text:
Madlin Schmelzer
email: madlin.schmelzer@uni-muenster.de

Stanislaw Schukajlow
email: schukajlow@uni-muenster.de

Strategies for fostering students' reading comprehension while they solve modelling problems

Keywords: Modelling, strategies, reading comprehension

A profound level of reading comprehension is essential for solving modelling problems, as a problem solver has to understand the real-life situation presented in the task in order to construct an adequate situation model, real model, and mathematical model. The aim of this paper is to present the theoretical grounds and a sample learning environment for fostering reading comprehension. In the first part of this article, we summarise research on reading comprehension while solving modelling problems, report on research on two strategies (text highlighting and selfgenerated drawing) that can help learners comprehend a modelling problem, and describe ways to implement both strategies in the classroom. In the second part, we present a learning environment that can be implemented to foster reading comprehension in the lower secondary classroom.

Introduction

Reading comprehension is important for learning in school and for students' everyday lives. As each school subject offers students different kinds of texts, fostering subject-specific reading comprehension strategies is necessary for learning and problem solving. In math classes, students have to read different kinds of texts such as proofs or word problems. We address word problems, which, in order to be solved, require a demanding transition from the real world to mathematics as modelling problems (Blum, Galbraith, Henn, & Niss, 2007). To solve a modelling problem, students need a profound understanding of the task, as a superficial combination of the numbers given in the task is not sufficient for finding a solution to a modelling problem. In this paper, we characterise the role of mathematical reading competency for the solution of modelling problems, summarise research on two strategies (highlighting and self-generated drawings) that might improve reading comprehension, and present a learning environment for fostering these strategies in the classroom.

Theoretical background for reading comprehension and modelling

Reading comprehension while solving modelling problems

Analyses of the cognitive processes that people engage in during mathematical modelling have often distinguished the following activities in the solution process: (1) The problem solver constructs a situation model on the basis of the information presented in the task and on his or her prior knowledge. The situation model reflects the learner's mental representation of the given situation. A profound level of reading comprehension is essential for constructing the situation model if a problem is partly or completely represented as text. (2) The situation model is then simplified and structured to obtain a real model that contains only the information necessary to solve the task. The problem solver needs a deep understanding of the problem to separate relevant from redundant information. (3) Mathematising the real model leads to the mathematical model. (4) Working mathematically then serves to produce a mathematical result, which is then used to refer back to the real situation given in the task by interpreting and validating the real result. In the following, we concentrate on understanding and simplifying/structuring, for which reading comprehension is critically important.

Understanding and simplifying/structuring are expected to be essential for solving the entire modelling problem. These expectations were supported by the results of the study by [START_REF] Leiss | The role of the situation model in mathematical modelling-Task analyses, student competencies, and teacher interventions[END_REF]. In this study, the strategies that students used in the construction of an adequate situation model and real model were found to have a positive influence on their modelling competency. Therefore, we conclude that it is important to support learners' reading comprehension by the use of strategies. Two strategies suitable for fostering reading comprehension in modelling problems are presented in the following chapter.

Highlighting and self-generated drawings for reading comprehension

Highlighting. Highlighting is a cognitive strategy that aims at directing the learner's attention to the specific words or sentences in a text. To highlight important information, the learner has to identify and select the relevant information.

The prompt "Use the highlighting strategy" is not sufficient by itself to support the learner's reading comprehension, as the quality of strategy use has been found to be important for effects on students' performance. The use of highlighting has to be regulated by students in order to be efficient, otherwise the learner might highlight too many words in the text or might forget to highlight some important information. In a study of college students, [START_REF] Leutner | Self-regulated learning with a texthighlighting strategy[END_REF] examined the effects of training the highlighting strategy with expository texts. They compared three groups: The first group did not receive any training and served as a control group, the second was given a strategy training in highlighting, and the third was given a combination of strategy and self-regulation training. The findings revealed that the third group outperformed the others in text comprehension. Thus, selfregulation seems to be an important factor for strengthening the appropriate and purposeful use of cognitive strategies such as highlighting. The combination of strategy and self-regulation training included the following elements: First, the learning goal was presented. Second, the participants observed how a fictive person applied the highlighting strategy. Third, they were given a strategy training in which the steps of the highlighting strategy were presented and then applied. Fourth, a self-regulation training followed, in which the steps of the highlighting strategy were recalled, and then the steps of the metacognitive strategy (monitoring, self-evaluation, and reaction) were introduced and applied.

Highlighting strategies can also be expected to be useful in the domain of mathematical modelling, as modelling problems with reference to real life often contain redundant information. In order to construct a situation model and a real model, students have to identify relevant information. In an exploratory case study by [START_REF] Leiss | The role of the situation model in mathematical modelling-Task analyses, student competencies, and teacher interventions[END_REF], some difficulties in the use of the highlighting strategy while solving modelling problems were observed. An analysis of students' solutions showed that some students highlighted all numbers written in numeric form given in the task, including numbers that were not needed to solve the modelling problem. Further, some students did not highlight numbers written in word form, even when these numbers were essential for the solution. These observations demonstrate the limited quality of the highlighting strategy by students and emphasise the importance of improving the quality of the use of this strategy.

Self-generated drawings. Another important cognitive strategy that can be applied to support a learner's reading comprehension and problem solving is the creation of self-generated drawings. Whereas highlighting is aimed at selecting the most important information, drawings are aimed at organising and visually representing information given in the text.

A study by [START_REF] Leopold | Science text comprehension: Drawing, main idea selection, and summarizing as learning strategies[END_REF] revealed the advantages of drawing activities for the comprehension of science texts. Students in grade 10 were instructed to read text paragraphs and then make a drawing that represented the main ideas of the paragraphs. To train the students to use the strategy, they worked on an example that demonstrated how to process the text with a related drawing. The results showed positive effects of the drawing instructions on students' science text comprehension. Drawing activities encourage students to construct a mental model and seem to offer a useful strategy for facilitating students' deeper understanding. In the domain of mathematics, positive effects of drawing activities were found on 3rd-grade students' word problem solutions [START_REF] Csíkos | The effects of using drawings in developing young children's mathematical word problem solving: A design experiment with third-grade Hungarian students[END_REF].

Drawing activities might also support the construction of a situation model in the context of mathematical modelling [START_REF] Rellensmann | Make a drawing. Effects of strategic knowledge, drawing accuracy, and type of drawing on students' mathematical modelling performance[END_REF]. Strategic knowledge about drawing was found to have a positive effect on modelling performance. This effect was mediated by the accuracy of the situational and mathematical drawings and emphasised the importance of the quality of the strategy for solving modelling problems. In addition, the study revealed that the accuracy of mathematical drawings is a strong predictor of modelling performance, whereas the situational drawing had only indirect influences on performance by facilitating the construction of a mathematical drawing. These findings suggest that self-generated drawings offer a strategy that is useful for fostering modelling. The most promising was found to be the generation of accurate mathematical drawings. Thus, when instructing students how to generate a drawing, teachers should pay special attention to the accuracy of the mathematical drawing (namely that it contains correct relations and all relevant numbers). Learners should be encouraged to generate a situational drawing if they do not succeed in drawing a more abstract mathematical model in their first attempt.

Even though highlighting and drawing seem to be useful strategies for fostering reading comprehension during mathematical modelling, they need to be taught in rich learning environments. In the following chapter, we present some learning environments that are appropriate for teaching these strategies.

Highlighting and self-generated drawings in learning environments for improving modelling

Several studies have investigated the effects of different learning environments on students' modelling competency. In the following, we present two studies that integrated highlighting and drawing strategies (among other elements) in their learning environments to foster modelling competency. [START_REF] Verschaffel | Learning to solve mathematical application problems: A design experiment with fifth graders[END_REF] revealed the positive effects of a certain learning environment on 5th graders' modelling and problem-solving competency. The learning environment contained the acquisition of an overall metacognitive strategy that involved five stages in the planning of the whole solution process. Eight strategies (e.g., "Distinguish relevant from irrelevant data" or "Draw a picture") were embedded in the first two stages (Verschaffel et al., 1999, p. 202). The distinction between relevant and irrelevant data is related to the highlighting strategy, as appropriate highlighting aims to make this distinction. Another condition under which the results were acquired was the instructional technique used in this study. It consisted of systematic changes between whole-class discussions and small group work. In both phases, the teacher encouraged the use of strategies and encouraged the students to reflect on their purposeful use in order to stimulate the regulation of strategy use.

A learning environment for modelling that included strategic elements was examined by [START_REF] Schukajlow | Scaffolding mathematical modelling with a solution plan[END_REF]. A scaffolding instrument called the solution plan with four steps was used in this study to support students' modelling activities. Strategic prompts were assigned to each step. As a whole, the solution plan served as a metacognitive planning strategy that was designed to guide students through the process of solving a modelling problem. Fostering reading comprehension was not the sole focus of the solution plan, but it included cognitive strategies that were aimed at improving reading comprehension (e.g., strategies such as "Look for the data you need and, if necessary, make assumptions!" or the strategy "Make a sketch!"; [START_REF] Schukajlow | Scaffolding mathematical modelling with a solution plan[END_REF]Schukajlow et al., , p. 1244)). Although the highlighting strategy was not explicitly mentioned in the solution plan, it is closely connected to the strategy of looking for relevant data.

The student-centred operative-strategic learning environment used in this study is characterised by a systematic change between individual work in groups and whole-class discussions. The whole-class discussions included presenting solutions and reflecting on the solution processes [START_REF] Schukajlow | Scaffolding mathematical modelling with a solution plan[END_REF](Schukajlow et al., , p. 1243). The study found that an experimental group that was taught the solution plan outperformed a control group that was not taught the solution plan in solving modelling problems. Furthermore, students in the experimental group reported more frequently using self-reported strategies than the control group.

On the basis of the theoretical and empirical findings on the effects of highlighting and selfgenerated drawing, we developed a learning environment for fostering reading comprehension. We describe this learning environment which will be approved in the next step of the project in the following section.

Learning environment for fostering students' reading comprehension while they solve modelling problems

Based on the theoretical grounds presented in the first part of this paper, the following learning environment was developed to foster 9th graders' modelling competence with special regard to the beginning of the modelling process, namely understanding, structuring, and simplifying. The aim of the learning environment is to improve students' performance in these sub-competencies by fostering their reading comprehension via trainings in highlighting and the use of self-generated drawings. Similar to some other studies that implemented strategy trainings (see e.g., [START_REF] Leutner | Self-regulated learning with a texthighlighting strategy[END_REF][START_REF] Schukajlow | Scaffolding mathematical modelling with a solution plan[END_REF], the duration of the teaching unit will be five lessons with a total of approximately 225 minutes1 .

The modelling problems that will be used in the present learning environment include text and can be solved by applying the Pythagorean Theorem as a mathematical procedure. The Pythagorean Theorem was chosen because of the importance of this mathematical procedure for national and international curricula. Before the beginning of the teaching unit that was designed to foster reading comprehension, students are expected to know the Pythagorean Theorem and to practise it on intramathematical problems. A sample problem Reaction time is shown in Figure 1.

Reaction time

During the 2016 European Championship, Germany played against Slovakia in the round of the last sixteen. With goals by Boateng (minute 8), Gomez (minute 43), and Draxler (minute 63) the German team won with a score of 3 to 0.

In the 14th minute, Germany was allowed a penalty kick after a foul by Slovakia. A penalty kick is shot from a distance of eleven metres from the goal, which has standard measures of 2.44 m in height and 7.32 m in width. The German penalty taker was Mesut Özil, and Matus Kozacik was in the Slovakian goal.

Unfortunately, the penalty kick was stopped by Kozacik so that Özil missed the chance to have an early score of 2 to 0 for Germany. His penalty kick was shot a bit too feebly and flew just over the ground to the lower right corner where Kozacik was able to deflect it away from the goal. Although the penalty kick was not shot very hard, the goal-keeper didn't have much time to react, as the football flew at a speed of about 80 km/h towards the goal.

Calculate how much time the goal-keeper had after Özil's kick to reach the position where he stopped the ball just before the corner of the goal.

Figure 1: Sample modelling problem Reaction time

In line with the solution plan study by [START_REF] Schukajlow | Scaffolding mathematical modelling with a solution plan[END_REF] and the study by [START_REF] Verschaffel | Learning to solve mathematical application problems: A design experiment with fifth graders[END_REF], the learning environment that we chose for our teaching unit includes systematically changing between individual work, group work, presenting solutions, and reflecting on the solution process as a class [START_REF] Schukajlow | Scaffolding mathematical modelling with a solution plan[END_REF](Schukajlow et al., , p. 1243).

In the first lesson, both of the strategies of highlighting and using self-generated drawings are introduced. The students are given the modelling problem Reaction time (cf. Figure 1) and are requested to highlight important information and to generate a drawing while doing their own individual work in groups, but they are asked not to solve the problem. The task requires the application of both the highlighting and drawing strategies.

After the individual work in groups, students present their highlighted texts and their drawings and describe how they proceeded in applying both strategies. To encourage a discussion about potential difficulties in the use of these strategies, the group that the teacher chooses to make the first presentation should be one that had difficulties with the generation of the highlighting or drawing. During the presentation and the subsequent reflection on the presented solution, the teacher should direct students' attention to typical problems that result from the misapplication of strategies. The teacher should then present the learning goal of the teaching unit to the students, namely to improve reading comprehension and the ability to solve modelling problems.

After that, the first two steps of the solution scheme (cf. Figure 2) are introduced. They save the results of the class discussion in written form and might also provide some advice that was not mentioned by the students. The solution plan by [START_REF] Schukajlow | Scaffolding mathematical modelling with a solution plan[END_REF] was adapted to better fit the aim of fostering reading comprehension and to guide the application of both of the strategies of highlighting and producing self-generated drawings. The solution scheme with the reading strategies of highlighting (integrated in step 1) and the creation of self-generated drawings (integrated in step 2) is shown in Figure 2. The entire solution scheme serves as a planning strategy for the whole solution process. In step 1, the highlighting strategy is presented. First, students are told to skim the task. This means they should get an overview of the task, which might contain a title, text, questions, pictures, tables, or diagrams. While skimming the text, students do not need to understand each word in detail. After skimming the text, the students are prompted to imagine the situation presented in the task. This might help to activate prior knowledge about the topic and facilitate reading the text in detail in the next step. These activities stimulate the understanding of the real situation. The highlighting is prompted after they read the question again, as the selection of relevant data depends on the question posed in the problem. After they finish the highlighting, students evaluate whether they highlighted only the most important information in the task and revise their highlighting if needed. The highlighting is aimed at helping students to simplify the given information.

In step 2 of the solution scheme, the students are asked to make a drawing and label it with relevant data from the text. These activities help them simplify and structure the information given in the text. During the following monitoring activities, students check whether their drawings contain all relevant data from the text and whether all mathematical relations are represented correctly. If students do not succeed in constructing an accurate mathematical drawing, they can first generate a less abstract situational drawing. At the end of step 2, students mathematise the information given in the drawing. In step 3, the students calculate a solution and obtain the mathematical result. They interpret, validate, and present the result in a final answer in step 3. The arrow pointing back towards step 1 indicates that the solution process might be restarted if the result does not fit.

In the first two lessons of the teaching unit, the students are familiarised with only the first two steps of the solution scheme in order to train their reading comprehension strategies and the subcompetencies of understanding, simplifying, structuring, and mathematising as part of the modelling process. In lessons 3, 4, and 5, students practise the entire modelling process by applying the entire solution scheme.

In line with Leutner et al.'s (2007) study, the current learning environment contains the same main elements to stimulate self-regulation. First, goal setting is realised in the first lesson. Students are confronted with a modelling problem that requires a profound level of reading comprehension. The teacher explains that the aim of the teaching unit is to learn strategies that support reading comprehension and to solve reality-related modelling problems. Second, instead of observing the application of the strategy by a fictive person, students analyse their classmates' highlighted texts and drawings in both the work done in small groups and the presentations involving the whole class. These practises are implemented in order to stimulate students' activities in the classroom. Third, the strategy training begins with a presentation of the steps that are necessary for highlighting and drawing and is followed by an application of the strategies while solving modelling problems. In contrast to Leutner et al.'s (2007) study, the self-regulation training is integrated in the strategy training. If requested, the teacher gives strategic advice by referring to the relevant steps of the solution scheme and stimulates reflection on the use of strategies during the individual work in groups and during the whole-class discussion when solutions are presented and reflected on. This process helps to encourage the use of strategies and to establish the solution scheme as a scaffold for solving modelling problems. In order to stimulate the self-regulation of strategy use, we included the prompts "Check your marks and change them if necessary" in step 1 and "Check your drawing" in step 2 of the solution scheme. Further, the validation of the results of solution problems is stimulated by the prompt "Check if your result fits approximately" in step 3 (cf. Figure 2).

Summary and future steps

In the first part of this paper, we discussed the theoretical background for reading comprehension and modelling. Based on the theory, we presented in the second part of the paper a learning environment to foster students' reading comprehension while solving modelling problems. This learning environment will be evaluated in a project for pre-service teachers. The pre-service teachers will obtain the material and instructions to implement the learning environment in their classrooms to gain practical experience in fostering reading comprehension in mathematical education. They will attend a seminar to prepare for the project and to reflect on the experiences they made while implementing the project.

Figure 2 :

 2 Figure 2: The solution scheme used in the learning environment

Proceedings of CERME10

[START_REF] Schukajlow | Scaffolding mathematical modelling with a solution plan[END_REF] measured effects after

minutes of total treatment.[START_REF] Leutner | Self-regulated learning with a texthighlighting strategy[END_REF] used a time of 150 minutes.[START_REF] Verschaffel | Learning to solve mathematical application problems: A design experiment with fifth graders[END_REF] used 20 lessons to teach eight strategies, so five lessons for teaching two strategies seemed appropriate for our teaching unit.Thematic Working Group 06Proceedings of CERME10

Acknowledgment

This project is part of the "Qualitätsoffensive Lehrerbildung", a joint initiative of the Federal Government and the Länder which aims to improve the quality of teacher training. The programme is funded by the Federal Ministry of Education and Research. The authors are responsible for the content of this publication.


domain: math shs

text:
Gemma Sala
email: gsala@ub.edu

Vicenç Font
email: vfont@ub.edu

Berta Barquero
email: bbarquero@ub.edu

Joaquim Giménez

Mathematical modelling in an archaeological context: Their complementarity as essential tool for inquiry

Keywords: Inquiry competence, mathematical modelling, co-disciplinary context, geometrical model, task design

This paper focuses on analyzing the potentialities of using a codisciplinary inquiry context to enhance the appearance of mathematical modelling in teaching and learning practices. We focus on an archaeological context where modelling becomes an essential tool to enquire into and progress on the study of questions emerged from the interplay between two main disciplines: Mathematics and History. We present the design, implementation and analysis of a teaching sequence based upon the ruins of a Roman theatre discovered in Badalona (Catalonia) to promote inquiry and modelling students' competences. The sequence was implemented in 2015 with 12 and 14-year-old students. The central question introduced (what kind of building could have been the discovered ruins) involved students in a reflective inquiry process that facilitate them to progress on modelling, at the time modelling facilitate that the inquiry process could follow.

Introduction

The main aims of our research are to (1) design codisciplinary teaching sequences to promote inquiry students' competence, (2) implement them and (3) analyze their affordances to enhance inquiry and modelling students' competences. In particular, this paper focuses on the design, implementation and analysis of a teaching sequence based upon an archaeological context-the ruins of a Roman theatre discovered in Badalona (Catalonia, Spain) and how modelling became an essential tool to enquire into several questions that emerged from this extra-mathematical context, in order to progressively build up the necessary subject knowledge to provide and validate answers.

As aforementioned, the sequence started with an initial situation that leads to the formulation of the specific initial question and became the core of the students' research. The starting situation which is presented to students is the discovery of some Roman ruins in a Badalona's suburb by the archaeologists' team of the city Museum. The archaeological report that is provided explained that these ruins could belong to an ancient Roman building, but what kind of building could it have been? Could be a theatre? A circus? An amphitheatre? In order to face these questions, it is essential to make students unfold their inquiry and modelling competences and help teachers to guide student's reflections and to organize inquiry activities so that (mathematical and historical) knowledge could emerge and be used.

In the research we present, the codisciplinary context here considered for the design of the sequence -an archaeological context adapted from a real research-presents several advantages, which will be further developed in the following sections. On the one hand, requires that Mathematics and History work together to face most of the questions that appear in the teaching and learning sequence. This no so common complementarity between both disciplines allows us to provide authenticity [START_REF] Vos | What is 'authentic' in the teaching and learning of mathematical modelling[END_REF] to the questions faced and to promote the blend of inquiry and modelling in different steps of the study, as also some previous investigations have shown [START_REF] Sala | Tareas matemáticas de contexto históricocultural para el desarrollo de la competencia de indagación en Primaria[END_REF][START_REF] Sala | A multidisciplinary approach to model some aspects of historical events[END_REF]. On the other hand, this codisciplinary context is enough close to students (as it is located in the same city that the school is) that facilitates that students can have access to the real ruins and to experts' answers about their real investigations, providing legitimacy to their inquiry process and responses. After describing the main theoretical frameworks upon which the design of the teaching sequence was proposed and analysed, we present the main research questions about how the use of codisciplinary inquiry contexts can help to break disciplinary boundaries (in particular between Mathematics and History) and about how their interaction leads to a rich complementarity between inquiry and modelling.

Theoretical framework and research questions

The research we present in this paper considers different theoretical aspects. First, it is assumed the notion of basic competence considered in Catalan curriculum. More concretely, we are interested in the notion of inquiry competence that, following [START_REF] Sala | Competència d'Indagació matemàtica en contextos històrics a Primària i Secundària[END_REF] and framed according to the Catalan curricula of Primary and Secondary school education, it is defined as:

The ability to mobilise the suitable knowledge and the appropriate resources that facilitate the development and application of a logical and critical methods -under the teachers' guidancein order to look for and find answers to problematic questions or situations in some school and/or out-school context. (Op. Cit. 2016, p. 64) In the same line than [START_REF] Artigue | Conceptualizing inquiry-based education in mathematics[END_REF] underlines, an important step in legitimising inquiry-based approaches was the publication of the National Science Education Standards (NSES; NRC, 1996), which called for students to do and know about scientific inquiry, and that teachers should foster the development of inquiry skills. When the inquiry-based approaches migrate towards mathematics, it seems important to consider approaches paying attention to establishing connections among mathematical and extra-mathematical world, such as modelling approaches. In this sense, some of the inquiry requirements (such as making observations, posing questions, examining sources of information, identification of assumptions, considering alternative explanations; NRC 1996, p.23) are closely related to some essential steps of the cycling process through which mathematical modelling is developed. So that, sometimes, it is difficult to find the differences between both processes (see for instance, the modelling and inquiry cycles presented in [START_REF] Blomhoj | Mathematical modelling -a theory for practice[END_REF] and the linked competences. In particular, as pointed in Niss, Blum & Galbraith. (2007, pp.3-8), modelling constitutes a competence in its own right, which needs to be developed through appropriate modelling activities. From our viewpoint, to promote inquiry mathematical modelling should be placed at the core of the mathematical (and scientific) teaching and learning practices to ensure the right and rich development of important enquire abilities. Also, in the other way round, to promote and ensure mathematical modelling practices, inquiry should nourish some essential steps that may make modelling successfully progress.

Second, we considered different theoretical elements to ensure that the sequence design could offer a rich and functional teaching and learning of mathematics. With this aim, the mathematical and didactic design quality is justified based on the three criteria of didactic 'suitability' -emotional suitability, ecological suitability and epistemic suitability-proposed by the onto-semiotic approach [START_REF] Godino | The onto-semiotic approach to research in mathematics education[END_REF]. Different aspects of its design considered justify the emotional suitability of the teaching sequence: the students could work with real data evidence -data from the research report of the discovery of real Roman ruins-; they could interact with the archaeologists' team of the city Museum and could share their results with them, etc. In turn, ecological suitability was justified by the curricula of these secondary-school students having a competency-based approach. Likewise the implementation allowed students to unleash relevant processes of mathematical activity, in particular processes of mathematical modelling that justifies the epistemic suitability or mathematical quality.

Finally, we also use the notion of research and study path (SRP) proposed in the framework of the anthropological theory of the didactic [START_REF] Chevallard | Teaching mathematics in tomorrow's society: A case for an oncoming counter paradigm[END_REF] and their main characteristics to design the sequence of tasks to achieve a high epistemic suitability (see also [START_REF] Sala | A multidisciplinary approach to model some aspects of historical events[END_REF]. The SRP proposal emphasises the necessary dialectics between 'research' or 'inquiry' (facing openproblems, problem posing, making observations, examining different media or source of information, etc.) and 'study' (attending teachers' explanations to provide mathematical knowledge, building up models, testing the validity of mathematical tools, etc.). As underlined in [START_REF] Barquero | Creating necessary conditions for mathematical modelling at university level[END_REF], mathematical modelling cannot be considered only as an aspect or modality of mathematical activity but has to be placed at the core of it. Modelling is not only as a way to make the functionality of mathematics visible, but also as a key tool for the functional construction and connection of mathematical and extra-mathematical knowledge. In this sense, inquiry and modelling have to be mutually enriched. On the one hand, modelling activity needs from more inquiry moments (posing questions, looking for external resources, looking alternative proposals, etc.). For instance, in the SRP the role of the questions and of question-posing is essential (as the starting point of an SRP with a generating question, powered enough to pose many derived questions) so that it is essential to help students develop some reflective inquiry gestures such as posing new questions along their particular study or using extra-mathematical context to validate mathematical models and make new questions emerge and thereby starting new loops in the modelling cycle. And, on the other hand, inquiry needs from modelling processes (when mathematical tools and models are build, used analysed, validate, etc.) to make the study progress.

Research questions

The aim of our research is to analyse if the implementation of the sequence we had designed, with features as the ones detailed above, could promote the inquiry students' competences and how modelling could become an essential tool to enquire into the inquiry questions that were emerging in order to provide and to validate answers.. Therefore, our main research questions that we focus on are the following: Can a teaching sequence based on the study of an archaeological problem in a codisciplinary context help students to face an inquiry that needs modelling processes as an instrumental tool to find, test and validate answers? Which is the relationship between modelling and inquiry in this kind of teaching sequence?

Design and implementation of the teaching sequence

As it has been introduced, the starting situation is based on the discovery of certain Roman ruins, some years ago, in the centre of Badalona (a city next to Barcelona, Catalonia) by the archaeologists' team of the Badalona's Museum. They concluded that these ruins could have been an antique public building belonging to the classical Baetulo (the Roman old name of Badalona), surely a theatre, and they explained their research in an article [START_REF] Padrós | La ciutat i la memòria: el teatre romà de Baetulo[END_REF].

The sequence was named «What are these ruins hiding? Investigating the Roman ruins of Baetulo» and all the sources, devices designed, links recommended and the worksheets that they had to follow were available in the blog designed by the authors: http://ruinesdebaetulo.blogspot.com/ We implemented the teaching sequence with a group of a 30 students (12-13 year-old) of a Secondary school education in Badalona during two weeks of June in 2015. The selection of the participants was intentionally chosen due to the facility that the first author of the paper had to the school, and the age and syllabus adequacy of the group class. Students worked during all sessions in 'inquiry teams' of three or four members, set up in the first session. They worked as real research teams, formulating hypothesis and doing tasks to validate them, discussing the partial results obtained during the process, finding points of agreement and writing an inquiry report to gather all the ideas, proofs and work done. Each team had on member in charge of explaining and defending their temporary report. Moreover, during all the study process, students got an inquiry guide to help them to progress in such a new activity.

From the archaeological reality to the emergence of questions

The didactic sequence was inspirited in a real archaeological investigation, a situation very close to a real extra-mathematical context, which was introduced to the students by the teacher and the first researcher paper signatory. The aim of the inquiry proposed to the students it is the same of the original aim research: discovering with which kind of building the ruins could be identified. In this sense, the context used to design the didactic sequence was an authentic situation [START_REF] Vos | What is 'authentic' in the teaching and learning of mathematical modelling[END_REF] because the situation introduced to the students is clearly not created for educational purposes even though some elements are included for educational purposes.

Therefore the students had to investigate -from real data, archaeological reports, and canons of Roman architects-what type of building the discovered Roman ruins could have been and its features. In the first session, they could explore the map of the zone where the ruins were located by the link in the blog. They also visited later the place of the discovery accompany by one of the archaeologists of the Museum of Badalona and asked questions, took photographs, and measurements. The current constructions in this zone, houses and streets, followed a curious curved shape-easily perceptible in the map. It indicated, surely, that all these constructions were built on top of the ancient structures.

The role of the context in the students' process of enquiring

Students looked up and investigated information about the Roman architecture to find what type of buildings had a curved part of their perimeter. They found few buildings that showed this feature at least. For instance, theatres were circular, amphitheatres were elliptic or circuses had a part circular and other part quadrangular. This fact generated some other questions that promoted the developing of the inquiry process because the students could formulate their first hypothesis about that kind of building the ruins could below: Which Roman building (theatre, circus, amphitheatre, etc.) shape would concur with the shape of the part of the Roman wall found? What do the geometrical shape of the partial Roman wall discovered determine?

In the first stage of the process, the problematic situation was introduced to the students by the History teacher and the first author. Then, first questions emerged and the students started to look up information about the public Roman buildings in the links, maps and books. At the end of this stage most of the students should have understood the context of problem. So, they could formulate their preliminary hypothesis and conjectures about which kind of public building the ruins could have been only based on the historical information found. So that, at this stage when they had only took into account this historical information, their assumptions included more than one building (all the buildings that could have a curved part). Each inquiry team had to write in their report the agreed hypotheses they reached.

Actually, the inquiry was based on a mainly discovery: a part of a Roman curved wall, a metre and half high. This partial wall belonged to a building that was the centre of the inquiry. It was the external wall of the public building and determining it shape could mean to know which type of Roman building it was. The students had to follow their inquiry from the study of this element and the context where it was find. The number of types of Roman buildings existing limited the quantity of different type of shapes the students had to consider; but other information deduced from de context directly as the orography and the dimensions of the place where the ruins were discovered, or the dimensions of the curved Roman wall discovered, also limited the options that the students could choice in order to formulate their hypothesis about the building the ruins would fit with. In this case, the context influence on the inquiry helps students to deal with a problem that would be very difficult to resolve from an only mathematical point of view and completely unachievable to their level.

Improving the first hypothesis: Building up models to systematise and mathematize the archaeological system

The second stage started with an important session that had the objective of discovering the geometrical shape that the Roman wall described (a circumference, an ellipsis, etc.) and test their first hypothesis. From this stage the teachers of Mathematics, Technology and Catalan Literature had involved in the sequence management. The results obtained from this central task provided enough data that allowed formulate sufficient plausible hypothesis about the type of public building.

The session in where the problematic situation began to be systematised and mathematized were placed on the public square next to the school, and was recorded and afterwards analysed. The students work with an exact representation of the part of the roman wall drawing on the ground. They could experiment with different ways to proof what was the geometrical shape that fit with the Roman wall. After few tasks in which students had to construct and drawing different curves on the ground, all the teams could check that the perimeter of the curve Roman wall fits with a circumference and so, they could conjecture about the building would have been a theatre. Also, this evidence lead students to think that it was necessary to find the radius of the circumference in order to know how large the building was. The students find the radius of the circumference (16 m.) with graphical methods, always on the ground of the square. Then they spread out on the square, drawing with their bodies the perimeter, to notice the likely real dimensions of the Roman theatre.

After this session they returned to the classroom and tried to explain the experimentation in their reports but it emerge the necessity to get a tool to draw the theatre properly. Moreover, the teacher introduced the book by Vitruvius (available in the blog), classical Roman architect who wrote several canons that Roman people followed to construct each type of public building. In this stage the technology had an essential role in order to allow students drawing their geometrical model of the theatre following the Vitruvius' instructions. The students constructed their model of the theatre using the software Geogebra.

Interpreting and validating mathematical models within the archaeological context

When each team had their model of the theatre finished could export the file as an image and pasted it on the map of the area studied, fitting it properly in the exactly site where the ruins was discovered (Figure 1). This task allowed students interpret their model considering the specific context and verify if their construction and hypothesis were suitable. To do the tasks in this second stage the students had to deal with information from the Mathematics -data related of geometrical shape of the wall, extracted from the activity on the square-and from the History -considering the features of the context in where the ruins were discovered-. Modelling, in that moment, became an essential tool that facilitated find answers.

Then, when the students were became competent researchers yet, it started the third stage of the sequence. They arrived at that point with an important collection of questions and doubts but in the following session they had the opportunity, firstly, to share it with other groups and, thereupon, preparing an interview to ask the archaeologist of the Museum. During this interview students could contrast and validate their results about the model selected. It was another especial time you could notice that in the real world different disciplines interacts in a natural way to find solutions of real . problems. After the session outdoors talking to the archaeologist, they could apply their geometrical model construct with Geogebra on maps with other ruins of theatres in Europe and thus could check how the same model fits there, too. The model also was useful counting how many people would fit in the theatre, the last task of this stage. Finally, the students finished writing the final report of their inquiry describing their process, the mathematical tools used, the result of verifying their hypothesis, new opened questions, etc.

Conclusions

We would like to stress the importance that the context and the initial question studied had to deeply involved students in the inquiry activity. Moreover, the possibility to work with real data, the facilities to get access to the ruins and to other real archaeological investigation facilitated that students faced the inquiry questions from a wider approach. What the design and implementation of the activity shows is that the context easily offered different kind of information, such as the one coming from the history, from orography, from geometrical models used to measure and drew plans. It facilitated that students considered all these kind of different nature information and tried to systematise it and to work with it in order to provide answers.

On the one hand, we have shown how the enquiring process carried out in the implementation nests a sub-process of modelling that appears as a tool to contemplate information emerged from the context that could become mathematized (such as: the measure of the curved wall, of the ratio, of the building perimeter or the possible use of the Vitruvius canon) in order to be able to progress in looking for specific answers. Thus, at some stages, mathematics appears as an essential modelling tool to look into systems, build up mathematical models, simulate and test them.

More concretely, in the task done on the square next to the school, students wondering how could know the geometrical shape of the whole Roman wall from the partial wall in the map. But this is a difficult mathematical problem to resolve-to find the geometrical curve from a part of it-because there could be a lot of solutions. Besides, the methods to finds these solutions only from the mathematical point of view, are totally beyond the powers of secondary school students'.

On the other hand, the dialogue with History limits the possible answers because the Roman buildings only had three relevant shapes: ellipsis (amphitheatre), circle (circus) or semi-circle (theatre). Due to the contribution of the historical information the problem became achievable at the students' level of mathematical knowledge and allows the beginning of modelling.

Last but not least, it was also very important the possibility that students know about the experts' work (the real research of archaeologists) and to realise that the process they follow are quiet similar to the process of enquiring and modelling that they were following. It also had an important impact on their motivation and on the perception they had about the usefulness of Mathematics and of modelling. The interaction with the archaeologists allowed students to validate their whole process of enquiring and their results.

Figure 1 .

 1 Figure 1. (a) Application the theatre model, which is construct with Geogebra, on the map studied from one of the inquiry teams report; (b) Detail enlarged

Proceedings of CERME10

Acknowledgment

The research leading to these results has received funding from the Spanish R & D project: EDU2015-64646-P, EDU2015-69865-C3-1-R and REDICE16-1520 (ICE-UB).


text:
Abolfazl Rafiepour
email: rafiepour@uk.ac.ir

Azam Karimianzade

Fifth-grade students construct decimal number through measurement activities

Keywords: Decimal number, measurement activities, realistic mathematics education, primary students

In this study, a teaching experiment was used in which fifth grade students developed their own knowledge about decimal number based on their prior knowledge and real life actual experiments. A Realistic Mathematics Education approach was used for designing this research. Participants in this study were 27 students from a primary school and they did not have formal instruction at school on decimal number until participating in the current study. These students engaged in some measuring activities during four sessions in two weekends. They discovered the idea of calibration of a measurement unit and tried to use this idea for measuring the length of objects. This study shows how real life experiments of students help them to calibration a measurement unit.

Introduction

Understanding of decimal numbers is very important for people who live in twenty first century because they use computer, calculator, digital monitor and other measurement activities in their real life experiments. Usually, people encounter with decimal number in their real life. For example nutrition facts written on foods, factors for buying products, vaccination card of children and … have a lot of things related to decimal number. All children encounter with decimal number before formal education at school, although its meaning is not understood. According to [START_REF] Bonotto | Artifacts: Influencing practice and supporting problem posing in the mathematics classroom[END_REF] connection between outside of school experiments of children and formal education can support students' conceptual understanding.

In traditional teaching, decimal numbers developed through place value. Usually students use tens and hundreds blocks for consolidating decimal numbers concept. Operations and procedures explain by teachers abstractly and then ask students to do some similar exercises. Indeed, students hadn't active role in developing their own knowledge and there is no connection between out of school students experiment and their math classroom activities.

It seems to we need to develop new approaches for teaching decimal number. One of the useful ideas in this regard is realistic mathematics education approach. In current study, we try to use this approach for finding some real contexts which are authentic for students and these contexts used for starting point for constructing mathematics by students themselves. Indeed, this paper will be introducing a teaching experiment in decimal number domain. During this teaching experiment, students do sequence of measuring activities to develop a measurement unit and calibrate it for measuring different lengths. Students also discover different representation for decimal number through these activities. The main purpose of this research was that show how students develop their own knowledge about decimal number in the context of real world with using their common sense and prior knowledge.

Literature review and theoretical framework

Decimal number is one of important topics in school mathematics which has plenty of application in students' real world experiments. But several studies show that students and even adults haven't good understanding about decimal numbers [START_REF] Moloney | Changes with age in students' conceptions of decimal notation[END_REF][START_REF] Steinle | Changes with age in students' misconceptions of decimal numbers[END_REF][START_REF] Lai | Children's thinking and misconceptions in decimal numbers[END_REF][START_REF] Sengul | An investigation of 5 th grade Turkish students' performance in number sense on the topic of decimal numbers[END_REF]. [START_REF] Lai | Children's thinking and misconceptions in decimal numbers[END_REF] show that procedural knowledge of students in decimal number was very good, but their conceptual knowledge in decimal number and decimal notation were so weak.

One of the important concerns of [START_REF] Lai | Children's thinking and misconceptions in decimal numbers[END_REF] is that do the mathematics teachers know how to deliver decimal conceptual knowledge to the students? [START_REF] Bonotto | From the decimal number as a measure to the decimal number as a mental object[END_REF] believe that students' difficulties in decimal number rooted in teachers teaching strategies which have no connection to real life of students. Indeed, students usually encounter with decimal number in format of some stereotype word problems. [START_REF] Niss | Introduction[END_REF] said word problems exist several centuries in school math curricula and used as application of mathematics, but these types of problems in fact are a pure math problem in cover of words. [START_REF] Greer | Modeling reality in mathematics classrooms: The case of word problems[END_REF] believes that word problems are artificial. [START_REF] Verschaffel | Taking the modeling perspective seriously at the elementary school level: Promises and pitfalls[END_REF] states that emphasis on word problem cause to suspend common sense during mathematical problem solving. [START_REF] Bonotto | On the relationships between informal out-of-school mathematics and formal in-school mathematics in the development of abstract mathematical knowledge[END_REF] in the line with realistic mathematics education, believe that engaging students into the contextual activities that related to their own personal life, help them to enhance their conceptual knowledge in mathematics and having positive attitude toward math. She mentions two below factors that separation between school mathematics and real life facts: Stereotype problems of mathematics textbooks and Classroom environment.

If teachers of mathematics wish to establish situations of realistic mathematical modeling in problem-solving activities, [START_REF] Bonotto | How informal out-of-school mathematics can help students make sense of formal in-school mathematics: the case of multiplying by decimal numbers, Mathematical Thinking and Learning[END_REF] proposes below suggestions:

 The type of activity to which teachers delegate the process of creating interplay between math classroom activities and everyday-life experience must be replaced with more realistic and less stereotyped problem situations;

 Teachers must endeavour to change students' conceptions, beliefs, and attitudes toward mathematics;

 A sustained effort to change classroom culture is needed. [START_REF] Bonotto | From the decimal number as a measure to the decimal number as a mental object[END_REF] maintains that children's understanding of decimal numbers can be fostered by classroom activities where learners can transfer their out-of-school knowledge and utilize familiar tools, such as the ruler, that they also use in out-of-school contexts. She believes it is possible to attempt an innovative teaching trajectory in which decimal numbers are introduced through contextualized measuring activities. Indeed, measuring activities requiring vast use of a ruler can offer children good opportunities to move toward the construction of a comprehensive understanding of decimal number and notation. For example, a study of [START_REF] Astuti | Learning one-digit decimal number by measurement and game predicting length[END_REF] shows that, if students use paper strip and calibrate it by themselves, then their understanding about decimal numbers and notation will be more developed.

Method

In this study, a teaching experiment about decimal number will be introduced. Based on the research aim, the type of the research is categorized as design research. This research contributes to develop a Local Instruction Theory (LIT) to support students develop the understanding of decimal number and notation. LIT has cyclic spirit [START_REF] Gravemeijer | Local instruction theories as means of support for teachers in reform mathematics education[END_REF] that in this study is prototyped by a Hypothetical Learning Trajectory (HLT) [START_REF] Simon | Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory[END_REF] which is elaborated and refined when conducting the design. The initial step of HLT in this study is developed based on the analysis of key areas of decimals from literature review, the analysis of Iran mathematics curriculum, and the analysis of the potential use of contexts and model based on the framework of RME.

Current study is part of larger study that investigates conceptual understanding of fifth grade students in several aspects such as constructing a measure, calibration a measure unite, familiarize with decimal and notation, comparison of decimal number, density of decimal number, and submission and multiplication of decimal number. This study concentrates only on constructing a measure, calibration a measure unite, familiarize with decimal and notation. Main purpose of this study was familiarizing students with concept of decimal number. In this regard five activities designed which related to students real life facts (see figure 1).

1. Select a measuring unit arbitrary and measure length and width of classroom board, approximately. Represent length and width of classroom board with mathematical symbol.

2. How we can get better approximation? Write your proposed method completely, then record measured length in a mathematical form.

3. Select an object that smaller than your measuring unit. How you measure the length of this object? Explain your method and write measured length.

4. In this week a one meter non-graded tape give to each group of students and ask them where do you hear about "half" concept? What is the meaning of "half"? What is mathematical symbol for that?

5. Divide non-graded tape in 10 parts. Then try to measure a selective object approximately, and then show it with new mathematical symbol (decimal number). This study conducted in a primary school in the beginning of school year (Fall 2014). Participants of this study were 27 fifth grade (10-11 years old) female students. These students had no formal program in these two days and all of them participate in this extracurricular class voluntarily. All students work on activities in group. Each group contains three students. Two types of complementary communications occur in this study: group discussion and whole class discussion.

During these teaching experiment sessions, second author and two other math educators record all communication of students in group. In the end of each session, students' group works collected also. Below considerations navigate activities designing process, data collection and data analysis.

 Using of non standard measuring units for measuring objects in the classroom;

 Measuring with high accuracy;

 Calibrating a measuring unit;

 Numerical representation of length of object with using calibrated unit;

 Importance of decimal division and decimal representation of length of object.

After each session, video record of session and students' group works and researchers note analyzed and use them for leading teaching experiment in next sessions.

Results

Results of first day (Sessions one and two)

The students used different tools for measuring the length and width of classroom board, such as notebook, math book and A4 paper size. As it was asked them to measure the length and width approximately, so they write these sizes as follow and in term of a complete unit. In fact, the extra parts were neglected. A group of students who chose their notebook size as measuring unit (module) had stated that the classroom is 23 and a half notebook size length. Using the term "half" showed that they know the decimals informally.

In the second activity, it was asked them to measure the length more carefully. In all the groups it was seen that they divided the measuring unit (module) to small sections. The difference between group operations is in the numbers of divisions and choosing denominator. The mathematical symbols which were used in this activity are as the length of a natural number plus a fraction or a Mixed number.

To guide students to a more accurate calibration, they were asked to measure the length of an object that is smaller than the length of their module and write its mathematical symbol in the third activity.

In this activity, student should choose smaller objects than unit. They chose the length of pencil lead packet, pen, notebook and etc to measure. The students' performance was divided in three classifications.

 Four groups of students neglect the previous division and creating the new one for their measuring. They changed the number of their part and explained that the length of chosen object is smaller than their unit, so they change the numbers of divisions. Indeed, they divided the length of measuring unit into larger equal parts.

 Three groups of students keeping their previous division. They just divide their previous division again.

 Two groups solve this problem in different way. They divide the unit into the five parts and then divide each part into the five parts again.

Maryam:

we divided our set unit into 2 parts. We divided each into 3

Researcher: can you represent the length of measurement in math symbols?

Aida: one second (one a half) and three of this part.

(i.e. 1/2 and 2/3of this part).

Researcher: would you please represent it in math symbols?

Aida: we should calculate it.

Maryam: we should add 1/2 and 2/3.

Marjan: no, it's not the 2/3 of the notebook length. It is 2/3 of one half of its length… It means its 5/6. (She shows it on the picture to her partners).

In fact, this group is faced with a challenge in calculating the length with mathematical symbols. They needed to be able to add the fractions in unequal denominator. Dividing the previous parts into same part numbers, the operation of one group was different. First, they divided the unit into 5 divisions and do it again for each part. They stated:

"Although we can re-divide each part into 5 sections and we repeat it again and again for smaller parts. But if we want to notate the length, it will be difficult. For example, we have to write one fifth plus two twenty fifth plus … so, probably the next denominator is 125. Again 125 times 5 … then set the common denominator…"

The fourth group operation was, writing the numbers in base 5. They stated interestingly that they can continue this trend. They expressed that for setting the denominator, it's necessary to multiply the denominator to 5, so it was obvious that they found a regular algorithm for approximating the length of object.

In the classroom discussion, which is performed at the end of the first session, students expressed that writing the mathematical symbol for the length is very important. They have some problems in irregular dividing of parts, so they choose the third method as the best way to calibrate the units of measuring.

Results of second day (Sessions Three and Four)

In the first activity in the second session, the students were asked to express that where in real life have seen the word half and their symbol "half" and its mathematical symbol. Some of the students' expressions were as the following ones.

 We usually say in the grocery store: I need 2.5 kg lentils or in my mother's shopping list it was written 1.5 kg beans.

 The house is 15.5 m length.

 The jar contains 2.5 litter of water.

 The volume of the coke bottle was specified 1.5 liters of water.

 We need 1.5 meters fabrics for making this shirt.

All these written expressions showed that the students have seen and heard of these decimal symbols during shopping, in parents' shopping lists, food labels such as cokes etc. and they understand this concept. The following classroom discussion was conducted:

Researcher: what does one half mean?

Student: it means a half.

Researcher: what does number 5 mean based on 0.5?

Fatemeh: half a kilogram means a half of one kilogram, it is 500 grams. Half a meter means a half of a meter, so it is 50 centimetres. You can find number 5 in both of them.

Zahra: in my opinion, one second is equal to one half, so ½=0.5.

Rezvan: ....so it means that there are many fractions which are equal to 1/2.... For example, ½=5/10=50/100 ...that all of them are half or a half.

Maryam: eureka, eureka. 1/2 is one out of two, so 0.5 is 5 out of 10.

Parisa: that's right teacher. 0.5 is 5 out of ten. It is the same as 50 out of 100 or 500 out of 1000. All are the same and mean a half.

Teacher: could you tell me that what 4 out of 10 looks like? Please, write down your answers on the paper (all the students wrote 0.4).

In this discussion, the teacher tries to help the students using their own knowledge of "half" concept in real world to find an equivalent phrase for ½ in mathematical world. Using the concept of one half and 1/2 and in regards to their knowledge about equal fractions, they can interpret 0.5 by their own and this process shows the horizontal mathematizing. In fact, these students are moving back and forward between the real and mathematical worlds for exploring this concept.

In the second activity in which a one meter ribbon was given to all the groups, all the students divided the ribbon into 10 parts. In fact, it was asked them to specify the decimetres on the ribbon and measure the object in decimetre or one tenth accurately. In other words, they realize the relationship between number 10 in denominator and decimal numbers go out one decimal place.

According to the class discussion in the previous session, the students divided decimetres into 10 parts for more accurate measuring and made the centimetres. They made a calibrated ribbon in centimetre accuracy. In fact, it was asked them to specify the decimetres on the ribbon and measure the object in decimetre or one tenth accurately. In other words, they realize the relationship between number 10 in denominator and decimal numbers go out one decimal place (figure 2). When the teacher pointed that the ribbon is one meter length while one of the students said: "That's interesting, we divided the ribbon into 10 parts, and re-divided it into 10 so it means we have one meter into 100 divisions. These are centimetres. We knew one meter is 100 centimetres but I perceive it now"

The process of making a calibrated tape measure helps the students to realize the relationship between meter, centimetre and decimetre.

Discussion and conclusion

In the first session of teaching test, the students are allowed to calibrate a measuring unit freely but they had no idea about how to calibrate the unit. In the results of this study, it was seen that they faced a challenge with writing the mathematical symbol for the length, in the other words, these challenges caused to create a regular algorithm for calibrating the unit of measuring. The results of this study showed that choosing a division in base 10 is not natural. In the experience of [START_REF] Astuti | Learning one-digit decimal number by measurement and game predicting length[END_REF], the idea of direct division in based 10 was provided by researcher but here it was tried to conduct the students to dividing the unit in base 10.

This study confirmed the Van de Wall ( 2001) quote as said "students' mind is not like a whiteboard as entering the class". As it was mentioned, although the students had not learnt the decimal numbers formally and before the research but they used them in their real life frequently or found them on some objects covers, their parents' notes and etc. also they had seen the decimal numbers and separator mark. As Freudenthal (1991) stated, using students' background experience and information could help them in learning decimal notation and its concept. Measuring the length of an object for many times and in more accurate ways help them to realize the meaning of the digits after the decimal point. In fact, when they divided one meter into 10 sections and re-divided it again and again it means that they realize the position and concept of decimal and centesimal scales. One of them expressed, for more accurate scales; we can divide the centimetres into 10 and make it smaller and smaller. In fact, he noted the millesimal position and could guess the decimal demonstration correctly.

This study represented that if the classroom environment changes and better information in regards to the students' real life experience are provided they can find new unknown mathematical structures by discussing and talking to each other and develop their mathematical knowledge.

Figure 1 :

 1 Figure 1: Activities of first weekThese activities implemented in four 80 minutes sessions in two continuous weekends' day. First three activities implemented in session 1 and 2 in the first weekend. Fourth and fifth activities implemented in session 3 and 4 in the second weekend.

Figure 2 :

 2 Figure 2: related to measuring the length of objects by calibrated ribbon in decimetre accuracy

Proceedings of CERME10


text:
Diego Lieban
email: diego.lieban@bento.ifrs.edu.br

Zsolt Lavicza
email: lavicza@gmail.com

Geometric modelling inspired by Da Vinci: Shaping and adding movement using technology and physical resources

Keywords: geometric modelling, GeoGebra, problem solving, creative thinking

We report on the experience gathered in a study using a geometric modelling approach based on dynamic geometry systems complemented by physical modelling. Our strategy intends to strengthen the interconnections between the current trends in Science, Technology, Engineering, Art and Mathematics Education. To help students to better understand how certain physical mechanisms work, some da Vinci machine prototypes were reconstructed and used as a starting point for this study. Building upon previous experience, our work currently concentrates on the analysis of connecting physical and digital resources and on how they contribute to students' creative thinking and problem-solving. We discuss the concept of geometric modelling, focusing on spatial thinking, joints and their movements. Further, we present some new manipulatives that are being tested together with digital applets and discussions from this practice.

Introduction

Geometric modelling has been used in a wide range of contexts, but continues to deserve more attention in mathematics education. The term "geometric modelling" usually refers to different digital techniques for representing specific objects or surfaces. This concept seems to be associated mostly with computer-aided design, the tasks of which are usually related to shape and to improving models in order to approximate real structures by means of sophisticated algorithms and software. This work intends to bring geometric modelling, supported by dynamic geometry systems (DGSs) and combined with physical resources, to the classroom. While an analogue model provides "hands-on experience" to students so they can comprehend certain mechanical movements (and their restrictions), a digital model forces them to develop suitable strategies for transcribing such actions. Students must therefore think about how to apply mathematical concepts properly in order to use them successfully in their digital models. In our case, creating a digital representation is not only a mathematical exercise, but also an opportunity for the students to refine and review their comprehension processes and to improve their physical models. In this modelling approach, students are faced with problem-solving in Science, Technology, Engineering, Art and Mathematics (STEAM) contexts.

Modelling is characterized as the branch of mathematics that deals with the translation of a real-world situation into mathematical language. In publications discussing mathematical modelling (e.g., [START_REF] Brinkmann | Problems for the secondary mathematics classrooms on the topic of future energy issues[END_REF][START_REF] Lingefjärd | Mathematical modelling and technology in teacher education -Visions and reality[END_REF], data are used to optimize particular processes or to develop algorithms for analysis or prediction. Geometric modelling approaches also concentrate predominantly on processes rather than on actual goals and concepts (e.g., [START_REF] Mason | Modelling modelling: Where is the centre of gravity of-for-when teaching modelling[END_REF]Henning & Keune, 2008;[START_REF] Siller | Modelling and technology[END_REF]. We use geometric modelling for both the virtual (on the computer screen) and physical (concrete manipulative) representation of objects, which allows both the objects' functionalities and the interactions between them to be analyzed. Exploring linked mechanisms by using either concrete manipulative or digital tools such as GeoGebra, students can examine mathematical ideas in order to improve their constructions. We focus on the process of using modelling to support geometrical relationships and vice versa. Our goal is that students gain a better understanding of how certain mechanisms work, beyond connect such STEAM areas. As a starting point, we used some mechanisms introduced by Leonardo da Vinci and shared them in an interactive GeoGebra book (see https://ggbm.at/AnHK7nCX). Although da Vinci invented them more than 500 years ago, their basic physical principles are still used in engineering and in education. Reconstructing these models does not only mean redoing what has already been done; it means connecting ideas and strategies for learning and using new resources based on some classical ideas, and maybe even improving them along the way.

In this paper, we present an activity developed within a vocational course in Brazil. Building upon this preliminary experience, we intend to investigate (in the form of a PhD project) how the combined use of physical and digital tools is of mutual benefit and promotes learning in science and mathematics.

Theoretical framework

Enhancing geometric modelling through Dynamic Geometry Systems

More than a "mathematical playground", dynamic geometry systems (e.g., GeoGebra) must be considered as a proper space for transforming students into explorersa platform where mathematical understanding takes on another dimension and goes beyond merely applying formulas. Numerous studies have advocated this (e.g. Schumann, 2004;Gawlick, 2005;[START_REF] Bu | Modeling for dynamic mathematics: Toward technologyintegrated aesthetic experiences in school mathematics[END_REF] and stress the importance of transcending traditional geometry courses before the increasingly sophisticated and widespread application of geometry in science and daily life. In particular, they highlight how DGSs have supported changes in mathematics teaching and learning as well as in professional mathematical practice. Among the perspectives associated with the DGS approach, we outline those that are closer to geometric modelling: problem-solving, concept formation, construction, measurement, visualization, exploring, and variation/animation. DGSs often grant access to information that would otherwise be inaccessible. For instance, in spatial geometry, GeoGebra enables users to see an object from different points of view and its cross sections. Such features provide an important link between 2D and 3D representation. Investigating whether some ideas established in 2D also work in 3D or how they should be adapted is a promising strategy for promoting spatial thinking. Consider an example where the reasoning based on circles in a plane can be extended to a sphere in space: We have a point given in a plane and a line at a certain distance d from the point. In order to mark those points on the line that are located at the same distance e > d from the point, one could use a circle in 2D, but this would not suffice in 3D space (a more appropriate approach would be to use a sphere in this case). The attempts to find a solution using DGS software enhance the students' conjecturing processesanother benefit of DGSs.

Teaching approach using analogues and DGS

In the didactic domain, [START_REF] Alsina | Less chalk, less words, less symbols more objects, more context, more actions[END_REF] proposed that students gain insights from the functionality of objects and thus engage themselves in a creative process because they can identify the potential or restrictions of a particular phenomenon. [START_REF] Gravina | Geometria Dinâmicauma nova abordagem para o aprendizado da geometria[END_REF] suggested that dynamic geometry can foster an approach to geometric learning in which assumptions are made from experimentation and the creation of geometric objects. In the same vein, [START_REF] Swan | The roles of modeling in learning mathematics[END_REF] stated that students can refine their own thinking by interacting with different representations of problem situations. In our case, the aligned and parallel modelling process led students to a practice where they had to hone their ideas in every construction step. Facing the need to convey their ideas, students tapped into their previously acquired understanding (an example is discussed in more detail further below).

Recent studies (e.g., [START_REF] Sinclair | Recent research on geometry education: an ICME-13 survey team report[END_REF][START_REF] Camou | High school students' learning of 3D geometry using iMAT (integrating Multitype-representations, approximations and technology) engineering[END_REF][START_REF] Lesh | Re-conceptualizing mathematics education as a design science[END_REF] support the positive effect of the design and implementation of a multi-representational approach to exploring 3D objects using crafts, computer technology, and paper-and-pencil methods. In this context, we seek to provide an integration of geometry with algebra and trigonometry (using the example of joints with circular movements) that goes beyond technical instrumentation.

In our case in particular, the use of mechanical principles provides the background for the modelling process, as can be seen from the diagram in Figure 1, which was adapted from De Sapio and De Sapio (2010): they considered the relevance of applying an approach to problem-solving at an elementary stage through constructing mechanical analogues to geometric problems. In this case, mechanical reasoning supports geometric reasoning. Note that we added the arrow in the opposite direction, since the reverse case (geometric reasoning supports mechanical reasoning) is equally possible, as shown in our study: On the one hand, mechanical reasoning was essential to discussing the proper ratio for a pulley system in one case. On the other hand, with the help of rotational simulations (i.e., geometric reasoning) by means of digital modelling, the students figured out how to build a functioning physical prototype in another case. Concerning the activity's driving, the activity is consistent with DeHaan ( 2009), who stated that some strategies can transform the lecture hall into a workshop or studio classroom (even partially), and stressed the use of computer-based interactive simulations as a promoter of creativity instruction.

In fact, researchers investigating creativity generally argue that projects tend to be more creative when the solution is redefined, revisited, and questioned numerous times during the process [START_REF] Lee | Creative thinking for 21st century composing practices: Creativity pedagogies across disciplines[END_REF]. Furthermore, there are many different ways of developing prototypes. The process of refining ideas and designs puts students on an unrestricted path. Siswono & Novibasari suggested that problem-posing activities using the "What's another way?" strategy could improve students' abilities in creative thinking (as cited in [START_REF] Siswono | Leveling student's creative thinking in solving and posing mathematical problem[END_REF]. In our study, the students discussed their different ideas, especially in digital modelling. In this phase, they considered various points of view and also tried to gain some insights to check whether their ideas were feasible or not in order to do make the model as simple as possible.

Methodology

Our experiment started in September 2015 at the Federal Institute of Education, Science and Technology in Brazil. It took approximately four months and had the form of a partial extra-class activity with two weekly meetings to follow the progress. The students (most of them were 16 years old) participating in the vocational (informatics) course were supported by two additional teachers (physics and mathematics). Although supervised by teachers, they chose their own topics to investigate. We first present the Da Vinci Rotatory Bridge Project (Figure 2) developed by 4 students. It was agreed that they should develop both physical and digital models in order to try to improve the joints of the existing mechanisms. Our intention was that, by comparing similarities and differences between the models, the students should be able to use one to support the other. No particular order was prescribed, but parallel development was suggested. Use of GeoGebra was also optional but recommended, since we were exploring it during class. In fact, da Vinci projects have been promoted since 2011, but this was the first time GeoGebra materials and GeoGebra 3D features were integrated. Particularly in the digital modelling process, the students concentrated on principles of rotation, translation, and spatial geometry. As a further development we are combining this activity with a new resource, 4Dframe1 , which is a flexible material and easy to manipulate. In the next example (Figure 3), we follow the development of a digital catapult in two versions. The second one is based upon the 4Dframe model. Since mechanical principles are highlighted in a simple way, it becomes easier for students to represent them in GeoGebra. Furthermore, elementary models with straws and connectors become part of the digital modelling more easily when only segments and points are used to represent the structures. The functional principle of the joints, however, is preserved. In addition, various colours are used to represent the corresponding elements and to contribute to the visualization and facilitate spatial comprehension.

Finally, another important benefit is that the digital model allows a wide range of representations to be created by simple cursor movements (in the example in Figure 4 by dragging the blue points). Using 4Dframe, the students can think freely about different possible solutions and the constraints arising for an eventual construction, and are engaged in a learning process involving critical and creative thinking.

Construction's ideas and discussion

In order to illustrate some basis used by students to support such models, we share some parts of them. The students were free to construct according to their own previous knowledge and were not required to use any specific content. However, if the need arose in a particular task, the students received proper support. In such cases we enjoyed discussing the problem at hand with them and, together, introduced new concepts or strategies, as presented below.

Given a circle with its centre in A (located on the x-axis) and an arbitrary radius2 we started the task (Figure 5). First students investigated the relative position of a point B on the circle in relation to its center while sliding along a line. Since the goal here was to simulate the movement of a wheel, they needed to implement rotational movement. To figure this out, they changed the definition of point B to (x(A) + sin(x(A)), y(A) + cos(x(A))). Naturally, this result was obtained after several attempts and discussions. Note that, in this case, the point B is a function of point A. In a 3D representation, the same logic is preserved, but this concept was totally new for the students: Each point then has three inputs. If a circle is perpendicular to one of the standard axes, then all points belonging to this circle have a constant input regarding to such axis. The other two coordinates repeat the idea from the previous 2D example. For instance, in the case below (Figure 6), B is given by (x(A) + sin(x(A)), 0, z(A) + cos(x(A))). The connected elements can be completed by rotation or symmetry.

Additionally, some principles applying to spatial coordinates were used to define proper points as a basis for such constructions. In the boat example (Figure 6), a first reference segment was built to guide the following marks through translations and reflections. In this activity, the students should preserve the coherence between analogue and digital model as much as possible. To illustrate this, we refer to the bridge example, where the relation between the turns of the driver pulley and the bridge was to be determined; this is an issue easily identified in the physical prototype but not as readily in the digital model. When the students realized that the models were not in agreement, they concentrated on the geometrical problem and concluded which ratio between the number of turns of the bridge and of the pulley should be appropriate. In contrast, when students modelled only the digital boat (in a previous experiment), they recognized the following misconception before building the physical model: if the paddle wheels spun together at the same speed, the boat would move only forward and backward. They then fixed this problem in the physical model. This suggests that it does not matter which model they build first as long as the whole model contributes to their experience and improves their learning process. When digital geometric modelling was the goal, students needed to use their knowledge of trigonometry and parameters (functional thinking) in order to establish links between elements to obtain the desired representation. Questions such as "If you want to change the direction of the rotation, what do you have to do?" often initiated their investigative process and sometimes became a challenge.

Conclusion

"Inviting" students to reconstruct historical models is one of many possible ways to teach mathematical concepts and to promote students' creative thinking processes. Students must decide how they can use their previous knowledge in order to solve a given task. In this way, students increase their autonomy and become more involved in their own learning processes. An important step of Pólya's heuristic strategies can be outlined: "If you cannot solve the proposed problem, try to solve first some related problem" (as cited in [START_REF] Schoenfeld | Solving the problem of powerful instruction[END_REF].

While we promote STEAM on the one hand, we introduce different dimensions of learning to students and enable deep learning driven by their own interests on the other. Problem-solving and geometric modelling can also become a basis for the integration of mathematical learning into trans-disciplinary educational frameworks, currently referred to as STEAM.

In the course of this study, students left some testimonials on Moodle. One such testimonial referred to some physics aspects: "I enjoyed two videos posted by the teacher that showed how da Vinci's bridge worked. They enabled a better understanding of the functioning of the rotatory bridge. I understood better how the system of ropes and pulleys works in order to reduce the workforce."

Feedback such as this and examples such as those reported above illustrate the students' interaction with different resources that supported them. They indicate that the geometric modelling approach can motivate and contribute to their learning process. We therefore seek to evaluate and promote this activity among teachers as part of our current research. We are now working on developing new resources that connect the physical and the digital world and will report on these at a further occasion.

Figure 1 :

 1 Figure 1: the solution's correspondence come out in both directions

Figure 2 :

 2 Figure 2: The digital prototype developed using the GeoGebra 3D feature (left) and the physical model made of wood (right) were developed in parallel

Figure 3 :

 3 Figure 3: Catapult evolution and becoming easy to represent

Figure 4 :

 4 Figure 4: Multiple representations promote creative thinking. By dragging the blue points in the digital model, students can create a range of possible solutions.

Figure 5 :

 5 Figure 5: Three-frame cycles represent the transition from a "dragging circle" to a "rolling circle"

Figure 6 :

 6 Figure 6: Developing a 3D representation in 3 steps

Proceedings of CERME10

Thematic Working Group 06Proceedings of CERME10

For some examples, see https://www.geogebra.org/m/xCxJUyyx.Thematic Working Group 06Proceedings of CERME10

The radius could also be controlled by a slider. In this case, the scale had to be in accordance with the remaining construction.Thematic Working Group 06Proceedings of CERME10