Maximilian Alexander Loch

Malte S Ubben

Embodied Cognition and Minds in Orbit

Wertepunk

Ausrich -tung

After conducting a pilot study whose purpose was to find out which exemplary mental models about the solar system exist among pupils, we expanded the study to a sample size of 100 pupils of classes 5 to 13, with 93 pre-tests and 78 post-tests.

We observed what impact the use of embodied learning could have during the Erasmus+ project "Aristarchus", which focuses on teaching astronomy basics with the "Human Orrery". This was done by conducting pre-and post-tests, the main component of which was to draw the solar system before and after the sequences carried out as part of the Aristarchus Project. With the results we inductively developed a set of items to classify the drawings, analyse their evolution and refine previously suggested classes. While the shape of a mental model can easily be captured with a drawing, the functional understanding can only be partially represented and will be subject to future studies. There are certain limitations to this method. One of the primary challenges is the difficulty in elucidating the functional understanding behind students' mental models through drawings. While capturing the "shape" of these models is feasible, gaining insight into the functional aspects proves more challenging. Unless we are lucky and clear indicators, like arrows for the movements, are included in the drawings, we cannot assume a student's functional understanding of the solar system. Furthermore, the sample size, encompassing 5 classes across 2 schools, is relatively small. This becomes evident in phenomena like the pronounced Earth Focus in Class 6.2. To enhance readability, the bars in the graph have been normalized, introducing a slight distortion due to single-digit fluctuations in the sizes of the pre-and post-test samples.

Design and Results

Limitations

Our focus will be the development of a reliable tool for grading the student drawings according to their shape, so the categorization will be a less subjective process and the categories more distinct from each other. Furthermore, we want to address the current inability to capture the functional understanding in the mental models by designing further tasks on our questionnaire with which we hope to capture the functional understanding. The main interest lies on the movement of the celestial objects in the solar system, followed by the understanding of the distances between and the sizes of the celestial objects. There is also the question as to why exactly this change in the mental models occurs. On average the students also show new knowledge that is not content of the intervention. To answer this, we will have a more detailed look with intermediate questionnaires, to understand the influence of the individual Sequences.

Multiple Orrery T1 Orrery T2

Orrery: The Orrery types can be viewed as two different versions of the same model, as they can appear quite similar, with the key distinction being that T2 usually, though not always, exhibits high details and a distribution of planets around the Sun. T1, on the other hand, tends to emphasize a strict order with a common lack of details. While an underlying functional reason is suggested, it has not been reliably measured yet.

Earth: The Earth-Focus, or simply Earth, places a strong emphasis on the planet Earth, featuring an above-average size, a significant number of details/colours, or a distinctive positioning of the Earth. This Type however is not always distinctive from the others as is shown here and the "Multiple" Example, where it is also quite prominent.

Multiple: This type typically features multiple planets on a single orbit, hence the name. Sometimes, there is an order resembling a division between inner and outer planets. In general, this model is evenly distributed across the age of the students. It could be seen as a transitional step between the Heap and Orrery models. It shows a higher understanding than the Heap, while still having some major flaws.

Heap: Contrary to the lacking "Functional Quality" of the model, as it lacks any sort of order or movement, this type typically includes all the planets with correct names and somewhat accurate colours and sizes. Yet, this model seems to be the first to disappear.

Heap

Building upon the pilot study, the dataset of which is incorporated into the current dataset, we executed an intervention in the German partner schools participating in the Erasmus+ Project "Aristarchus." The intervention comprised four embodied learning sequences featuring the Human Orrery. During these sequences, students were instructed on the movement of planets, the definition of years, the day/night cycle, and the shape of orbits in a hands-on, embodied manner. While we are currently developing a reliable tool for objectively categorizing and grading the collected drawings, the categorization of the drawings has been done manually by us after Mayring (2014). These categories are elucidated on this poster, with further details available in Loch and Ubben (2023).

Analysing Figure 1 reveals several noteworthy observations. Following the intervention, there appears to be a trend toward disfavouring "weaker" models such as the Heap, and notably the emphasis on Earth, which seems more pronounced in younger students before the intervention. Another observation is that older students tend to commence with a higher percentage of T1 and T2 types, with some weaker types entirely absent in the sample. This implies a natural developmental progression or external influence, as these students have not undergone formal Astronomy education in school. Furthermore, T1 types appear to dominate among older students, while T2 types show a decrease, in contrast to younger students where they exhibit the most significant increase.

Earth

Outlook

  Figure 1: Shown are the 5 different classes from the Sample with the Shape-Types of the students mental models. T2 and T1 represent the Orrery types. M is the "Multiple Planet" Type, H the "Heap" and E the "Earth"-Type. The last two bars are the summary across all students. All graphs are normalised which can lead to slight distortions.


Claude Pasquier

BIBLE : a system for design and Management of Context-controlled Documents

Keywords: structured document, context-controlled document, variable document, automatic design, client-selective mailing, inheritance, reusability

come   L'archive ouverte pluridisciplinaire

Introduction

International standards for the representation of structured documents like ODA [1S08613 89] or SGML [ISO8679 86] are well adapted for the design and the generation of long and sophisticated documents like books or technical documentation. But, in the tertiary industry, most documents are intended for clients. Their constitution depends on the client profile.

By the term client profile, we mean all information which may be used to govern the choice of the structure, the content or the presentation of a document. This information can refer either the recipient of the document, its writer or a particular context like economic position.

For high volume client-selective documents, the traditional document processing model (figure 1) is inappropriate. Volumes are such that cost prohibits the creation of a specific document for each person who receives a mailing, yet the same document may not be applicable for each targeted reader.

Moreover, current standards don't include the concepts of reusability, modularity [Quint 89] or inheritance, even though they have an object-oriented approach. But these concepts are necessary to modelize more than one kind of document.

The group Caisse des Dépots et Consignations is especially concerned by these problems since thousands of documents are produced every day. Therefore, a project named BIBLE was launched in 1990, to define a new architecture for the production of documents.

In the first section of this paper, we explain the differences between a traditional document processing model and the processing model used in BIBLE which permits the semi-automatic design of documents. We introduce in this section a new kind of document representation called context-controlled document. In the second section we describe context-controlled documents. In the third section, we discusse the creation and modification of document models which are used to guide the editing and formatting of documents.. Finally, we introduce in the fourth section, the concept of inheritance in order to provide commun structure for all document models.

A new document processing model

In traditional systems for the manipulation of structured documents, a document is represented by a logical structure created in accordance with specifications hold in a generic structure (called DTD in SGML). The layout structure of the document is generated by a formatting process from the logical structure and some presentation rules. The set of these rules is also called generic layout structure. After that, an imaging process produces the final image of document adapted to the output medium (figure 1).

To automate high-volume client selective mailing, it is necessary to introduce an intermediate step between editing and formatting. This step permits choice of content and structure based on the client profile. We call this new phase the automatic design (figure 2).

Between this new phase and the editing process, we need a specific logical structure with variable parts. The choice of document parts is governed by data issuing from client profile. We use the term context-controlled to express that The tree-like structure of a document is represented by a list of expressions describing the content of each object. For example,

Body = {Paragraph#1 Paragraph#2 Paragraph#3}
Means that the object named Body contains the tree paragraphs between the brackets.

In this representation, documents are described by a fixed indiscriminate logic of subordination.

Variable subordination

A context-controlled document must also contain instructions on how to use variables from the client profile in order to automatically design documents. These instructions may be very complicated because the client profile is used in different ways: it may be printed as is. For instance, the name and the address of the recipient of a letter is simply copied from the database to the document, it may be used to choose content. For example, the form of polite address is computed according to the sex and the family condition of the recipient, it may be used to govern both structure and content. For instance in an insurance contract, some clauses or chapters can be determinated in accordance with client characteristics.

Attaining variable subordination

The use of external data is denoted by $<var>. Like in the sentence :

We let you know that your banking account number $BankAccNumber is in debit balance of $BankAccBalance ECUs.

In addition, to express control structures in the context-controlled document, two other constructors are used:

The choice between several elements, denoted by CASE $<var> OF.

e.g.: FormOfAddress = CASE $sex OF { "Male" : "Mr" "Female" : CASE $married OF { "Yes" : "Mrs" "No" : "Ms " } }

The repetition of an element, expressed by the construct LIST (<lowerLimit>, <upperLimit>) OF. LowerLimit and upperLimit are either a constant or a variable. e.g.: BranchsReports <--LIST (1 .. $nbBranch) OF BranchReport

Linking to the database

To generate structured documents from a context-controlled document, we need to make a link between variables used in the document and data issuing from a database. This is made with the command :

GENERATE < Context Controlled Document >
(<list of variables used in documents > ) USING <list of variables from database >

In combination with a database language like SQL [ANSI 86], this allows us to select the documents to produce. For instance, to send a follow-up letter to all late-payers:

The logical models

The control structures in context-controlled documents use the data from the client profile. Some of these control structures are specific to one document. They are mentioned by the user in the editing process. But other control structures concern a type of document. For instance, the determination of the polite address is the same for all letters. So it can be specified in the letter model. The same approach applies to the name and address of the receiver or the date.

When a document is created, control structures are simply transmitted from the model to the document. In this way, the writer can focus his full attention on the essential of adocument, its content: The absence of an element is represented by the NULL constant. The content of some elements can also be specified in the grammar.

The document model of Letter is expressed in modules. Body and Name are indeed described outside the model. In this way they constitute independants structures which can be reused in the description of other documents.

The layout models

The image of a document is built up from the logical model, with the help of presentation rules [Quint 88]. These rules can be associated with each logical object, or better still, be described separately. With the latter solution, we can have different presentations for the same logical model : a screen layout and a paper layout for example.

From the layout point of view, a document is simply a gathering of boxes [Knuth 86]. Each box contains its own properties, like height, width, border size. The grammar of layout model specifies hierarchy between boxes.

The layout model of a document can be described in a modular way, like the logical model. However, this is not sufficient to allow us to reuse boxes. We must indeed take their position into account.

In the ODA standard, box position is specified in the box itself with the attribute Position (ISO8613-2 p.65) which prohibits reuse of the box. To solve this problem, we mention positions in the grammar, when boxes are used. In this way, box are context independant and can be used in several documents at different places.

This allows us easily to build layout models of documents by placing boxes on the page.

The position and dimension of boxes is expressed either by fixed coordinates or with a set of constraints [Franchi 89] such as separation from other boxes, alignment, dimension suitable for content, etc. For instance, the layout model of a letter can be described by :

Letter-lay = { FirstPage = { AT <POS > FirstPageHeaderBox = { AT <POS >NameBox AT <POS >AddressBox AT <POS >DateBox } AT <POS >FirstPageBodyBox} OtherPages = { AT <POS >OtherPagesHeaderBox = { AT <POS >DateBox2} AT <POS >OtherPagesBodyBox}}

Linking logical models with layout models

In order to be complete, the description of a model must contain links between its logical and layout aspects.

In the project BIBLE, we have chosen to express relations separately, in the form of a set of links:

Name --> NameBox Address --> AddressBox Date --> DateBox, DateBox2 Body --> FirstPageBodyBox:OtherPagesBodyBox
This example shows the tree different possibilities: Address -> AddressBox means that the logical object Address must be displayed in the box named AddressBox. If AddressBox can't hold the content of Address (after reshaping if the box dimension is variable) then the content of Address is truncated.

Date -> DateBox, DateBox2 means that the logical object Date must be displayed in DateBox and in DateBox2. As for Address if a box can't hold the entire Date, it is truncated.

Body -> FirstPageBodyBox:OtherPagesBodyBox means that the content of Body is displayed in FirstPageBodyBox. If Body can't fit in FirstPage-BodyBox, then the rest of the content is displayed on OtherPagesBody-Box.

The management of several models

All models are gathered in a document database. Modularity allows us to create new models by reusing existing modules. But this concept doesn't provide for modification of shared modules in a document neither does it introduce a sufficient structure for modelling large numbers of models.

The limits of reusability

In section 3.1 we said that modularity allows us to reuse structured objects. This is true, but modularity has its limits.

Suppose that we want to create a new model of letter in which paragraphs will be grouped in sections.

To create the structure of NewLetter we can reuse Header of Letter. For describing the Body, we have two solutions : either we rewrite it entirely, or we copy and modify Body from Letter.

In the two cases, the result is the same : Body of Letter and Body of NewLetter are described in an independent way. We lack all advantages of reusability. For instance, if we latter want to modify all letters in order to put the writer's name under his signature, we need to modify more than one model.

The use of inheritance

We have solved this problem by using inheritance. The structure of NewLetter can indeed be expressed in terms of modifications of the model of Letter. In this way, all modifications done within the structure Letter will be incorporated automatically in the structure of Newletter. So to add the name of the writer on all letters, we only have to modify a single model.

In BIBLE, we have chosen to generalize this concept by decreeing that each document be created by specialization. BIBLE defines a single, root document represented by an empty structure. Thus all documents are structured in an tree like manner (Fig. 3).

Using an object-oriented approach

BIBLE uses the concepts, of instanciation and inheritance of object-oriented programming field. Both the modelling of the document database and the functioning of editors can be described by using an object-oriented approach.

The structure of each model can be gathered in a class. Thus the contextcontrolled documents generated from models are described by instances of classes. As in all object-oriented languages we describe the behaviour of instances in classes and the behaviour of classes in metaclasses [Murata 89].

We describe the functions of our syntax-directed document editor in classes and the working of our model editor in metaclasses.

Summary

This article is primarily concerned with the generation of documents from context-controlled documents and the design of document models. The weakness of this part is the use of a set of links to describe the matching between the logical and the layout aspect of a document. We are trying to replace this system by a dynamic matching between the two structures. We are also working on the definition of a new structure, that we'll call context-controlled layout structure, to allow a variable control of the document's presentation However the most innovative aspect of our project concerns the organization of a set of models.

At the present time, we are defining and experimenting the concept of inheritance between structured objects. We are studying the possibility of using multiple inheritance. For instance, creating a new model of a letter which inherits both the Logo defined in LetterWithLogo and the Section used in NewLetter (see section 4.1 & 4.2).

Figure 1 :

 1 Figure 1: A traditional document processing model

Figure 2 :

 2 Figure 2: The model of document processing in BIBLE

  GENERATE LatePayerFollowupLetter ($name $address $date $bankAccNumber $bankAccBalance) USING SELECT c.name c.address p.date c.bankAccNumber c.bankAccBalance FROM (Client c, Parameters p) WHERE c.bankAccBalance < 0 3 The design of document models Models are used in the editing process and the formatting process. They are created and modified by a model editor.

  " "Female" : CASE $married OF { "Yes" : "Mrs" "No" : "Ms"} $name}In this example, the structure of a document is expressed by the grammar[Furuta 88] [Ingold 88] which uses the following constructors [Quint 88] : The aggregate, represented by a list of elements delimited with brackets. e.g : Letter = {Header Body} The choice between several elements denoted by the keyword CHOICE. e.g: Paragraph = CHOICE {TEXT GEOMETRICGRAPHIC} The repetition of an element, expressed by the construct LIST [<lowerLimit> .. <upperLimit>] OF. e.g : Body = LIST [1..*] OF Paragraph The star denotes an unspecified limit.

Figure 3 :

 3 Figure 3: Example of a model hierarchy

Acknowledgements

I am grateful to the members of Reaseach Department of Informatique CDC who kindly read primary versions of this paper and made many constructive comments. I also wish to thanks Prof. Paul Franchi-Zannettacci for his valuable advices and Mr. Kent Hudson for his help.


Laurence Noirez

Patrick Baroni

Patrick Baroni Are

Are there really viscous liquids?

Send this form to: oegr@boku.ac.at

Are there really viscous liquids?

Laurence Noirez 1 , Patrick Baroni 1

1 Laboratoire Léon Brillouin (CEA-CNRS), Université Paris-Saclay, CE-Saclay, 91191 Gif-sur-Yvette Cédex, France A solid body deforms under shear while a liquid flows. The absence of mechanical resistance to a shear stress indicates a liquid behavior. However, is this common definition really satisfying?

Looking at the mesoscopic scale, we show that liquids and melts are actually endowed with shear elasticity that it is hidden in standard rheological measurements [START_REF]Probing sub-millimeter dynamic to access static shear elasticity from polymer melts to molecular fluids[END_REF][START_REF] Zaccone | Explaining the low-frequency shear elasticity of confined liquids[END_REF]. To prove the elastic nature of "liquids and melts", a new series of experiments highlight that ordinary liquids and polymer melts (glycerol, Polypropylene glycol, liquid water ...) are able to emit a modulated thermal (hot and cold) signal when submitted to a mechanical shear stress (figure 1) [START_REF] Kume | Strain-induced violation of temperature uniformity in mesoscale liquids[END_REF]. Thermal response is a collective effect. It is similar to the thermo-elasticity, coupling the deformation of an elastic (solid) body to its temperature and vice-versa. Such properties require long-range elastic interactions that are absent in current viscoelastic descriptions.

Our approach is an invitation to consider liquids as not fundamentally different from solids, highlighting splendid collective effects [START_REF] Kume | Strain-induced violation of temperature uniformity in mesoscale liquids[END_REF][START_REF] Kume | Collective Effects in Ionic Liquid [emim][Tf2N] and Ionic Paramagnetic Nitrate Solutions without Long Range Structure[END_REF]. More than ever, a phononic interpretation is relevant, opening the way to coupling with surface boundaries, whether solid [START_REF] Warburton | Identification by Inelastic X-Ray scattering of bulk alteration due to Liquid Wetting submitted[END_REF] or biological and defining an extended third zone.

2nd Annual Meeting of the Austrian Society for Rheology, November, 17 th , 2023 JKU Linz ABSTRACT FORM

  ÖGR: c/o Department of Food Science and Technology, University for Natural Ressources and Life Science, Muthgasse 18, 1190 Wien

	Yes: I agree that my abstract is provided to all conference participants in the abstract
	booklet ______X______
	No: I do not agree ___________
	Name: __Laurence NOIREZ_____________________
	email of presenter: __laurence.noirez@cea.fr_______________________________________

ABSTRACT: 300 words including title, authors and affiliation(s). Use character size 10 Pt. and paste in your text (*.doc) below.


Elisa Miragliotta

Carola Manolino
email: c.manolino@univda.it

Andrea Maffia

Figural component in geometrical reasoning: The case of a blind solver

Keywords: Geometrical reasoning, figural concepts, spatial reasoning, blindness, gestures

In this paper, we use the Theory of Figural Concepts to describe the geometrical reasoning when solvers cannot rely on visual information. We analyse a case study of a blind mathematician solving a geometrical task. Qualitative analyses allow observations of an effective figural and conceptual blending and problematizing the visual and spatial aspects that seem to be encapsulated into Fischbein's description of figural components.

Introduction

Researchers in mathematics education point out the tension between visuospatial and logical aspects of geometrical objects (e.g., [START_REF] Mariotti | Images and concepts in geometrical reasoning[END_REF]. Scholars agree that effective geometrical reasoning requires the interplay of theoretical and pictorial domains. From a developmental point of view, although gradually logical organisation becomes more important, visuospatial aspects maintain a role in geometrical reasoning, even at a more advanced stage (Sinclair & Gol Tabaghi, 2010). Solvers might differently elaborate representations of geometrical objects because of ineffective visuospatial abilities or visual impairment. Historically, as Sinclair and Gol Tabaghi state, "visual intelligence has been much more prominent than that of motor intelligence" (2010, p. 225). This raises questions about the exploratory power of the lens provided by traditional frameworks on geometrical reasoning. To explore these questions, the case of born blind solvers seems particularly fruitful for conducting this investigation. From the educational point of view, geometrical experience can be grounded on spatial experience but, in absence of visual perception, spatiality might be differently experienced and the hands become critical channels through which "the world is sensed, processed, and conceived" (Healy & Fernandes, 2014, p. 125). We assume that, while solving a geometrical task, a blind solver relies on figural components, as sighted solvers do, despite the fact that the figural component has not been developed through visual-perceptual experiences. Here, we intend to explore to what extent the theoretical lens provided by the Theory of Figural Concepts [START_REF] Fischbein | The theory of figural concepts[END_REF] can describe the geometrical reasoning of a blind solver. With this general aim, we report the analysis of a task-based interview with a born blind mathematician while she solves a geometrical task.

Theoretical framework and research aim

Visual and spatial aspects within Fischbein's Theory of figural concepts

Building on the research in cognitive psychology on imagery, [START_REF] Fischbein | The theory of figural concepts[END_REF] makes a distinction between two different mental entities: the concept that is ideal and general, and the image that is conceived as "a sensorial representation of an object or phenomenon" (1993, p. 139, italics in the original) and is "based on the perceptive-sensorial experience, like the image of a drawing" (1993, p. 148).

Although the Theory of Figural Concepts (TFC) seems to consider primarily visual perception -e.g., pointing out that a geometrical figure is not a mere concept, but a "visual image" too (p. 141) -the definition of image presented above is broad enough to include other sensorial experiences. Acting upon geometrical figures requires an image that is "not sensorially perceived but thought, the genuine object of our geometrical reasoning" (p. 143). Indeed, psychologically, geometry deals with objects intrinsically and simultaneously having a dual nature [START_REF] Fischbein | The theory of figural concepts[END_REF]: as concepts they are controlled by a definition within a mathematical theory, but they maintain the figural aspects of images. These conceptual and figural components provide complementary contributions to a third mental entity, the figural concept, that is the actual object upon which geometrical reasoning is realized, a crucial achievement of geometry learning. In particular, the conceptual components refer to the ideal theorization of a geometrical object as part of an axiomatic system (as learned by the solver); on the other hand, the figural components refer to the "spatial properties (shape, position, magnitude)" (1993, p. 143) of a figural concept. Ideally, these two components are harmonically fused in the figural concept, but more frequently one of the two dominates the other. Effective geometrical reasoning needs to work on figural concepts, as it was constructed by discarding the visual aspects: I do not intend to affirm that the representation we have in mind […] is devoid of any sensorial quality (like color) except space properties. But I affirm that, while operating with a geometrical figure, we act as if no other quality counts. (Fischbein, 1993, p. 143, emphasis in the original) In other words, a figural concept maintains only the spatial aspects of the perceptual experience. For instance, in the case of a rectangle we can refer to its global appearance as a whole, but what counts in geometry are the mutual relationships among its parts (e.g., sides, diagonals). As learners, we might have experienced "rectangularity" by perceiving rectangular objects, but for conceptualizing the rectangle as a geometrical figure we must discard many perceptual features. Nevertheless, researchers have shown that while sighted solvers are exploring the resolution of a geometrical task, they rely on spatial and visual aspects for locally and globally controlling the figure, respectively [START_REF] Miragliotta | From figural to theoretical control (and back): a first proposal for framing the interplay between different controls[END_REF]. Our hypothesis is that while solving a geometrical task, blind solvers rely on figural components, as sighted solvers do, even though the figural component is not visual in nature.

Figural and conceptual blending in geometric reasoning

Geometrical reasoning can be described as a dialectic between figural and conceptual components [START_REF] Mariotti | Images and concepts in geometrical reasoning[END_REF] and the effectiveness of geometrical reasoning reveals the harmonic blending of these two components. For instance, let us consider the following task proposed by [START_REF] Fischbein | The theory of figural concepts[END_REF].

In a circle of centre O, consider two perpendicular diameters AB and CD. Arbitrarily choose a point M on the circumference and consider the projections of M onto the two diameters: the segments MN and MP, respectively. What is the length of the segment PN? (a possible representation in Figure 1a)

The task seems to require a strong theoretical effort. Since the length of MP and MN depends on the position of M, by recalling the Pythagorean Theorem the solver might embark on a long process of calculation at the end of which the figural counterpart might be lost. On the other hand, focusing only on figural aspects and imagining moving M along the circumference, the conceptual counterpart might be lost and PN might be perceived as having different length depending on the location of M.

According to TFC a coherent solution is not reachable by considering the image and the theoretical constraints independently, but "by a unique process in which a distilled figure is considered, revealing logical relationships" (Fischbein, 1993, p. 142). The solution process is supported by an interplay between figural and conceptual aspects: looking at the given theoretical constraints (i.e., orthogonality between two diameters, projections of M on the diameters, M on a circle) the solver can deduce that there are three right angles, then MNOP is a rectangle. PN is the figural component of the figural concept 'diagonals of a rectangle'. Possibly, the solver might conceptualise MNOP as a prototypical rectangle; the drawing could suggest this interpretation. Now the solver can theoretically recall the rectangle's properties and figurally introduce the segment MO, which is congruent to PN being a diagonal of the rectangle. Then, conceptually controlling the figure, the solver can notice that MO is also the figural component of another figural concept: the radius of the circle. So, the conceptual and figural blending allows a simultaneous understanding of MO as the figural counterpart of different figural concepts. The solver can conclude that PN is equal to the radius The role of gestures in the mathematical practices of blind solvers [START_REF] Healy | The role of gestures in the mathematical practices of those who do not see with their eyes[END_REF] have highlighted that blind solvers use gestures for communicating, reasoning upon, and haptically perceiving (representation of) mathematical objects. While we acknowledge that literature provides different definitions for what gestures are, for the sake of this research report, we will consider gestures as those "movements of the hands and arms that are produced when engaged in effortful cognitive activity" (Alibali, 2005, p. 309); indeed, our analysis is focused on the different ways in which an object can be explored using the hands (by touch and manipulation) during a problem solving activity, which is considered as an instance of effortful cognitive activity [START_REF] Alibali | Gesture in spatial cognition: Expressing, communicating, and thinking about spatial information[END_REF]. We conceive gestures as a "tool for seeing" since hands can provide access to the mathematical content as eyes do for sighted solvers, but they are also "tools for thinking", since haptic perception and conceptualization are intertwined: "Like the eyes of the sighted, the hands are moved in an intentional manner […] in order to perceive -and at the same time conceive -the object" (Healy & Fernandes, 2011, p. 159). The resulting conceptualization is potentially different from a visual one. Indeed, "whereas vision is synthetic and global, touch permits a gradual analysis, from parts to the whole" (Healy & Fernandes, 2011, p. 159) and therefore the two perceptions of a same object will not highlight the same properties. In tune with [START_REF] Healy | The role of gestures in the mathematical practices of those who do not see with their eyes[END_REF], we assume that the substitution of one tool (eye) to another (touch) would differently empower cognitive processes. Considering the twofold role of gesture as described above, in this study we look at blind solvers' gestures as windows onto their cognitive processes during the resolution of a geometric task, in order to observe the possible role played by haptic perception within the figural and conceptual blending of the blind solver's geometrical reasoning.

Research question

Starting from the assumption that, during the resolution of a geometrical task, a solver might recall the most suitable figural and conceptual components of the figural concepts in focus, we intend to explore to what extent the theoretical lens provided by the TFC intervenes in the interpretation of the geometrical reasoning of a blind solver. We address the following research question: What is the role played by the figural components (if any) in a blind solver's geometrical reasoning?

Methods

We design an instrumental case study aimed at describing a blind solver's effective geometrical reasoning. We conduct a task-based interview [START_REF] Goldin | A scientific perspective on structures, task-based interviews in mathematics education research[END_REF] with Monica (fictious name), while she solves the geometrical task reported in the previous section. Monica is in her thirties, works as a business analyst, holds an honours degree in mathematics, and reads braille instead of using speech synthesis due to being blind since birth. At the beginning of the virtual meeting where the interview took place Monica was told that she could use whatever equipment or artifacts she wanted. She chose to use the rubber pad, in addition to her braille display. The rubber pad is used to make relief drawings; normally it is a 23x32 cm board. It consists of a wood and metal base, covered with a layer of rubber on which plastic sheets are clamped. The image is drawn with a pen (even without ink); when pressure is applied to the sheet, the plastic forms a relief perceptible to the touch.

The data consists of the video recording of the interview, from which frames and the verbatim transcription were retrieved. For the interview, a webcam was set up on Monica's hands. Therefore, the frames all depict her hands at work on the rubber pad. The data analysis followed a multimodal perspective, by focusing on all the solver's productions. Speech, gestures, and their blending were diachronically and synchronically analysed. Independent analyses of the transcripts were conducted by the three authors; the cases of disagreement were discussed until a consensus was reached.

Results of the analyses

In this section, we describe Monica's resolution path, by reporting on the most telling excerpts in terms of the contribution of conceptual and figural components. After reading the given construction, Monica manifests the need to have an "accurate drawing" which seems to be a personal need to exercise figural control upon the geometric situation by producing a perceptually accessible representation. This drawing needs to be isomorphic with the figural components of her personal figural concepts. She decides to perform a freehand drawing. Table 1 shows the corresponding excerpt. With both hands, she touches the drawing, comprehending it between her hands. Then, one hand touches the intersection of the diameters while the other touches the rest of the drawing.

Finally, she touches the whole drawing with one hand.

All of Monica's drawing processes follow a common path: one hand holds the pen and two fingers of the other hand follow the graphic mark as it is generated (e.g., first frame in Table 1, line 1); then Monica moves the whole hand over the newly obtained drawing (e.g., second frame in Table 1, line 1). The first gesture seems to be used to produce the "accurate drawing", while the second gesture acts as a sort of figural control for the accuracy being sought. The latter hand movement echoes the moment when a sighted solver decides to glance at the global drawing as it is so far. Therefore, during the drawing process, the haptic interaction is local when Monica follows the trace-mark she is drawing, while it becomes global when she is checking the final and whole drawing.

After the drawing is performed, Monica places two fingers at specific positions to check the local consistency of the drawing. This is evident in line 3, when she focuses on the diameters with one hand and on their intersection point with the other. This kind of static gesture reveals the figural components in focus at a certain moment. The claim at line 3 supports our interpretation of Monica's need to have a drawing that incorporates and "well" mirrors the figural component of the figural concepts she has recalled so far. Indeed, the drawing Monica has produced contains a mistake (see Figure 1b): the two lines are perpendicular but one is not a diameter; hence, the circle is not split into four equal sectors. Here we see the interplay between the conceptual and figural components: Monica seems to be quite aware of the conceptual aspects required of her drawing by the problem, but she does not find a trustworthy figural counterpart in the drawing. Then, Monica starts exploring the configuration. She holds with two fingers N and P (Figure 1b), while she claims: "I must calculate the length of the line segment NP". We interpret this gesture coupled with the utterance as a focus on the figural component that is requested in the task. She adds "of this one" following with one finger the possible trajectory of the unrealized segment (see Figure 1c), while the other hand holds the circle that marks the boundaries of the configuration. This gesture externalises the inner process of imagining an object that is not present in the drawing. After this brief exploration, she traces the segment NP using the two hands as she did before (same process as in Table 1, line 1) and she starts exploring the new configuration using two hands (Figure 1d). This global gesture corresponds to a broader focus that is gradually narrowing on the graphical traces within the sector DOB when the gestures move there. This interaction with the drawing suggests to us that Monica is looking for figural regularities that suggest a solution. The haptic perception shifts from local to global and then from global to local, revealing a continuous change of focus.

Monica continues the exploration frenetically moving her fingers on all the segments (NO, OP, MP, MN, NP) in the sector DOB. We interpret the speed of these gestures as a sign that she is looking for a sub-configuration to be further explored. During this exploration, the gestures slow down when Monica seems to find an interesting configuration. Indeed, we observe a new two-handed gesture:

with one hand she touches the angle at N, and she uses two fingers of the other hand to hold M and the segment PN at the same time. Now the focus is local and global at the same time. This moment turns out to be crucial for the following development of the resolution path.

In the following, the frenetic movements on the triangle PMN reveal her need to explore further due to a lack of conceptual components ("I don't know anything"). Monica needs "a more accurate thing", that is a new drawing (Figure 1e) where all the figural components, that she considers essential for obtaining an accurate rectangle, are carefully traced. Then a new exploration starts. She draws PN: she puts two fingers on P and N; then she moves the P-finger to the N-finger following the pen.

She moves the two hands on the rectangle and touches all the segments. Monica frantically explores a specific portion of the drawing, focusing on the sides of the rectangle MNOP. The haptic exploration is driven by the need to gather "accurate" conceptual components through the figural ones that the drawing embeds. Differently from the previous explorations, gestures are focused on the mutual spatial relationship between the figural components (see line 4).

Suddenly, she reaches a new insight into the solution (Table 2, line 5) as if during the exploration she has developed a different conceptualization of the figural component OM. Indeed, we infer that, so far, OM was narrowly conceptualised as a diagonal of the rectangle (see "this diagonal") and now it assumes another conceptual status (see "like if it were a radius") while the figural component remains the same. Therefore, OM is seen as the figural component of two different figural concepts: the circle radius and the diagonal of a rectangle. In this process, figural and conceptual components are deeply intertwined: the control of conceptual components of the figural concept at stake has suggested focusing on the mutual relationship between the radii and the sides of the rectangle; through the haptic exploration coupled with the absence of tracemarks inside the rectangle Monica develops a twofold conceptualization of OM as part of different figural concepts. Indeed, this absence allows Monica to focus more deeply on the rectangle and the radii, and only after on the diagonals. Moreover, the figural component of the recalled figural concept plays an important role precisely because she talks about OM in absence of a physical support but using an ostensive lexicon (see "this diagonal") as if the figural component is embedded into the drawing. Finally, the gesture-speech coordination reveals how the figural concepts are recalled in their figural components (see the gesture at line 5) and only then in their conceptual components (see the utterance at line 5). The last two utterances (line 7-8) show the intervention of conceptual components for concluding that PN is always equal to the radius.

Table 1 :

 1 First excerpt from Monica's interview, time proceeds from left to right let's do the diameters. One… …and two. We said they must be perpendicular.Yes, […] Well, it didn't come out so well, but theoretically I quite have it in mind.What is doneWith one hand she draws a line from a point on the circumference on her left and moves to the right and then the other hand moves along the trace:She moves a whole hand on the drawing:. She selects a point and draws a vertical trace from below to above. With one hand she draws and then the other hand moves along the trace: Then, she moves the whole hand on the drawing.

Figure 1 :

 1 Figure 1: (a) An instance of task's representation; (b-e) Monica's drawing and exploration process

Table 2 : Second excerpt from Monica's interview

 2

	Line	4	5	6	7	8
	What is said	[...] I don't have accurate information.	Ah! But, but… Then there's this thing: this diagonal in the rectangle	is like if it were a radius.	And then also this other diagonal, which is the segment I wanted to calculate,	because I am talking about a rectangle and the rectangle has to equal diagonals.
					will be necessarily
					equal to the radius,
	What	She fixes a right-hand	She moves twice a	She
	is	finger on the point of	finger between O	draws
	done	the circumference and	and M.	MO
		moves repeatedly two		using
		fingers of the left		two
		hand on pairs of		hands, as
		segments MN, MP,		usual.
		ON, OP.

Acknowledgement

The study was partially funded by INdAM (GNSAGA project CUP_E55F22000270001).

Discussion and conclusions

We focused on geometrical reasoning of a blind solver in order to shed light on the role played by the figural component. The reported case study allows us to describe an effective interplay between figural and conceptual components and, in retrospect, reconsider the multifaceted aspects that are encapsulated into TFC's description of figural components. The qualitative analysis provides an insight into Monica's geometrical reasoning and confirms our hypothesis on the role played by figural components, upon which Monica frequently relies during the task resolution. Much evidence supports this claim: from the start Monica creates a drawing that mainly supports figural elaboration; then the investigation is driven by conceptual aspects, but it is devoted to exploring the figure to gather additional properties. The variety of Monica's gestures manifests these different dimensions of haptic perception. Finally, the decision to redraw the configuration manifests the need to better control the mutual relationship between figural components. Monica seems to conceive the conceptual elaboration as reliable enough for solving the task, but also misaligned with the figural components embedded into the drawing. The new drawing allows her to recompose the break between the figural component, haptically perceived, and the conceptual component. Overall, the figural component seems to be essentially supported and exploited using drawing and gestures.

In this study, gestures have a significant multifaceted and intertwined role both for the solver and for the researchers. As for the latter, the analysis of gestures allows us to identify a shift of the solver's focus and infer ongoing processes. This provides evidence of the importance of the spatial dimension which is embedded into the figural component within the TFC. Indeed, the analysis of the haptic interaction with the drawing reveals the crucial role played by a narrow focus on the mutual relationship between graphical traces (e.g., the simultaneous focus on the right angle and the hypotenuse in the right triangle or the exploration of the rectangle's sides), more than the analysis of sighted solvers' processes, where researchers have less access to the specific elements in focus. Broadly speaking, blind solvers' interviews offer a rich situation for learning more about geometrical reasoning precisely because the perception is haptic and directly observable by the researchers. As for the solver, Monica combines the use of different gestures -with one/two hands and one/two fingers -which correspond to different aims: following the drawing process; checking the figural consistency of the drawing; anticipating the position of unrealised figural components; maintaining the focus on a specific object; investigating the mutual relationships between figural components.

We have observed other dimensions of Monica's gestures that can be summarised in three dichotomies: global and local, static and dynamic, fast and slow gestures. It seems that Monica uses a skilful combination of these dimensions to manipulate figural components while harmonising the conceptual components she has developed and, thereby, navigate the solution process. Although further investigations are needed, our hypothesis is that Monica strategically combines different gestures in line with previously developed geometrical problem-solving heuristics. In this sense, gestures are not just "tools for seeing", but crucial "tools for thinking". Despite starting from different theoretical assumptions on cognitive processes, our results are consistent with Healy and Fernandes' findings on gestures as "part of thinking" (2014, p. 142). We had assumed that the haptic perception would differently empower cognitive processes; this point is well demonstrated by the spatial nature of Monica's exploration and conceptualization.

From the theoretical point of view, TFC turns out to provide a suitable interpretative lens to analyse effective geometrical reasoning also when solvers do not have access to visual information. Monica provides a telling example of what it could mean to work on a "purified version of the image" (Fischbein, 1993, p. 148): to be effective the solver's figural component has to be spatially determined and as little as possible visually biased. Therefore, Monica's interview allows us to stress that the visual aspects are less crucial for geometrical reasoning than spatial ones. However, other visually impaired solvers could still rely on their visual past experience; so, other research is needed to unfold personal differences in figural and conceptual blending in the case of visual impairments. From the educational point of view, since Monica provides an example of an effective figural-conceptual blending not rooted in a visual experience, we would like to stress the importance of designing geometrical activities to promote the development of the spatial aspects of geometrical objects. We conjecture that diversifying the channels of access to the geometrical information is a useful design principle for strengthening the teaching-learning processes for each and everyone.


Jacoliene Van Wijk
email: j.vanwijk@uu.nl

Rogier Bos

Anna Shvarts

Michiel Doorman

Activate students? Let them fold! Mathematical paper folding in secondary education in France and Germany

Keywords: Haptic motion, mathematical paper folding, origami, real classroom experience, representations

Though the potential of mathematical paper folding for learning mathematics is known, this practice is still rare in mathematics classes. To inform the design of mathematics lessons enriched with folding, we investigated via a literature review what mathematical paper folding is done in secondary education and why. The research question that we explore in this paper is: What reasons do teachers from France and Germany report for implementing mathematical folding activities in authentic classroom situations? Through a grounded theory approach, we found reasons like: 'to activate students by letting them manipulate paper' and 'to visualise mathematics'. Teachers state that folding allows for dynamic representations that support the transition from informal to formal mathematics and the practice of skills. We summarise these findings in two major categories of reasons: 'to activate students by providing folding tasks' and 'to grow mathematical understanding by folding'.

Introduction

In the Netherlands, the word for tinkering -freubelen -refers directly to the last name of the German pedagogue Friedrich Fröbel . He advocated the use of paper folding to teach the first principles of mathematics to young children. To explain and explore mathematics, paper folding-or origami-is still under development today. There are many ideas and examples of mathematical folding and origami activities for the mathematics lessons [START_REF] Haga | Origamics, mathematical explorations through paper folding[END_REF][START_REF] Hull | Project origami, activities for exploring mathematics[END_REF][START_REF] Meyer | Origami with explanations, fun with folding and math[END_REF][START_REF] Row | Geometric exercises in paper folding[END_REF] from primary school [START_REF] Golan | Origametria and the van Hiele theory of teaching geometry[END_REF] to university courses [START_REF] Demaine | Geometric folding algorithms, linkages, origami, polyhedra[END_REF]. Research on origami tasks in mathematics education concerns topics like geometry [START_REF] Boakes | Origami instruction in the middle school mathematics classroom: Its impact on spatial visualisation and geometry knowledge of students[END_REF][START_REF] Golan | Origametria and the van Hiele theory of teaching geometry[END_REF], axiomatizing [START_REF] Nedrenco | Learning how to axiomatise through paperfolding[END_REF] and logic. Serre and Spreafico state, in the context of their research on teaching mathematical logic with origami, that it can "…motivate students to learn, and actively participate during the lessons, and make it easier for them to grasp complex abstract concepts" (2018, p. 225).

Mathematical folding is known to have great potential, but if we look around, we see that only few teachers use it in secondary school mathematics classes. To understand how one could argue for the scaling up of folding in classroom, we research the motivations of those who already successfully implement it. This study is part of a long-term research project, aiming to explore how mathematical folding activities can be used effectively in mathematics lessons, and what support teachers need. As we want to connect with the teaching culture of secondary education, we investigate what mathematical paper folding is done in secondary education and why. We did a first survey in the Netherlands, going through the digital archive of 95 volumes of Euclides, the magazine of the Dutch association of mathematics teachers (NVvW) [START_REF] Van Wijk | Paper-folding in mathematics classes in the Netherlands, part 1[END_REF][START_REF] Van Wijk | Paper-folding in mathematics classes in the Netherlands, part 2[END_REF]. Analysing this preliminary survey, we introduced eight categories: geometry, symmetry and ratios, optimisation problems, folding for understanding functions, folding for convincing, folding logically, didactic considerations, and remaining topics (e.g., 'fun'). These categories are still quite diverse. In this next, expanded survey, including other European countries, we bring a specific focus: the reasons for folding in classroom mentioned by teachers. In this way we have a starting point for developing mathematics lessons enriched with mathematical folding, in line with current teaching practices.

Theoretical framework

The word origami stems from the Japanese words ori and kami, meaning folding and paper, and at present, origami is a worldwide phenomenon. It covers activities from folding as a hobby to scientific research on protein folding, from folding as an astonishing artform to an efficient means of (un)folding solar panels for satellites. Mathematical paper folding is part of origami, and the emphasis is on the haptic process of (un)folding, in relation to a mathematical problem, to enable reflection while creating crease lines, areas and shapes, and to analyse the result mathematically (based on [START_REF] Friedman | The material reasoning of folding paper[END_REF][START_REF] Haga | Origamics, mathematical explorations through paper folding[END_REF][START_REF] Nedrenco | Learning how to axiomatise through paperfolding[END_REF]. Mathematics educators use origami, or mathematical folding, in a variety of ways, but there is a lack of research investigating if folding is effective as a mathematics teaching tool [START_REF] Boakes | Origami instruction in the middle school mathematics classroom: Its impact on spatial visualisation and geometry knowledge of students[END_REF].

For the coding of reasons mentioned by teachers we used a grounded theory approach [START_REF] Boeije | Analysis in qualitative research[END_REF]. As far as we know, research into professional reports by teachers from secondary education on mathematical folding has not been done before. The grounded theory approach allowed us to explore their reports with an open mind. Besides that, we want to use the results from this study to design classroom interventions. Grounded theory and staying 'close to the data' can make this possible. Before we elaborate on our method, we summarize available literature from experts in the field of teaching mathematics with origami (Table 1). The outcomes of our study are compared with this summary. The known examples of the use of mathematical folding stem from expert origami teachers at university and primary school. For example, successful applications of origami and mathematics in university are lecture series by Demaine, Hull, and Meyer. Demaine uses origami as a basis to teach mathematics and computer science [START_REF] Demaine | Geometric folding algorithms, linkages, origami, polyhedra[END_REF]. Meyer has a very different audience. She teaches the course 'general education mathematics', which includes many students who fear mathematics. According to Meyer, a course based on origami "is accessible (…) and students get a peek at more complex math" (2020, min. 9-11). A successful application of origami to learn geometry and skills in preschool is done by [START_REF] Golan | Origametria and the van Hiele theory of teaching geometry[END_REF], via the project Origametria. A special detail to mention here is that Golan started using origami without a mathematical goal in a special school (grade 1-10), as a remedial teaching tool for children from Israel and Palestine. [START_REF] Vardi | The art of origami at Broshim school[END_REF] state that origami is "enriching emotionally, expanding mentally and developing motorically" (p. 4). These examples show the application of mathematical folding for students with various learning requirements: from students with special needs, to students with math anxiety, and mathematically gifted university students. If we look at several books on origami for mathematics education, only the book of [START_REF] Tubis | Unfolding mathematics with origami boxes[END_REF] contains a chapter on "Why teach origami?". They start this chapter with the philosophy of education of Fröbel. He propagates paper folding for mental development, and to informally introduce geometry. Tubis and Mills mention the positive mathematical experience, interaction and cooperation, patience and perseverance, manual dexterity, spatial visualization, mathematical thinking and language and communication skills. Next to that, they list 17 mathematical concepts related to origami and provide tips for the teacher. This book is one of the few that is aimed at 7 th to 10 th grade. Like in all books, most space is reserved for describing folding tasks. From the books for college and university level, Haga mentions the great interest, enthusiasm, and eagerness of the students for paper folding (2008, p. 1). [START_REF] Hull | Project origami, activities for exploring mathematics[END_REF] states that paper folding enables active and discovery-based learning, with hands-on participation. Meyer and Mukoda (2021, p. vii) state on abstraction: "Mathematics can be abstract, but here our fingers are doing the work: for example, creating a triangle that is congruent to another triangle.", and they mention involvement of the student. We can summarize the remarks from the expert origami and mathematics teachers related to "why teach origami" in Most experts give affective reasons, name the use of hands for folding, and write about learning mathematics. Tubis and Mills stand out with their list of skills. This first summary in Table 1 provides some answers to "why teach origami" from the expert teachers. These teachers have in common that the instructors could dedicate a series of full lessons to mathematical origami topics of their choice. This situation would not stand for secondary schools, where teachers are bound to a certain number of teaching hours, a prescribed curriculum, a prescribed textbook, and so on. As practice shows that secondary school teachers cannot teach full courses based on mathematical folding, we want to investigate via a literature review what mathematical paper folding is done in secondary education and why, despite the limits of school practice. To create a more international perspective, we have chosen to include two other European countries in our review. The research question that we explore in this paper is: What reasons do teachers from France and Germany report for implementing mathematical folding activities in authentical classroom situations in secondary school?

Method

We reviewed articles on mathematical paper folding written by teachers from Germany and France, using a grounded theory approach. We will elaborate on the coding process in the next paragraph. In the sample we included contributions from the French and German mathematics teachers' societies, written by teachers. The French teachers' society, founded in 1910, is called APMEP and is aimed at teachers from kindergarten to university. The German teachers' society, founded in 1819, is called MNU and is aimed at secondary education teachers for mathematics, biology, chemistry, physics, computer science and technology. Following referrals from these contributions, we included some commercial journals for mathematics teachers, aimed at secondary education. The articles from the MNU and Mathematik Lehren were available by membership. We checked if the authors teach, or have taught, in secondary education by googling the authors. We only included articles written in the country's language. We list a summary of the inclusion criteria of the associations of mathematics teachers and journals:

France: Search words: plier, origami. Teachers' society: APMEP. Journals: PLOT (Until 2017), Bulletin Vert (Until 2017), Au fil des Maths (from 2018), Tangente Éducation. Websites: www.apmep.fr, www.tangente-mag.com.

Germany: Search words: papierfalten, falten, origami. Teachers' society: MNU. Journals: MNU-Journals, ML: Mathematik Lehren, PM: Praxis der Mathematik in der Schule1 (Until 2017). Websites: www.mnu.de, www.friedrich-verlag.de.

Process of coding

After we made the summary (Table 1), we coded again from scratch, first for France, and then Germany. We started with 130 French articles. After a first check, like filtering duplicates and articles with the word "multiplier", 72 articles remained. For the first round of coding for France we uploaded these articles from the sample into MAXQDA. We read the articles, and filtered again many articles for several reasons, e.g., not aimed at secondary education, not written by secondary education teachers, or off-topic, like tinkering. Eleven articles remained. The full coding process, with categories in italics, is summarized in Figure 1. The first step, the open coding procedure, means getting a first impression of the data, break it up data into smaller parts, and make first categories. We started off coding texts about mathematical topics (what is folded) and texts with reasons, tips, and stories of folding in the classroom: the category why folding. The category mathematical topics remained throughout the coding process, and many subcodes, from "proof" to "fractions" and "hexaflexagons" were added. In the axial coding step, we made connections between categories and related categories to subcategories. The why folding topic had to be split up. A part became learning from mathematical folding. This topic contained codes like "cooperation" and "recollect prior knowledge". The other part was qualities of folding, where specific qualities were addressed like "visualizing symmetry", and "activate the students". At the selective coding procedure, we determined important categories to answer the research question. Whe made a distinction between the reasons for using mathematical folding in the classroom and the more practical tips for teachers, like "use a flat surface and colored paper". We renamed learning from mathematical folding, as all categories are about learning, into phases in a learning process. For triangulation, we discussed the categories at each phase and co-coded two articles in phase 1 and 2. When we collectively agreed on the categorisation, we ran the different coding cycle steps for the German articles (180 from the sample, 55 in MAXQDA and 12 remaining articles for axial coding). After selective coding of the German and French articles we jointly discussed the various categories with all four authors and agreed on the classification, names, and descriptions of the categories.

Results

As we are doing explorative research from which we want to use the results to design classroom interventions, the result of our study is a completed codebook on the data of France and Germany.

We aggregated all information that explained teachers' choices for folding activities, resulting in four categories: 1. Reasons for using folding mentioned by mathematics teachers, 2. Phases in a learning process supported by mathematical folding, 3. Mathematical topics covered with mathematical folding, and 4. Notes for the teacher. To answer the research question, we will only present data from category 1 and 2 (made bold in Figure 1) in Table 2 and3. 3 2

Reasons for using folding

Table 2: Reasons to use folding mentioned by mathematics teachers

An example of 1.3: "Au-delà de l'aspect manipulatoire qui permet d'impliquer plus facilement les élèves, des propriétés au programme peuvent être visualisées et travaillées grâce aux pliages." Beyond the manipulative aspect, which makes it easier to involve the pupils, properties in the curriculum can be visualized and worked on thanks to the folding. (Aoustin & Brilleaud, 2021, p. 6).

Phase in a learning process Description of phase FRA # frag.

GER

# frag.

Transition from informal to formal

Folding allows students to take a step in the learning process, and/or to ground it, towards more formal mathematics. An example of 2.1: "Im Anschluss daran können diese Entdeckungen formuliert, überprüft und begründet sowie eventuell verallgemeinert werden." Subsequently, these discoveries can be formulated, verified, and justified, and possibly generalised. (Kaufmann, 2016, p. 12)

Discussion

Using the presented results, we can provide more clarity on the question about reasons for implementing mathematical folding activities in authentical classroom situations in secondary school reported by teacher from France and Germany. There is an overlap in Table 1 of the experts and Table 2 and 3 of the secondary school teachers, in (sub)categories like skills, visualisation, manipulation and involvement of the students. Based on this overlap, we conjecture that experience with teaching mathematical folding tasks of secondary school teachers is in line with what 'professionals' in the field of mathematical paper folding indicate. We can also see some interesting extra categories from France and Germany. The articles from France and Germany are written to inform other teachers, and maybe, next to that, to convince other teachers to use folding in the mathematics lesson. The subcategory 1.5 'To provide an alternative perspective on mathematics' shows the reader that there are alternative representations of mathematics besides paper and pencil tasks. Subcategory 1.7 'To solve problems beyond to ruler and compass' points at an alternative way of doing tasks. It is only coded in the French articles, and the references to this feature of folding were mostly used at the start of an article. This seems to justify the advantages of mathematical paper folding in the classroom compared to 'formal mathematics'. Some more pedagogical reasons from the secondary education teachers are the categories like 'To ground understanding' 1.4, and 1.8 'To encourage reflection'. In some categories the growth in learning mathematics is emphasised, like in 2.1 'Transition from informal to formal' and 1.6 'To implement low floor, high ceiling tasks'. A quote from Hull points to seeing the folded paper as the 'experimental laboratory' of students:

The math ceases to be an abstract entity, only existing in their [students] mind. It becomes tangible, something they can hold in their hand and count or use to compute data from which patterns, conjectures, and theorems flow. (Hull, 2013, p. xiv) This argument by Hull might indicate that 'switching' between the different representations can add educational value, even for students who already have considerable abstract knowledge.

This research is based on articles on authentic classroom experiences from teachers. It was hard to find articles from teachers who described real life situations in the classroom with mathematical folding. Many articles were discarded because they were not written by teachers in secondary education, or the lessons were not taught there. Based on the number of articles used of this research, we have some remarks. Though we have used multiple sources, we cannot claim, for example, that the activation of students is only the result of the folding activity, but that it (partly) might be a result of 'doing something new' during the lessons. We are aware that in other (European) countries, e.g., Italy and Belgium, mathematical folding is applied in the classroom. For our purposes, the present study is sufficient.

Conclusion

Though several reasons for using paper folding were mentioned in the earlier literature, our research on authentic situations allowed us to uncover several additional ones. We divided the group of reasons found into possible categories. This allows us to align them with the theoretical grounding of learning mathematics. Trying to summarize all reasons in a condensed way, we suggest considering the first category of reasons as 'To activate students by providing folding tasks', as all subcategories describe actions of the students or features of the folding tasks. The second emerging category was the phase in the learning process where mathematical folding is implemented. We suggest characterizing these reasons as 'To grow mathematical understanding by folding', as teachers indicate that folding supports students from informal, tangible, mathematics to more formal mathematics, while also working on their skills.

In the reviewed articles, teachers show great enthusiasm for folding in the classroom. From this research, we have a good impression of how teachers perceive and use mathematical folding. Further research should more clearly highlight how students experience mathematical folding, and how the potential of folding can be embedded in classroom practice to engage all students in mathematical activities and support mathematics learning.

Figure 1 :

 1 Figure 1: Summary of the coding procedure

  encourage reflection Folding encourages students to (mathematically) reflect on the topic at hand while folding.

Table 1 : "Why teach origami" based on Tubis and Mills

 1 Table 1, consisting of four categories:

	Affective reasons	Skills	Manual skills	Learning mathematics
	Positive mathematical	Interaction, cooperation,	Manual dexterity	Spatial visualization, mathematical
	experience (TM), Interest,	patience, perseverance,	(TM), Fingers do	thinking, mental development,
	enthusiasm, eagerness	language,	the work (M)	Informal introduction of geometry.
	(H), Involvement (M),	communication skills	Hands-on	(TM), Active and discovery-based
	Students with special	(TM)	participation (H)	learning (H).
	needs (M, G)

(TM), Haga (H), Meyer (M) and Golan (G)

Table 3 : Phases in a learning process supported by mathematical folding

 3

	6	29

Only following referrals (from MNU reviews)

Acknowledgments

We acknowledge funding from the Dutch Research Council (NWO), grant number 40.5.22395.009.