f(x) in x 0 . As is positive, the function is increasing, but in A and B we see that decreases => I exclude them. Moreover, doing the sign of the function 2/(e x +2)>0 we notice that the function is always negative, because of tf'(0)=2/9 he minus in front of the function (otherwise positive). Then we exclude the graph C, since from 0 onwards it is positive. The choice then falls on graph D, correct.

[Escludo i grafici A e B poiché abbiamo calcolato e 2/9 è la pendenza della retta tangente a f(x) in x 0 . Essendo positiva la funzione è crescente, ma in A e B notiamo che decresce => li escludo. Inoltre facendo il segno della funzione 2/(e x +2)>0 notiamo che la funzione è sempre negativa, per via del meno davanti alla funzione (altrimenti positiva). Allora possiamo escludere il grafico C, poiché da 0 in poi è positiva. La scelta ricade allora sul grafico D, corretto.]

Student 2: The exact graph is D since by computing f(0) = 2/3 I found the point where the function cuts the y axis, that is 2/3. Furthermore, by computing f'(0) = 2/9 I see that the function in zero is increasing because the derivative in 0 is positive.

[Il grafico esatto è il D poiché calcolando f(0) = 2/3 ho trovato il punto in cui la funzione interseca l'asse y cioè 2/3. Inoltre calcolando f'(0) = 2/9 noto che la funzione in zero è crescente poiché la derivata in zero è positiva.]

Student 3: Surely graph C does not correspond to f(x) as it passes through the origin. Graph A does not correspond to f(x) since it is decreasing in x=0 whereas f'(0) = 9/2 which is positive, so f(x) for x=0 is increasing. Graph B does not correspond to f(x) since it is decreasing in x=0 whereas f'(0) = 9/2 which is positive, so f(x) for x=0 is increasing.

The graph that most likely represents f(x) is D.

[Il grafico C sicuramente non corrisponde a f(x) in quanto passa per l'origine. Il grafico

A non corrisponde a f(x) in quanto è decrescente in x=0 mentre f'(0) = 9/2 che è positivo quindi f(x) per x=0 è crescente. Il grafico B non corrisponde a f(x) in quanto è decrescente per x=0 mentre f'(0) = 9/2 che è positivo quindi f(x) per x=0 è crescente. Il grafico che più probabilmente rappresenta f(x) è il D.]
Student 4: Graph C does not correspond to f as it passes through the origin. I know that in f'(0) the derivative is positive, so its function must be increasing in that point. In A and B this does not happen. So the graph of the function is letter D.

[Il grafico C non corrisponde a f in quanto passa per l'origine. So che in f'(0) la derivata è positiva, quindi la sua funzione deve essere crescente in quel punto. In A e B ciò non avviene. Quindi il grafico della funzione è la lettera D.]

Student 5: We exclude letter C at once since for x=0 it passes through point (0; 0) while f (0) ≠ 0. If we consider that f'(0)>0 holds, the function is increasing in 0 so I exclude letters A and B, as they represent functions decreasing in 0 and a differentiable function decreasing in a point has a non-positive derivative in that point.

[Escludiamo subito la lettera C perché per x=0 passa nel punto (0; 0) mentre f(0)  0. Considerando che vale f'(0)>0 la funzione f è crescente in 0 quindi escludo le lettere A e B, in quanto rappresentano funzioni decrescenti in 0 e una funzione derivabile decrescente in un punto ha la derivata non positiva in quel punto.]

A priori analysis

Students are requested to find which graph out of a set of four corresponds to f and to justify their answer. Teachers are requested to evaluate those explanations. The relation linking the answer to the data is that of logical consequence: the fact that graph D corresponds to f is a logical consequence of the data. In other words, students have to illustrate a mathematical, semantic relation by means of a verbal text, with the option of adding symbolic expressions and other signs.

Basically students had at least three kinds of data: (1) those available from the definition of f;

(2) those available from the four graphs;

(3) the 'closure' condition "Among the following graphs one corresponds to f ". The sentences that are logical consequence of the definition of f (in the context of classical mathematics) are of course a countable infinity. Of course we expect that students will focus on a small choice of them, such as, for example, f(0) = 2/3 or f (0) < 0 or f'(0)=2/9. As regards graphs, the sentences obtainable are infinite too. The process is a bit more complex as the interpretation of graphs anyway requires some conventions. Properties like domain, continuity, differentiability and passage through given points cannot be inferred from the graph alone. At any rate, it is crucial for science students mastering multiple representations. The competencies required to devise counterexamples showing that properties of the kind mentioned above cannot be inferred by the graphs are not included in mathematics curricula for science undergraduates. So there is no alternative to compromise, in relation to the teaching goals. In our classes we usually accept that students, when dealing with a graph, adopt naive interpretations of properties like domain, continuity and differentiability. We usually require a more critical attitude towards properties such as the behavior of the function out of the range represented and the choice of the units of the axes. All of these aspects can be explained by means of examples, problems and activities that are within the reach of our students. So we expect that from the graphs the following properties are inferred.

Graph A: the function associated (fA ) is negative and decreasing in (-3, 0); -1 < fA(0) < -0.5 Graph B: fB is negative in (-3, 2); -1 < fB(0) < -0.5 Graph C: fC is negative in (-3,-1), positive in (1, 3), increasing in (-3, 3); fC (0) > -0.5 Graph D: fD is negative in (-3, 3) and increasing in (-3, 3); -1 < fD(0) < -0.5

The closure condition (see above) is perhaps the most important piece of information, as it makes reasoning by exclusion possible. Without this piece of information the problem would not be solvable. As a matter of fact, in many cases (even though not in all) it is possible to decide that a given graph does not correspond to a given formula, but it is never possible to be sure that a graph corresponds to a given formula. A Cartesian graph, for example, represents a function only in a limited range and in an approximate way. For any graph given in a real interval, there are infinite functions that might be associated to it. In a wide range of school problems, and in particular in many of real world ones, data are almost never usable at once, but they need anyway to be identified and extracted and afterwards they require some basic competencies in order to be used.

Also, a seemingly primitive property like f(0) = 2/3 requires to operate a replacement and apply some arithmetic operations, which implicitly involve a number of properties (and competencies).

Even the recognition that the function f D is negative in the range displayed requires some acquaintance with the conventions usually adopted to draw graphs, and some linguistic competence to master the meaning of words like 'negative' without confusing it with 'decreasing' or with other words. In the same way, recognizing f D as 'increasing' requires some knowledge of the appropriate definition.

Criteria

In order to analyze teachers' comments, we have taken into account the following aspects.

1. The aspect of the argumentation they have focused on (e.g., mathematical, linguistic or logical correctness, communicative adequacy, explicitation level of logical relationships). 2. The overall evaluation of each argumentation, with special to contrasting evaluations. 3. Any other specific remarks about mathematical, linguistic and argumentative aspects.

Outcomes

Of the 9 mathematics teachers only one (Flavia) considered the texts from the viewpoint of argumentation, while 7 of them almost exclusively focused on mathematical content. One of them focused on both mathematical content and language. Also Maria, the Science teacher, and Alberto, one of the Philosophy teachers, considered aspects related to argumentation. In particular Maria took into account the explicitation of the links among the parts of the texts produced by the students. Sara, the other Philosophy teacher appeared more interested in language. As far as mathematical content is concerned, the two Philosophy teachers did not make any remarks.

The teachers provided different and even contradictory evaluations of the argumentations, especially as far as Student 2 is concerned. Ada, Franca, Oriana and Sara place it among the best, whereas Aldo, Carlo, Fabia, Giulio, Lisa and Maria place it among the worst ones. Claudia and Giorgio considered the five texts substantially equivalent. The first group of teachers appreciated the conciseness of the argumentation of student 2. On the contrary, some of the other teachers found the argumentation incomplete. Giulio commented on this text in a different way compared to the other teachers. He claimed that "it is impossible to find -2/3 on the graph, unless the unit is given divided into three parts, which is not the case, since on the y-axis there are multiples of 0,5 only". He also claimed that, even this would be possible, "it would not be enough to exclude graphs A and B".2 From the interview, we gathered that Giulio thought that student 2 had stated two relevant facts but had failed to relate them with the goal of the argument. The text of student 3 has been criticized by some teachers as "redundant". As a matter of fact, the student has explicitly excluded three cases to select the fourth. The structure of his argument is logically much more acceptable than explanation 2, which is compatible with wrong interpretations of logical consequence.

The most common objection raised by some mathematics teachers concerns the association of the sign of the derivative at a point to the monotonicity of the function at that point, whereas speaking of intervals would have been more correct. For example, Lisa, commenting on explanation 4, writes: "Once more we find confusion between pointwise derivative and monotonicity of the function in an interval". Nonetheless, the same teacher, when summing up, states: "The reference to pointwise derivative, which is common to all explanations, could be accepted if one assumes that the students mean to refer to a neighborhood of the point".3 Some of the criticism involves aspects related to style, such as the use of plural forms ("We exclude …", "… we consider …") or of expressions like "letter D" in place of "graph D". For example Ada comments: "In the conclusion expressions like 'the graph of the function is letter D' do not sound well". Some other remarks involve the (linguistic) confusion the between "graph" and "function" or between a point and its y-coordinate, labelled as "imprecisions". On the other hand, Aldo, Fabia and Maria spot problems that severely affect the understandability of the text, such as the lack of subjects or of connectives. Fabia speaks of explanations "linguistically inadequate for undergraduates". Student 3 has been criticized for the use of "probably", even though with different motivations. Giorgio thinks it is a mathematical error, for in classical mathematics a sentence is either true or false. Oriana claims that "it cannot be accepted in any case". On the other hand, Ada, Fabia and Lisa interpret it as lack of confidence. For example, Lisa writes "the student is not fully convinced that the others can be excluded".

The specific issue of argumentation has not been dealt with much. Aldo is the only one who explicitly analyzes the structure of arguments, in a classical way, in terms of premises-conclusions, possibly related to the model of Aristotle's syllogisms. In his opinion "In no argumentation are premises and conclusion properly identified", although he regards the arguments overall acceptable. Fabia, Giorgio and Maria underline the topic of the explicitation of the links among statements and remark that the argument of student 3 is the most explicit of the lot.

Final remarks

Most of the mathematics teachers involved have focused on mathematical content. It seems that their view of mathematics is still oriented to knowledge rather than to competence or reasoning. In other words, their view seems sharply acquisitional rather than discursive. Moreover, some of the objections raised involve aspects (such as the questions related to pointwise differentiability) that are not crucial for students who need a basic mathematical education only. The same teachers seem not regard argumentation as a relevant part of mathematical instruction. This suggests to teacher educators that there is still much attention to pay to teachers' interpretation of mathematics education and to familiarise them with the learning potential of argumentation. It suggests also that teacher formation should focus on argumentation as a component of mathematical competence, rather than as an additional subject to be taught. A nice discussion, at the educational level, of these topics, has been provided by [START_REF] Freudenthal | Logic as a subject or as an attitude?[END_REF], who clearly related the logical topics in mathematics education to language. The interdisciplinary character of argumentation is damaged if it is narrowly interpreted according to the schemes of some content domain, such as mathematical logic, philosophy or law. Most of them seem to regard linguistic competence as conformity to some pattern rather than adequacy to some functions or goal. This is the case of teachers criticizing the use of plural forms or of expressions like 'letter D'. These may not correspond to some stylistic patterns but surely do not severely damage the readability of the text. Moreover, some teachers seem to give value to concise texts rather than to more extensive ones. It is possible that high school teachers' expectations of undergraduates' linguistic competence are too high and that they assume that undergraduates can fluently use literate registers, i.e. the more educated varieties of language. This unfortunately does not often happen. The use of plural forms and of 'probably', which has been criticized by some teachers, seem to be related to the interpersonal metafunction of language rather than to the ideational one. In other words the students using such wordings did not mean to express some specific content but to position themselves in relation to the instructors or even to the other students. On the other hand, only three teachers notice some more substantial weaknesses of the texts, such as the lack of subjects or of connectives, which severely damages the accessibility of the text. In our experience of undergraduate teaching, this is a major problem with a number of students. This is not just matter of style, but, in a framework assuming that language plays a major role in the development of thought, it is a major obstacle for students, who lack the means for effectively representing and compacting information and verbally expressing the organization of their thinking process. It seems that some teachers do not recognize the importance of the language demands of a formal proof and do not relate them to the field of mathematical instruction. Verbally expressing argumentation can allow students to become aware of its logical structure, but the language means to do that, such as connectives and their combination within sentences, are crucial [START_REF] Prediger | Learning to meet language demands in multi-step mathematical argumentations: Design Research on a subject-specific genre[END_REF]. We have a feeling that some of the teachers involved had too high expectations about the mathematical competence of the students involved, as far as they were at university rather than at high school. In the future we need to improve the methodology to reduce the bias caused by the teachers' view of undergraduate studies, for example by asking them to evaluate argumentations produced by high school students of their or other schools. We propose also to expand the number of teachers involved (including both mathematics and other subject teachers) and to use these results in teacher education activities.

A sentence P is a logical consequence of a set  of sentences if any interpretation satisfying all of the sentences of  satisfies P as well.

From f(0) = -2/3 one can exclude graph C only. To exclude graphs A and B more information is needed, such as f'(0).

This difference is not relevant to this problem as all of the functions involved are continuously differentiable.