text:
Miriam Amit
email: amit@bgu.ac.il

Yelena Portnov-Neeman
email: yelenapo@post.bgu.ac.il

'Explicit teaching' as an effective method of acquiring problem solving strategiesthe case of 'working backwards'

Keywords: Explicit teaching, strategy, working backwards strategy

published or not. The documents may come     'Explicit teaching' as an effective method of acquiring problem solving strategies-the case of 'working backwards'

Introduction

Since [START_REF] Polya | How to solve it[END_REF], who claimed that students who control many strategies will become more effective and intelligent problem solvers, other researchers have advocated integrating problem solving strategies into school mathematics [START_REF] English | Development of children's strategic and metastrategic knowledge in novel mathematical domain[END_REF][START_REF] Steiner | A Microgenetic analysis of strategic variability in gifted and average-ability children[END_REF], especially for talented students [START_REF] Lee | Gifted and Talented High School Students Self-Regulated Motivation and Learning Strategies[END_REF]. The development and use of strategies is definitely not intuitive, and students need proper instruction, guidance and encouragement in order to systematically implement strategies in different domains -especially in problem solving (Tishmen, Perkins & Jay, 1996). Unfortunately, teachers face immense difficulties when it comes to teaching strategies in an effective way [START_REF] Zbiek | Teaching Strategies to Improve Algebra Learning[END_REF]. This fact led us to look for an effective method to teach (talented) students some basic problem solving strategies. Based on interviews and talks with "professional mathematicians," we decided on four such strategies: Trial and Error, Proof by Contradiction, Working Backwards, and Recursion, which were taught to students using the "Explicit Teaching method." In this paper we focus on just one example, showing how Explicit Teaching can be used to teach students the "Working Backwards Strategy" for solving non-routine mathematical problems.

Explicit teaching method

Explicit Teaching is a systematic methodology that is currently used primarily to teach reading and mathematics. This method is described as "highly organized and structured, teacher-directed, and task-oriented" [START_REF] Archer | Explicit instruction: Effective and efficient teaching[END_REF]. All stages of the learning process include mediation between the teacher and the learner, in which the teacher transmits an external understanding of certain information to the learner, who then processes that pre-determined understanding [START_REF] Olson | Psychological theory and educational reform: how school remakes mind and society[END_REF]. Nevertheless, using Explicit Teaching does not predetermine or confine learners' thinking; on the contrary, it can help them become more active solvers and foster independent thinking [START_REF] Portnov-Neeman | Mathematics Education in a Connected World[END_REF]. The methodology consists of a five step model (Figure 1). The steps described below are performed sequentially by the instructor in order to efficiently pass on specific information to the learner with as little ambiguity and room for error as possible [START_REF] Rosenshine | Syhttps://youtu.be/585Rm84qz5wnthesis of research on explicit teaching[END_REF].

Orientation: Each lesson begins with a clear instruction about the purpose of the lesson. Learners need to understand what they going to learn and how it connects to previous lessons.

Presentation:

The lesson material is divided into small units that fit the learners' cognitive abilities. The teacher uses a model or schema to guide them through their problem solving process.

Structured Practice:

The instructor gives a direct and detailed explanation of the problem solving using the model or schema that was presented in the previous step. During this phase, it is critical that the instructor asks learners questions and encourages class discussion in order to check and assess their understanding of the material and clarify any confusion.

Guided Practice: In this practice the instructor addresses individuals' questions and misconceptions one-on-one, and tailors responses to meet the individual needs of each learner. Students can work in small groups in order to develop their ideas together and help each other with the new material.

Independent Practice: In this step, learners are asked to complete an assignment on their own and without assistance. They are not expected to have a flawless understanding of the lesson, but they must understand the steps involved in the process. This step should continue until learners gain full independent proficiency with the materials. 'Working Backwards' is a useful and efficient strategy for solving problems in many aspects of our lives, in which an achievable outcome is known, but we have not yet determined the path towards achieving it (Newell & Simons, 1972;[START_REF] Portnov-Neeman | Mathematics Education in a Connected World[END_REF]. When dealing with word problems, for example, the information given in the problem can appear like a complex list of facts, so it is sometimes helpful to begin with the last detail given [START_REF] Wright | Wright's kitchen table math: A step-by-step guide for teaching your child math[END_REF]. The Working Backward strategy is illustrated in Figure 2 and explained step by step in detail below:

1)

Read the problem from beginning to end and identify all its components and steps.

2) Check the final outcome of the problem.

3) From the final outcome, start reversing each mathematical operation in each step until you reach the beginning of the problem. For example, reverse the adding operation and replace with a subtraction operation.

4)

After reversing every step, resolve the initial state of the problem.

5)

Check the answer by starting from the initial state and working through the steps to see if the final outcome is achieved [START_REF] Amit | Kidumatica-mathematical excellence. Culture, thinking and creativity methods and applications[END_REF]. b) The ability to use and implement the Working Backwards Strategy?

Context

The study was conducted in the framework of "Kidumatica". Kidumatica -the math club for excellence and creativity -is an after school program for talented students in the 5th to 11th grades who are interested in mathematics, but require further tools to reach their full potential [START_REF] Amit | The "Kidumatica" project -for the promotion of talented students from underprivileged backgrounds[END_REF]. Fifty-seven (N= 57) 6 th grade students were divided in two groups: an Experimental Group (EG = 30 students) and a control group (CG = 27 students). Over a period of six months, these students learned different mathematical strategies, including the Working Backwards Strategy. The EG learned through the Explicit Teaching method, while the CG was taught using the traditional school one. None of the students in this study had been research subjects in previous studies involving the Working Backward Strategy, and none had learned the strategy before.

The 'Explicit Teaching' Group (Experimental Group)

Students in this group studied all the strategies by means of the Explicit Teaching method. Each strategy, including the Working Backwards Strategy, was taught for four weeks by one of the researchers, according to the model illustrated in Figure 1. The teacher had an integral part in the lessons. She clearly and explicitly outlined what the learning goals are for the student, and offered clear, unambiguous explanations of the skills, information and the problem solving process. As the lessons progressed, the teacher's role reduced, until students were able to solve problems independently. It was like riding a bicycle, were the instructor gradually releases his hold of the bike and the child rides off by herself. The first lesson started with an explanation of the strategy, including its importance as well as where and how it should be implemented. The teacher showed the students the model of the strategy and explained the role of each step in the solution process. The following lessons were dedicated to structural, guided and independent practices. During the structured practice the teacher gave a direct and detailed explanation of the problem solution using the Working Backwards model (Figure 2). The teacher encouraged discourse between the students and asked questions to assess their understanding and clarify any confusion. In the guided practice, students worked in smaller groups or by themselves on different working backwards problems. The teacher walked around the students and addressed individuals' questions. When the teacher felt confident enough of a student's abilities, that student was allowed to start working individually and begin the independent practice step. At the independent practice stage, students were asked to complete several assignments using the working Backwards Strategy, and to solve complex problems on their own.

Traditional teaching control group

The control group studied the Working Backwards Strategy for the same period of time as the EG, but they studied the strategy in the traditional school method. This group differed from the EG in the following ways:

1. Lessons were mainly dedicated to students' work. The teacher's part was smaller than in the EG. Her role was to give short explanations about the lesson activities. She did not use the word "strategy" in her explanations, or explain that a special approach is needed for solving working backwards problems. Most of the lesson was dedicated to independent time, so that students would develop their own strategy toward those problems. It was important that students draw their own conclusions, create their own conceptual structures, and assimilate the information in the way that makes the most sense to them.

2. The teacher did not show the model of the strategy and did not name the strategy explicitly. Instead, students could develop their own model and meaningful name based on the teacher's examples and their own experience.

3. The practice process in the CG was mainly independent, in contrast to the three levels of practice used in the EG. That led to less room for discussion and collaborative work between the students, unlike the EG, where time was allotted for these during the structured and guided practice.

Data collection and analysis

Data was collected via pre-and post-tests, students' products, short interviews during and after the lessons, and teacher's notes. The pre/post-tests were administrated to both groups before and after the learning program. Both tests included working backwards problems. This paper will discuss two problems from the pretest (problems 1 and 2 below), and two from the posttest (problems 3 and 4). At the end of each test, students were asked to write what method they had used to solve the problems.

Card

The purpose of the pre-test was to determine the homogeneity of the two groups. The post-test examined the effect of the teaching methods at the end of the learning process. A five point scale was used to rank students' answers (5 points = full and correct answer, 0 points = no answer). For example, if students identified all the steps, calculated each one by doing the opposite mathematical calculation and wrote the final answer correctly, they received 5 points. Figure 3, for example, shows a five point solution for the "Weight Problem." The problem has three steps: (1) Jenya was seven kilograms heavier than Gaby; (2) Gaby was twice as heavy as Cobi;

(3) Cobi was 15 kilograms lighter than Adi. The student calculated the weight of each person by working backward through every step of the problem. Figure 4 shows an example of a 2 point solution. The student started with the last detail given and calculated Gaby's weight correctly. However in the next two steps he did not reverse the mathematical operations and got an incorrect answer.

Findings

Findings from the pre-test showed that in both problems, there was no significant difference between the groups, which indicates that both groups had the same level of homogeneity. After six months of learning strategies, the average scores in the post-test for both problems were higher among the EG than the CG. In Table 1 we can see a significant difference in the post-test between the two groups in both problems. Figure 5 indicates that students' ability to recognize the strategy improved after the learning process, but that both groups had similar results in the pre and post-test. Our qualitative analysis of students' solutions revealed that the EG students reversed the mathematical operations much better and more easily than those in the CG, and were thus able to solve the problem correctly. Moreover, while the EG explicitly stated the name of the strategy they had used when asked, the CG students were very creative in naming the strategy, coining names such as, "going in through the back door", "reverse manual" etc. Finally, the EG students used the model of Working Backwards Strategy in a very efficient way, sometimes adjusting the model to make it easier to use.

Discussion

Strategies are undoubtedly an important tool for goal-directed procedures in problem solving.

Introducing them at a younger age can improve learners' mathematical ability [START_REF] Polya | How to solve it[END_REF] as well as their understanding and thinking skills [START_REF] English | Development of children's strategic and metastrategic knowledge in novel mathematical domain[END_REF]. To achieve this goal, it is important to use a specific teaching method (Tishmen, Perkins & Jay, 1996). In this study, that method is the Explicit Teaching method, through which we introduced the Working Backwards Strategy. The study examined the effect of this method on students' ability to recognize and solve working backwards problems. Fifty-seven sixth graders were divided into two groups, an experiment group (EG) that studied with the Explicit Teaching method and a control group (CG) that studied with a traditional school one. The strategy was unfamiliar to both groups and the findings from the pre-test showed that both groups had a similar starting point. At the end of the learning process, the group that studied explicitly showed higher results than the control group. The structured steps of the Explicit Teaching helped the students to have a better, clearer understanding of the strategy [START_REF] Anhalt | Mathematical Modeling: A Structured Process[END_REF]. Qualitative analysis revealed that students who studied explicitly were more much resourceful in their solutions. They understood how the strategy works, adopted it and changed it to make it easier to solve. We believe that this ability developed due to the discourse and the collaborative work in the structured and guided practice. We saw how students' understanding of the strategy and its use improved over time. They asked more questions, listened to other students' answers and learned how to avoid misconceptions. In addition, the integral role of the teacher in this method helped students gradually to build their confidence. Thus, these students were more prepared to work on working backwards problems by themselves. Our previous study showed that teaching explicitly can help students become active learners and foster their independent thinking [START_REF] Portnov-Neeman | Mathematics Education in a Connected World[END_REF]. The current study supports this conclusion, showing that Explicit Teaching did not limit students' thinking by fixing it on a particular process. On the contrary, students understood the core principle of the Working Backwards Strategy and then applied it creatively in whatever way seemed best to them. Both groups showed improvement over time in their ability to recognize when and why the Working Backward Strategy is needed. The percentage of students that recognized the strategy before and after the learning process was similar. This is very encouraging, since it may indicate that the teaching method does not affect students' ability to recognize strategies. We can assume that with additional practice, all of the students could potentially master strategies and develop their understanding and their strategic approach to problem solving, but this has to be tested and researched. In this study, we experimented in 'laboratory conditions' with talented students, and found that the method works. Further research is needed to confirm its effectiveness outside of the Kidumatica Mathematics Club, in the 'real world' of education.

Conclusion

Mathematical strategies are complex concepts to learn and understand, and we as educators must search for the most effective way teach them. In this study, we used a systematic and structured methodology called Explicit Teaching, and found that students who studied with this method had higher scores than students who studied with a traditional school method. Introducing strategies like these to students is important, since they can help students evolve into better thinkers and develop their ability to solve problems. We believe that strategies can and should be introduced from a younger age so they can be developed over time. We have found that younger children are capable of acquiring the basic tools. Given time, they will be able to develop their tool kit of strategies further, and eventually master them all. It is our obligation as educators to teach our students how to use strategies correctly, and the sooner we do so the better.

Figure

  Figure 1: Model of Explicit Teaching

Figure 2 :

 2 Figure 2: Model of the Working Backwards strategy (Amit, Heifets & Samovol, 2007)

Figure 3 :

 3 Figure 3: Example of a five point answer to the weight problem

Table 1 :Figure 5 :

 15 Figure 5: Amount of students from the EG and CG that recognized the Working Backwards Strategy in pre-and post-tests

Students' ability to recognize the strategy

	100%			90%	83%	88%	81%
	80%
	40% 60%	37%	41%
	20%	13%	11%
	0%
		EG-pre test	CG-pre test	EG-post test	CG-post test
		Card problem	Mango Problem	Weight Problem	Baseball Problem

Proceedings of CERME10


Marine J Herrmann
email: marine.herrmann@m4x.org

Samuel Somot

Relevance of ERA40 dynamical downscaling for modeling deep convection in the Mediterranean Sea

Relevance of ERA40 dynamical downscaling for modeling deep convection in the North-Western

Mediterranean Sea

Introduction

Deep convection is a key process for the Northwestern Mediterranean sea (NWMS) oceanic circulation and ecosystem. This process shows high interannual variability whose understanding and modeling still need to be improved. Atmospheric conditions, namely strong northerly winds and significant heat loss, are the principal drivers of winter deep convection, and hence account for much of its variability. These atmospheric conditions (Tramontane, Mistral, cyclogenesis of the Gulf of Genoa, ...) are themselves mainly driven by the local orography of the Mediterranean coast (Pyrénées, Alpes, Massif Central, Rhône valley) that induces physical processes such as lee cyclogenesis or wind channeling. It is therefore essential to use an appropriate atmospheric forcing method when performing numerical studies of deep convection. Currently, high resolution atmospheric fluxes are only available for recent years (ECMWF analysis at 50 km resolution available since 1990, used by [START_REF] Béranger | Seasonal variability of transports through the Gibraltar, Sicily and Corsica Straits from a high resolution Mediterranean model[END_REF]).

The 1986-87 winter was cold and windy in the NWMS, leading to a strong bottom openocean convection event [Mertens and [START_REF] D R A F T Mertens | Interannual variability of deep-water formation in the Northwestern Mediterranean[END_REF]] that was monitored during a specific campaign (MEDOC87, reported by Schott and Leaman [1991] and [START_REF] Leaman | Hydrographic structure of the convection regime in the Gulf of Lions : winter 1987[END_REF], hereafter SL91 and LS91). This period is therefore particularly appropriate to evaluate the capacity of an oceanic model to represent deep convection [START_REF] Demirov | On the relationship between the water mass pathways and eddy variability in the Western Mediterranean sea[END_REF]Herrmann et al., 2008] as well as the impact of atmospheric forcing resolution on this modeling problem, as suggested in those studies.

We carried out a spectral dynamical downscaling of the ERA40 reanalysis [START_REF] Simmons | The ERA-40 project plan[END_REF]

Dynamical downscaling of the ERA40 reanalysis

ERA40 is widely used to analyse the variability of oceanic and atmospheric circulation over Europe during the last decades, in particular for the period before the 1990s, i.e.

prior to the high resolution ECMWF analysis. Covering the 1958-2001 period with a spatial resolution of ∼125 km, this reanalysis is indeed one of the most homogeneous, high-resolution and long-term databases available to force ocean models. Moreover, since the data assimilation system constrains the temporal behavior of the atmospheric flow, synoptic events follow the observed chronology. It was however suggested that its resolution is not sufficient to realistically simulate the formation of the deep waters involved in the Mediterranean thermohaline circulation [START_REF] Josey | Changes in the heat and freshwater forcing of the Eastern Mediterreanean and their influence on deep water formation[END_REF][START_REF] Demirov | On the relationship between the water mass pathways and eddy variability in the Western Mediterranean sea[END_REF]Herrmann et al., 2008]. We tested this hypothesis by increasing the ERA40 resolution with a downscaling method.

The downscaling method

Several statistical or dynamical methods aimed at downscaling a reanalysis were developed. [START_REF] Sotillo | A high-resolution 44-year atmospheric hindcast for the Mediterranean Basin: contribution to the regional improvement of global reanalysis[END_REF] performed a dynamical downscaling of the NCEP reanalysis

D R A F T January 18, 2008, 9:18am D R A F T X -5
[ [START_REF] Kalnay | The NCEP/NCAR 40-year reanalysis project[END_REF] over the Mediterranean basin with a limited area model forced at its lateral boundaries and with spectral nudging over the central area. We also applied a dynamical downscaling based on spectral nudging. The method was described in detail in [START_REF] Guldberg | Reduction of systematic errors by empirical model correction : impact on seasonal prediction skill[END_REF]. The principle is to use a high resolution atmospheric model in which small scales can develop freely and large scales are driven by ERA40. The synoptic chronology then follows that of the reanalysis while the high resolution structures of the atmospheric flow are created by the model.

The ARPEGE-Climate model [START_REF] Déqué | High-resolution climate simulation over Europe[END_REF] is a global and spectral Atmosphere General Circulation Model whose grid can be stretched over the area of interest.

We used a version with a pole located in the Tyrrhenian Sea and a resolution of ∼50 km over the Mediterranean Basin, as in [START_REF] Sotillo | A high-resolution 44-year atmospheric hindcast for the Mediterranean Basin: contribution to the regional improvement of global reanalysis[END_REF]. A nudging term was added to the equations of the temporal evolution of the prognostic variables (temperature, velocity and surface pressure). We chose a wavelength of 250 km as the limit of the waves driven by ERA40, and followed [START_REF] Guldberg | Reduction of systematic errors by empirical model correction : impact on seasonal prediction skill[END_REF] for e-folding times used for the nudging of the different variables. ERA40 fields were updated every 6 hours and linearly interpolated in between. The ARPEGE-Climate simulation covers the whole ERA40 period. Air-sea fluxes (water, heat and momentum) were extracted from both simulations (ERA40 reanalysis and ARPEGE-Climate simulation) and interpolated onto the ocean model grid (Section 3).

Comparison of atmospheric datasets

To underline the differences between the two datasets, we compared the new dynamical downscaling dataset (HRAF for High Resolution Atmospheric Forcing) with the initial

D R A F T January 18, 2008, 9:18am D R A F T
ERA40 dataset (LRAF for Low Resolution Atmospheric Forcing) over the Gulf of Lions.

For this purpose, we examined the daily time series of the net heat loss (HL), net water loss (WL), buoyancy loss (BL) and wind stress (τ ) averaged over the LION area (defined on Fig. 1c).

First, the spectral dynamical downscaling technique does not change the temporal chronology of the downscaled reanalysis. Indeed, both datasets are well correlated in time for τ , HL and WL (Fig. 1a). Each LRAF intense event is well represented in HRAF. Tab. 1 summarizes these correlations for the LION area over a wide winter period (December -April). Daily correlations are higher than 0.8 except for WL, due to the weaker ability of forced climate models to simulate the precipitation chronology [START_REF] Caya | Internal variability of rcm simulations over an annual cycle[END_REF].

This lower correlation for WL does not have a significant impact on the chronology of BL since BL is mainly driven by HL. Indeed, we computed a 0.99 time-correlation between BL and HL in both datasets, vs. 0.53, resp. 0.59 between BL and WL in LRAF, resp.

HRAF.

Second, over the area of interest and for the winter period investigated, the dynamical downscaling slightly increases the winter average air-sea fluxes (Tab. 1).

Third, extreme air-sea fluxes during intense winter events are more pronounced in space and in time in HRAF than in LRAF. For HL, WL and BL, the value of the 90th percentile increases more than the time-averaged value (Tab. 1). This is even stronger for the maximum value. The upper end of the corresponding density functions increases consequently more than the average, as seen on the quantile-quantile plot for the three variables (Fig. 1b). This is also true for τ despite the inconsistent behavior of the maximum value which

D R A F T January 18, 2008, 9:18am D R A F T
is due to a slight temporal shift between HRAF and LRAF at the beginning of the E2 event (see Fig 1a). This very intense wind event is indeed concentrated in one day in LRAF but is split in two days in HRAF. A two-day average shows that HRAF is actually more windy than LRAF for this event. The upper end of the τ density function also increases more than the average (see Fig. 1b) even if the maximum value has a strong impact on the change of the 90th percentile value (Tab. 1). The lower end of the WL density function is explained by a rainier E2 event in LRAF than in HRAF, probably due to differences in model physics. The upper end of the density functions corresponds to the main synoptic weather events observed during this winter (LS91), characterized by regional Mistral or Tramontane winds (Fig. 1a). During these events, the impact of the resolution increases and the orography forcing becomes predominant in HRAF, leading to large differences with the initial reanalysis. Temporal extremes are therefore more pronounced in HRAF. The resolution enhancement also increases the intensity of small spatial scale structures in HRAF, as illustrated on Fig. 1c for a particular day. The relative difference in the HL winter mean between the MEDOC point (5 • E, 42 • N), known to undergo maximum heat loss, and the LION area is indeed equal to +12% in LRAF and +26% in HRAF.

Due to a lack of in-situ data, it is generally difficult to validate air-sea fluxes. For the 1986-87 winter, Mertens and [START_REF] D R A F T Mertens | Interannual variability of deep-water formation in the Northwestern Mediterranean[END_REF] however provided values integrated from December to February: 2.5 10 9 J.m -2 for HL and 1.5 m 2 .s -2 for BL. For HRAF (resp. LRAF), we obtained for the MEDOC point, 1.9 (resp. 1.6) 10 9 J.m -2 and 1.0 (resp. 0.85) m 2 .s -2 .

D R A F T January 18, 2008, 9:18am D R A F T

Despite the large uncertainty associated to the Mertens and [START_REF] D R A F T Mertens | Interannual variability of deep-water formation in the Northwestern Mediterranean[END_REF] data, HRAF dataset is closer to the observed data than LRAF.

Impact of atmospheric forcing resolution on deep convection modeling

We used the atmospheric datasets compared above to force a regional ocean model.

Results of both simulations (HRAF and LRAF) were compared with data obtained during MEDOC87 to show the influence of atmospheric forcing spatial resolution on open ocean deep convection modeling and emphasize the interest of the downscaling method.

The oceanic model: SYMPHONIE

The 3D primitive equation eddy-resolving ocean model SYMPHONIE was used by Herrmann et al. [2008] to study the influence of the oceanic model spatial resolution on the modeling of the 1986-87 winter convection in the NWMS. We used the same configuration. This free-surface model is based on the hydrostatic assumption and the Boussinesq approximation. We used a 3-km orthogonal horizontal grid and a 40-level hybrid sigmastep coordinates system. Vertical diffusivities are calculated using the [START_REF] Gaspar | A simple eddy kinetic energy model for simulations of the oceanic vertical mixing: tests at station Papa and long-term upper ocean study site[END_REF] second-order closure scheme. In the case of unstable stratification a non-penetrative convective adjustment algorithm is used. At the surface, the model is forced by air-sea fluxes (heat, momentum and water flux). The water flux is the simple sum (without relaxation or correction) of the evaporation-minus-precipitation flux extracted from the ERA40 and ARPEGE-Climate atmospheric simulations, and of the Rhône river runoff (UNESCO RivDis database [START_REF] Vörösmarty | Global river discharge database, RivDis, UNESCO, Paris, international Hydrological Program[END_REF] climatological monthly values). Two regional simulations were carried out between September 1986 and September 1987, forced with the LRAF and HRAF datasets respectively. The eddy-permitting oceanic model OPA D R A F T January 18, 2008, 9:18am D R A F T described in [START_REF] Somot | Transient climate change scenario simulation of the Mediterranean Sea for the 21st century using a high resolution ocean circulation model[END_REF] was used to perform two simulations over the whole Mediterranean sea using HRAF and LRAF air-sea fluxes. Monthly averaged results of each OPA simulation provided lateral boundary conditions and initial conditions for the regional simulations. The consistency of this forcing strategy was demonstrated in Herrmann et al. [2008].

Time evolution of the convection event

The LION area (Fig. 1c) entirely covers the convection zone. We examined the evolution of the maximum mixed layer depth (M LD max ) over the LION area during the 1986-87 winter for both simulations (Fig. 1a). As justified in Herrmann et al. [2008], the mixed layer depth was defined using a threshold value of 4 cm 2 .s -1 for the vertical diffusion coefficient. In LRAF M LD max does not exceed 800 m: deep convection does not occur.

In HRAF M LD max begins to increase on 11th January and reaches 1200 m on 27th

January. This first deepening is due to a mid-January intense Mistral event (E2, Fig. 1a), in agreement with the observations. By examining CTD and velocity profiles, LS91

and SL91 deduced that this event had initiated deep convection before 23rd January.

During early February, M LD max shallows to 500 m, as observed by LS91 and SL91. The with observations [START_REF] Gascard | Mediterranean deep water formation, baroclinic eddies and ocean eddies[END_REF][START_REF] Testor | Post-convection spreading phase in the Northwestern Mediterranean Sea[END_REF]. These structures play a major role during the convection event: they are responsible for the stabilization of the convection zone extension and DW characteristics, and contribute importantly to DW transport, therefore influencing the whole NWMS thermohaline circulation [START_REF] Testor | Post-convection spreading phase in the Northwestern Mediterranean Sea[END_REF][START_REF] Demirov | On the relationship between the water mass pathways and eddy variability in the Western Mediterranean sea[END_REF]Herrmann et al., 2008]. The circulation during the convection event is therefore represented correctly in HRAF, and not in LRAF.

Conclusions

We chose the well-observed 1986-87 winter to evaluate the capacity of an eddy-resolving oceanic model forced by high resolution atmospheric fluxes to represent deep convection.

Using ERA40 downscaling instead of directly using ERA40 fluxes, we obtained a better representation of the 1986-87 deep convection event: its temporal and spatial characteristics are in better agreement with the observations, and the circulation mesoscale processes, which play an essential role during deep convection, are accurately represented. To our knowledge, this is the first time that deep convection and associated mesoscale structures were simulated within a high-resolution realistic framework for both the oceanic model and the air-sea fluxes. This downscaling method enables us to correct the underestimation of spatial and temporal extreme air-sea fluxes in ERA40. As shown in this study, this appears to be crucial when modeling oceanic deep convection, a major circulation process in the NWMS. Detailed sensitivity tests should be performed to confirm this hypothesis.

Other years and longer periods between 1958 and 2001 can now be studied using ERA40

  to obtain a high resolution long-term dataset following the observed D R A F T January 18, 2008, 9:18am D R A F T chronology over the 1958-2001 period. Twin numerical simulations of the 1986-87 winter were performed using an eddy-resolving oceanic model forced at the surface by two sets of atmospheric fluxes: the first directly extracted from ERA40 and the second produced by the downscaling. The objectives of this study were to evaluate the influence of the atmospheric forcing spatial resolution on convection modeling and to underline the relevance of the downscaled ERA40 atmospheric fluxes in the framework of realistic oceanic modeling.

∼

  third atmospheric event E3 induces a second deepening phase of the mixed-layer that reaches 1240 m on 23rd February. A brief but intense Mistral event (E4) occurs in the first few days of March after which M LD max reaches the bottom. This represents a small delay (11 days) compared to LS91 and SL91 observations. M LD max shallows to 1270 m, and then exceeds 2000m again after E5. Restratification begins on 26th March but is briefly interrupted by E6 until 5th April when it restarts once again. In HRAF, bottom to M LD max ≥ 2000 m, occurs and the mixed layer evolution is consistent with the atmospheric forcing and with the observations. When bottom convection occurs, the water column is fully mixed and has homogeneous characteristics, corresponding to deep water (DW) temperature and salinity properties. DW characteristics observed by LS91 were 12.76 • C, 38.44 psu, 29.11 kg.m -3 . In LRAF no deep convection occurs therefore no DW is formed, and the surface density never exceeds 29.06 kg.m -3 . Also using a 125 km resolution atmospheric forcing,[START_REF] Demirov | On the relationship between the water mass pathways and eddy variability in the Western Mediterranean sea[END_REF] obtained DW whose density did not exceed 29.05 kg.m -3 and attributed this to the low spatial resolution of the atmospheric forcing. In HRAF DW characteristics are 12.78 • C, 38.44 psu, 29.10 kg.m -3 , similar to the observed values.3.3. Deep convection spatial characteristicsLS91 defined the convection area as the region where sea surface salinity (SSS) exceeded 38.40 psu. The corresponding observed region on 21st February, when bottom convection occurred, is shown on Fig.1d, as well as the mixed layer depth computed in LRAF on 8th February, when M LD max is maximum, and in HRAF on 15th March, when bottom convection is established (Fig.1a). In LRAF, as already observed, no deep convection occurs. In HRAF, though the convection area seems slightly underestimated, the size and position of this area, centered around (5 • E, 42 • N), is in better agreement with the observations.Herrmann et al. [2008] carried out a simulation using corrected ERA40 fluxes: a 130 W.m -2 correction was added homogeneously to HL from December to February to correct its underestimation in ERA40. They obtained deep convection but the convection 10-15 km and of maximum orbital velocity ∼ 10 cm.s -1 . These eddies advect DW out of and stratified water into the convection area. These characteristics are in agreement

Figure 1 .

 1 Figure 1. Atmospheric and oceanic characteristics during the 1986-87 winter for LRAF

Table 1 .

 1 HRAF values (mean, 90th percentile, maximum and time correlation with the

	D R A F T	January 18, 2008, 9:18am	D R A F T

Acknowledgments. We would like to thank M. Déqué, C. Estournel, F. Sevault, M.

Lucas and A. Pinari for their careful rereading and useful discussions.

X -11

area was too large and shifted to the southwest. This overestimation was very likely due to the homogeneity of the correction. The mean HL difference over the same period between LRAF and HRAF is indeed 20 W.m -2 for the LION area and 40 W.m -2 for the MEDOC point, respectively 70% and 85% smaller than the 130 W.m -2 correction: the downscaling method allows us to correct more accurately the atmospheric fluxes spatial and temporal extremes without overestimating HL over the entire region.

Thus, using downscaled ERA40 fluxes enables a more accurate representation of the size and position of the convection area and the characteristics of the DW formed than using raw or corrected ERA40 fluxes. Rather than an average increase in fluxes, the enhancement of meteorological extremes may be the determining factor in this improvement. Indeed, previous numerical studies showed that extreme atmospheric events [START_REF] Madec | The effect of thermohaline forcing variability on deep water formation in the Western Mediterranean sea: a high resolution three dimensional numerical study[END_REF][START_REF] Artale | Role of surface fluxes in ocean general circulation models using satellite sea surface temperature: Validation of and sensitivity to the forcing frequency of the Mediterranean thermohaline circulation[END_REF] and space variability, through the strengthening of the wind stress curl [START_REF] Madec | Large-scale preconditioning of deep-water formation in the Northwestern Mediterranean Sea[END_REF], promote deep convection.

Circulation during the convection event

The main current in the NWMS is the Northern Current that flows southwestward along the coast and across the Gulf of Lions shelf. In winter, it develops intense mesoscale meanders of a few tens of km [START_REF] Millot | Circulation in the Western Mediterranean Sea[END_REF]. For both simulations this current is visible on Fig. 1d. However, the current flowed steadily in LRAF, whereas in HRAF it is unstable with 30-50 km meanders. [START_REF] Gascard | Mediterranean deep water formation, baroclinic eddies and ocean eddies[END_REF] observed that mesoscale structures corresponding to baroclinic instabilities developed at the periphery of the convection area.

Such meanders are present at the southern periphery of the convection in HRAF, but not in LRAF. In HRAF, they evolve into eddies of vertical scale larger than 1000 m, of radius D R A F T January 18, 2008, 9:18am D R A F T


domain: math shs

text:
Sophie Joffredo-Le Brun
email: s.joffredolebrun@gmail.com

Mireille Morellato
email: mireille@syrah.fr

Gérard Sensevy
email: gerard.sensevy@espe-bretagne.fr

Serge Quilio
email: serge.quilio@gmail.com

Continuity of the students' experience and systems of representation, an example of teaching and learning in mathematics first grade students

Keywords: Elementary school mathematics, continuity of education, semiotic representation

Our study is based on a national research project called Arithmetic and Cooperation at Elementary school (ACE). The main objective of this research is the designing of a curriculum for first grade students. This communication focuses on the analysis of an extract from a lesson proposed by this curriculum. During this lesson, the students work on the notion of difference, which is introduced with the help of several systems of representation. These systems are already known by the students. The analysis of the extract shows that the past and future situations can be related to each other. We point out that the students' continuity of experience can produce knowledge growth. This growth specifically occurs when the teacher's uttering activity directs the students' actions while they use the systems of representation. This uttering activity allows a reshaping / remodeling /modification of former knowledge through the systems of representation.

This paper focuses on the continuity of epistemic experience in mathematics. We argue that systems of representation constitute a prominent way of achieving such continuity. We sketch the role of a specific system of representation (the number line) to build of the number sense at first grade. This research is based a larger French national research, arithmetic and cooperation at Elementary School (ACE). It offers a complete arithmetic program to 6-7 year-old-students (First grade). The conception of this curriculum relies on available scientific knowledge in different areas (Cognitive Neuroscience, Science of Education, Developmental Psychology and Didactics of Mathematics). A specific part of the conception of the curriculum is built within what we call a cooperative engineering (Sensevy, Forest, Quilio, Morales, 2011;Morales, Sensevy, & Forest, in press). This engineering consists of two spheres. The sphere 1 gathers a multi categorical team (PhD, teachers of study classes, researchers, teacher's trainers, pedagogical advisors) and the sphere 2 is constituted by 120 experimental classes. In the first year of the experiment (2011)(2012), the sphere 1 designed the mathematical situations of the curriculum. These situations were implemented in the « study classes », and redesigned on line in the course of the implementation process. The first year of the experiment, the sphere 1 designed eleven modules corresponding to forty-five sessions. The second year of this experiment, this curriculum has been implemented in 60 experimental classes (versus 60 control classes) and in 120 experimental classes (versus 120 control classes) the third year (2013)(2014). The experimental classes' involvement in this curriculum and their feedbacks allowed numerous improvements of the initial design proposed by the research team situations. This cooperation between researchers and teachers showed a willingness to create a didactic continuity in student's experience through the use of representation of the number systems that are present in the progression throughout the year. We assume that in the joint action between students and teacher, the systems of representation may authorize the continuity of the student's experience. But, how precisely this can be built in joint action between teacher and students? Here we can rely on Dewey's conception of continuity. "The principle of continuity of experience means that every experience both takes up something from those which have gone before and modifies in some way the quality of those which come after" (Dewey, 1998, p.27).

The Joint Action Theory in Didactics

Our analyses will build on the Joint Action Theory in Didactics (JATD) originated in comparative approach in didactics [START_REF] Sensevy | Le sens du savoir. Eléments pour une théorie de l'action conjointe en didactique[END_REF][START_REF] Sensevy | On the Nature of the Semiotic Structure of the Didactic Action : The Joint Action Theory in Didactics Within a Comparative Approach[END_REF]Ligozat, 2009;[START_REF] Tiberghien | The construction of physics knowledge in a classroom community from different perspectives[END_REF][START_REF] Venturini | Analysis of conditions leading to a productive disciplinary engagement during a physics lesson in a deprived area school[END_REF]. Among the theoretical tools provided by the "JATD", we use mainly the contract/milieu dialectics and the reticence/ expression dialectics. When facing a new problem, students are confronted to what we call to a milieu [START_REF] Brousseau | Theory of Didactical Situations in Mathematics[END_REF][START_REF] Sensevy | Milieu[END_REF], as the epistemic structure. This milieu can be seen as the state of problem, what "has to be known" (Sensevy et al, 2015). Students have to face a rather enigmatic set of elements that they have to relate in order to build a system of meanings, in the knowing of what has to be known » [START_REF] Sensevy | On the Nature of the Semiotic Structure of the Didactic Action : The Joint Action Theory in Didactics Within a Comparative Approach[END_REF]. For example, for someone who has to do something with a representational system, the milieu is a specific symbolic organization of the system of representations itself. The milieu offered opportunities of enquiry, in which students have to connect elements of knowledge. They deal with this milieu by relying on the knowledge built in the preceding didactic joint action, the didactic contract, what "is already known". This relationship between contract and milieu is a dialectal one, because the understanding of a given milieu depends on the nature of the contract that guides the student's efforts.

In order to enable the student to learn, teacher enacts strategies to engage student's action. Interactions between teacher and students are determined by the didactic contract [START_REF] Brousseau | Theory of Didactical Situations in Mathematics[END_REF]. In fact, the teacher knows the knowledge that students will have to learn. But he must not tell directly all what he knows. Thus, he has to make choices, in his teaching, about the equilibrium between saying/showing (expression) and remaining tacit/hiding (reticence). This is the reticenceexpression dialectics. The two dialectics (contract-milieu and reticence-expression) are entangled, in that expression or reticence can be oriented to "contract " ("what is already-there"), or "milieu " ("what has to be known").

The research on the using of manipulatives and representations focus on the necessity of enabling the students to rely first on manipulative and concrete "objects", then to study iconic (analogical) representations of numbers [START_REF] Bass | Quantities, numbers, number names, and the real number[END_REF][START_REF] Schmittau | The development of algebraic thinking: A Vygotskian perspective[END_REF][START_REF] Davydov | Logical and psychological problems of elementary mathematics as an academic subject[END_REF] then to write down equations in canonical form. This process seems very close to the tradition in Chinese textbooks [START_REF] Bussi | Towards cultural analysis of content: Problems with variation in primary school[END_REF]Sun, 2011;[START_REF] Ding | Transition from concrete to abstract representations: the distributive property in a Chinese textbook series[END_REF] and can be thought of as "concreteness fading" (McNeil et al., 2012;[START_REF] Fyfe | Concreteness Fading in Mathematics and Science Instruction: a Systematic Review[END_REF]. In this communication, we will try to show how in the new situation in which the notion of the difference (subtraction) is introduced, the systems of representation of number can guarantee a kind of continuity of experience. In fact, the "translational principle for representations systems" in a representational game (Morales, Sensevy & Forest, 2016) can allow students to understand the concept of the difference between two numbers.

Methodology

To discuss these questions, we focus on a specific moment of teacher's practice in a study class, with an experimented teacher, who belongs to the research team (sphere1). The data were collected in december 2013, in a first grade classroom of a French primary school. The twenty five students of this classroom were aged 6-7. This study follows a qualitative approach.

In this extract, the students collectively search the difference between two additive writings with two terms. In this communication, we focus on the introduction of a new piece of knowledge, the notion of difference between two numbers. In the preceding sessions, the students orally compared the production of two hands ads (students showed a number on the two hands, the statement) and a launch of two six-sided dice. The statement wins if it was bigger (in some cases smaller) than the two-dice throw. Then, students compared the two additions with two terms in reference to the situation of the "Statements" (fingers and dice). The result of this comparison was written in the form of a quality or an inequality with the mathematical signs « =, ≠, >, < ». These two additions were represented in two number lines to solve or prove the comparison, as we can see below (figure 1).

Figure 1: An example of comparison between 2 + 4 and 5 + 3 on the number line

The choice of this extract is motivated by the following reason: this extract shows how the continuity of student's experience could be developed through the use the systems of representation. This extract can be considered as a mesoscopic level of the description, the pivotal level (Sensevy et al., 2015), which allows relationship between what preceded and what is going to follow. So, we can analyze the didactic transactions hic et nunc. We can characterize and describe the motives and the forms that directed teacher and students 'action. This description can show the teacher's strategies to make the didactic time forward.

Analyze Presentation

This part of the curriculum "ACE" is organized around a connected series of situations. The initial situation of this curriculum is the situation of the "Statements game" (fingers and dice). One die (marked with standard dot patterns for 1-6) is about to be thrown. Beforehand, the students use their fingers to make a "statement" (for example, a student shows two fingers on her right hand, and three fingers on her left hand). The die is thrown. The students compare their statement with what is indicated by the die. If the sums are equal, the pupils have won. After this oral comparison, students compare an addition (two terms ≤ 5 with a number ≤ 6). The progressive complexification of the situation guides students to increasingly rich comparisons: the number of hands (students) is increased, so the number and the nature of dice (1 to 10 dice are played with), the rules of the game are changed (for example a pupil no longer wins because he has the same number as in the statement, but because he has a lower or higher number). These connected situations should allow the students to build a real mathematical experience, particularly in the handling of representation and symbolic writing systems, as we will show in what follows. A number line is also introduced on which students represent the numbers. Indeed, the students manipulated a concrete object (the fingers) and they translate fingers by an iconic (analogical) representation of number (the number line) and wrote down equation in canonical form (2 + 4 = 6). For example, a student shows two fingers on his left hand and four fingers on his right hand, then she draws these numbers on a number line:

And she writes down this addition: 2 + 4 = 6

To understand various properties of numbers, students had to compare different representations of the same mathematical reality to become progressively able to recognize the differences and the similarities between these representations.

Since the beginning of the year, students acquired knowledge related to compare numbers. This comparison is performed first orally with the production of «two or three hands ads" and a launch of six-sided die. Then, students compared an addition in two or three terms with a throw of dice. They used the mathematical signs « =, ≠ <, > » (for example, to compare 3 + 1 et 5, students write 3 + 1 < 5). This situation become more complex when students have to compare two additions in two terms. Finally, students deal with the question of the subtraction on the basis of the comparison between two additions, in the continuity of the previous situations. The study is accompanied by the use of the number line.

The students have built a semiotic knowledge to represent the comparison between two numbers. The number is seen as a measurement. It refers to "the quantity of fingers" in two hands. The number line shows the number like a length measurement. This ancient knowledge is the didactic contract, the habits of action with which teacher and students are going to approach the new knowledge, the difference between two numbers. The extract of the session that we chose introduces the difference between two numbers from the comparison of two additions and the terms "larger than and smaller than". The difference is a gap between two numbers, two length measurements. Four episodes will be analyzed. Here is a synoptic view of this analyzer.

Episodes Content

Episode 1 Presentation of the instructions by the teacher Episode 2 a) Comparison, looking for the difference between "1 + 3" and "1 + 1". b) Proposal of two students: (tdp 15) "1 + 3" is larger than "1 + 1". c) Proposal of another student: the difference between "1 + 3" and "1 + 1" is 3.

Episode 3 a) The difference (tdp 53). b) Introduction of the two hands by the teacher to confirm the difference 3.

Episode 4 Introduction of the number line by the teacher to search the difference between "1 + 3" and "1 + 1" (tdp 68).

Table 1: Extract division

We abstract the four episodes and provide a short analysis.

Search for the difference between two numbers (episode 1, 2 3)

The teacher asks the students to look for the difference between "1 + 3" and "1 + 1". He presents the instructions like this:

Teacher: Today, we are going to begin a new game. It is always a game with statements.

But today what we are going to make, it is to compare ours statements. We look for which is the larger statement, the smaller statement but that I would know how much more and how much less (…) we are going to find, this calls in fact the difference.

Actually, the students meet difficulties to find the difference between "1 + 3" and "1 + 1. They compare the two numbers and look for the larger number or the smaller number with the term-byterm strategy or by computing. They do not focus on "difference". Confronted to the difficulties of the students, the teacher suggests to illustrate « the two additions "1 + 3" and "1 + 1" » by a statement with both hands. The following picture (figure two) shows such a statement comparison. Unfortunately, this translation between the mathematical symbolic writing and the hands statements in a game of representation does not bring the students to produce an adequate answer. "The semiotic habit" of the contract, which considers fingers as instruments to compare numbers, impedes a new designation of numbers, the difference.

Using the number line

Therefore, the teacher introduces in a milieu two number lines on which students have to represent both additive writings. A student writes a first bridge above the first interval of the number line (hence representing the number "1") and a second bridge above three intervals (between the second and a fourth graduation, hence representing the number "3"). He writes down above these bridges the numbers 1 and 3. Then, he draws below the number line a bridge of four intervals and writes down the sum number (4). On the second number line, the same student draws two consecutive bridges and a bridge of two intervals, signifying the sum number (2) The students provide answers different answers: "three", "four" and "two". The teacher asks one student to show how he knows that his statement is largest.

Student: because here is the two [He slides his finger from the third graduation (number "2") on the first number line to the third graduation (number "2") of the second number line] It's a part of four…

The student draws a bridge between the second and the fourth graduation on the first number line. The teacher's expression encourages a translation between the symbolic writing and the number line. But, in the same time, she's reticent because she doesn't say how to draw these additions on the number lines. The teacher thus refers to the preceding contract. She just says: "What do you see in the two number lines? Do you see if the 1+3 is largest than 1+1?" This question can be seen as a ''milieu-oriented situation'. The action of the student is moved toward the effective representation of the difference on the two number lines. But, though this expression, the teacher is reticent because she remains silent when the student searches the difference. She indicates where the student must look but the research stays under the responsibility of the student. The number line affords to show concretely the gap between the two numbers, the difference between the two length bounded by the bridges, the sum of the two additions. The students investigate an instrument, the number line, on which they know already how to play (in the situation of comparison). By using this instrument, they achieve not yet explored potentialities of this semiotic system. In particular, the number line shows a number included in another.

Results

The students investigate different systems of representation to find the difference between two numbers: symbolic writing, concrete representation with the hands, number line. All these representations are known by the students. First, the symbolic writing is translated by statements on the two hands. Then these statements are translated into two number lines in what we may call a translation game, which is a particular representational game [START_REF] Sensevy | Le sens du savoir. Eléments pour une théorie de l'action conjointe en didactique[END_REF]Morales, Sensevy & Forest, in press). The preceding semiotic knowledge is "re-experienced" by the students. In this way, we can say that the systems of representation are instrumental (Dewey, 1938[START_REF] Dewey | Experience and Education[END_REF] in that they allow investigating new knowledge. The semiotic habits are accommodated in a new situation, a new knowledge, allowing the continuity of experience. However, the re-experience of a semiotic system for introducing a new knowledge requires a subtle enunciative work for the teacher, given that the different strategic systems in teacher's action can be a contract oriented or a milieu-oriented transactional activity. In this communication, we have shown how the expression-reticence game of this teacher enable her to introduce in the milieu a system of representation (the number line) already known by the students (contract) in order to understand new properties of this system (then considered as a milieu). This teacher's strategy enables the students to investigate the difference between two numbers while leaving them the responsibility of this enquiry (Dewey, 1938[START_REF] Dewey | Experience and Education[END_REF].

Discussion and conclusion

However, these results ask to be worked. It is necessary to explore on a long duration this continuity of the experience of the students in mathematics through the systems of representation (Joffredo-Le [START_REF] Brun | Continuité de l'expérience des élèves et systèmes de représentation en mathématiques au cours préparatoire, une étude de cas au sein d'une ingénierie coopérative[END_REF]. In particular, it is necessary to note that such continuity can be built only through a real epistemic continuity of the knowledges within the curriculum. The design of such a curriculum has to be performed through the effective experience of the teachers, within an iterative process, as it is the case in the ACE research.

Figure 2 :

 2 Figure 2: Translation of "1 + 3" and "1 + 1" by two statements

  the statement makes four as Neil shows us. On the second line, what are you going to draw?

Figure 3 :

 3 Figure 3: the representation of "1 + 3" and "1 + 1" on the number lines Teacher: What do you see in the two number lines? Do you see if the 1+3 is largest than 1+1?

Figure 4 :

 4 Figure 4: Representation of the difference on the number line

Proceedings of CERME10


text:
Francesca Ferrara
email: francesca.ferrara@unito.it

Giulia Ferrari
email: giulia.ferrari@unito.it

Diagrams and mathematical events: Encountering spatio-temporal relationships with graphing technology

Keywords: Graphing technology, movement, diagram, event, problematic

This paper presents the diagrammatic activity of some secondary school students exploring motion through graphing technology, which captures a pair of space-time graphs on a single Cartesian plane. Focus is on a written task about the connections between two imaginary movements and (between) the corresponding graphs. Drawing on a vision that conceives mathematical thinking as a place of events instead of objects, we discuss three unexpected diagrams for how they bring forth inventive and speculative possibilities for learners to encounter and problematize spatio-temporal relationships, rather than seeing them as ways of being mistaken.

Introduction

In this paper, we deal with the issue of how students might learn about a representational system in which temporospatial relationships are the ground for the mathematical doing. Our interests are also in how visual, proprioceptive and kinaesthetic aspects of experiencing these relationships might move the learning of mathematics in unexpected and unconventional directions. We follow here de Freitas (2013) in rethinking mathematics as the place of events, instead of objects, where creativity and contingency prevail and the problematic-rather than the axiomatic-better capture the vitality of mathematical activity. The idea is that deduction "moves from the problem to the ideal accidents and events that condition the problem and form the cases that resolve it." (Smith 2006, p. 145). Thus, mathematics is concerned with the occurrence of events more than with the existence of objects, and attention is on the material encounters with the mathematical.

In this perspective, we present an activity that was carried out with grade 9 students working with graphing motion technology to study function. In particular, the technology requires that two remote controllers of the Nintendo Wii game console (Wiimotes) are moved at the same time in front of a sensor bar, and it displays two space-time graphs on the same Cartesian plane. In the graphs, space is given by the distance of each controller from the bar. Thus, the software captures the movement of the Wiimotes over time. Our focus is on a written task that asks the students, divided into groups, to draw a space-time graph related via movement to a given graph. The task is called "Rob and Bob". In it, Rob and Bob are the names of two little robots that are imagined to be moving the controllers in front of the sensor. The graph associated to Rob's movement is given on paper, together with the instructions with which Bob is supposed to be moving with respect to Rob. The students are expected to complete the task adding Bob's graph on the Cartesian plane. We will discuss how three different graphs are presented as solutions to the task from different groups, and we will develop how we think that these are significant in terms of the novel mathematical meanings that the students are articulating. In the meanwhile, we will also draw attention to aspects of the experience with the technology that might support this novelty, raising issues about the role of perception on the one side, and about the features of the technology on the other side.

Theoretical highlights

The representational system we refer to in this paper is the (space-time) Cartesian plane, which our students encounter through activity with the technology. However, we want to trouble traditional ideational assumptions that conceive such system as inhabited by mathematical figures or functions that, in their essence, are representations (particular instantiations or attributes, concrete instances) of some form, inert, transcendent, abstract and disembodied. In fact, claims de Freitas (2013), "the process of instantiation fails to capture the creative and material act of individuation that is entailed when we do mathematics." (p. 586). We instead embrace an animate vision of the mathematical drawing/creation or act of drawing/creation of a figure or a function as event-structured, full of potentiality, temporality and movement, immanent, contingent to material circumstances, and incidentally subject to transformation. This positions us in the broad discussion on the theorising about the embodied nature of mathematics thinking and learning, which attempts to look at knowledge in non-representational ways and to overcome body/mind Cartesianisms (see e.g. [START_REF] Nemirovsky | Playing mathematical instruments: Emerging perceptuomotor integration with an interactive mathematics exhibit[END_REF][START_REF] Sinclair | Generations of research on new technologies in mathematics education[END_REF][START_REF] Ferrara | Perceiving and creating in the mathematics classroom: A case-study in the early years[END_REF][START_REF] Roth | Growing-making mathematics: A dynamic perspective on people, materials, and movement in classrooms[END_REF]. According to this view, learning is much more about encountering concepts than about recognizing concepts. [START_REF] Cutler | Bodies of learning[END_REF] might argue that thus the challenge is to treat learning as an ontological rather than an epistemological problem, staying away, we would add, from opposing the mathematical and the physical. The issue of representations is crucial here. As [START_REF] Sinclair | Generations of research on new technologies in mathematics education[END_REF] points out, it is not that "symbols, diagrams, programming languages and even gestures" (and any other system, we would add) "do not at times function to re-present mathematical concepts and relations", rather "they are inevitably bound up with bodies and discourses and thus potentially poised to open up new meanings." (p. 174, emphasis in the original). Our own reading of this makes sense as regards our commitment to a mobile view of mathematics and mathematical doing that tries to escape concrete versus abstract and matter versus thought divides [START_REF] Ferrara | Agency and assemblage in pattern generalisation: A materialist approach to learning[END_REF]. In [START_REF] Ferrara | Perceiving and creating in the mathematics classroom: A case-study in the early years[END_REF], these divides are challenged through a vision of perception and creation in/of mathematics for which perceiving is conceiving, thinking is acting and creating is learning. The work of philosopher Gilles [START_REF] Châtelet | L'enchantement du virtuel[END_REF]Châtelet ( , 1993Châtelet ( /2000) ) on inventive diagramming was provocative to us in considering the centrality of mobility or virtuality to bridging the physical and the mathematical. The virtual is the necessary link between the two realms. [START_REF] Roth | Growing-making mathematics: A dynamic perspective on people, materials, and movement in classrooms[END_REF] also draws on Châtelet to underline how one way of thinking about dynamic systems is just in terms of the virtual. We can better understand this if we take the examples that [START_REF] Châtelet | L'enchantement du virtuel[END_REF] makes about historical contributions of new ideas by Leibniz and Abel. Leibniz theorised differential calculus thinking of points as if they were alive, as powers of explosion ("puissances d'explosion"), while Abel saw the curve not as fixed but in terms of its power of receiving intersections ("comme puissance à recevoir des intersections"). The virtual restores concepts to mobility, granting them inventive force and power. For de Freitas (2014), Châtelet shows us "how we might study a particular practice for how its lines of flight flourish and act generatively in unfolding new intensive dimensions." (p. 290, emphasis in the original). The virtual is that which nourishes encounters with mathematics, linking the concrete and the abstract and allowing recoding the indeterminate contours of the sensible and the intelligible. This has to do with the potentiality or virtuality that is always entailed in perception: "We never just register visual information from that which is in front of our eyes: we see potentiality, relationality, mobility, occurrence. Students are not seeing an object; they are seeing an event" (de Freitas, 2014, p. 298).

In this paper, we take this perspective to look at the students' mathematical encounters with spatiotemporal relationships, focussing on the material and virtual dimensions of these encounters.

Method and activity

The activity, which is the focus of this paper, is part of a classroom-based intervention [START_REF] Stylianides | Seeking research-grounded solutions to problems of practice: Classroom-based interventions in mathematics education[END_REF] aimed at introducing the concept of function through the use of graphing technology. The wider research had the main purpose of investigating how learners might articulate meanings for functional relationships through modelling motion, and how their embodied activity with the technology might affect these meanings. A class of 30 grade 9 students and their regular mathematics teacher participated in the study, which lasted for a period of about three months with weekly sessions. During this time, the students worked on individual tasks, in groups of three people and in pairs of groups, taking part in class discussions. The authors were both present in the classroom and two cameras were used to film the mathematical activity of the students during all the sessions. Data for the research analyses are based on the films and students' written productions and diagrammatic activity. A microethnographic methodology (e.g. Streeck & Mehus, 2005) is essentially chosen for studying interactions and discourse in the classroom through strands of semiotic and representational activity over short periods of time, drawing attention to the material circumstances of the mathematical events.

The technology the students used in the case we consider here is WiiGraph, an interactive software application, which leverages two Wiimotes to display the space-time graphs of two users moving the remote controllers in front of a sensor bar. WiiGraph provides several challenges and composite operations, including shape tracing, maze traversal and ratio resolution. Choosing the plain visualisation (Line), the software captures the distances of the controllers over time and two graphs appear, in real time and with different colours, on a single Cartesian plane. Figure 1 shows a case of this type of visualisation for a 30-second default time and two students who move the controllers. Visual and bodily (especially proprioceptive and kinaesthetic) interactions partake in the students' encounters with the graphical system in relation to experiencing spatial and temporal aspects with the technology. We will not refer to other types of graphical activity, since this is the one of interest in the case of Rob and Bob.

The task was given in a written form to the class during the second session and did not imply direct use of the technological devices. In the first session, the students explored Line and its graphical potential, became acquainted with the devices and started discussing about pairs of functions (for example, horizontal or slanted straight lines), with graphs originating in real time and projected on an interactive whiteboard. The activity of Rob and Bob was designed with the purpose of unfolding the slope/speed relation (early insights emerged out of class discussion in the first session), and how it may reveal relationships between two space-time graphs (functions).

Rob and Bob

The task was faced by the students divided into groups of three people, and followed by a class discussion led by one of the authors. It focuses on an imaginary experience with WiiGraph in which two little robots move (the controllers) in front of the sensor bar, but only the graph associated to one robot's movement is given (Figure 2a). The text of the task is the following:

Rob and Bob are two little robots, which can be taught to move in front of the sensor very precisely. Suppose that, in response to Rob's movement, WiiGraph produces the line below (Figure 2a). Imagine that Bob also moved: it started together with Rob, at the same distance from the sensor, but moved at a double speed and in the opposite direction.

• Which graph would WiiGraph show for Bob's movement?

• Did Rob and Bob meet again after the start? Justify your answers.

The task has an unconventional nature with respect to the representational system offered by the technology, because it does not ask the students to merely reason on the model to motion, or motion to model, shift. Instead, information about the missing graph is given in terms of the relationships between the two robots' movements ("double speed", "opposite direction"), so that the students are moved to think about the relationships between the two graphs (double slope with opposite sign), through their perceptual and bodily experience with the tool. In addition, the simultaneity of the two movements, which by the way recalls the usual way of using the tool, is embedded in information about the starting instant/point ("it started together with Rob", "at the same distance"). The given graph is that of a piecewise function made up of four pieces, which capture alternate ways of moving by Rob: stepping further from the sensor for the first five seconds, stopping for the next fifteen seconds, returning to the starting position in other five seconds, and stopping for the last five seconds (Rob keeps constant speed in each time interval). We expected the students to complete the Cartesian plane drawing a graph like the one in Figure 2b. It is the graph of a piecewise function again made up of four pieces, defined on the same sequence of time intervals as the given graph. These pieces correspond to four ways of moving by Bob: getting close to the sensor for the first five seconds, stopping for the next fifteen seconds, returning to the starting position in five more seconds, and stopping for the last five seconds (however, Bob is supposed to cover double space with respect to Rob, according to the constraint of moving at a double speed. Of course, this is true when he moves, and trivially when he does not, since the distance covered is null).

Instead of looking at the expected graph as the correct one and speaking of difference in terms of being mistaken, we dwell on different unexpected solutions emerged from the groups about their potential to bring forth new relational possibilities for the two robots' movements as well as for the pair of graphs. In the next section, we take these solutions as the problematic actualizations of the mathematical events that the groups encounter in solving the task. It is this idea of novelty that speaks directly to inventive mathematics and makes students alive to their engaging with the task.

Graphs and discussion

The groups worked on the task for half of the time, then they took part in a collective discussion in which their graphical solutions were compared. Only one group drew the expected solution (Figure 2b), while eight out of ten created one of the three unexpected lines shown in Figure 3 (For the sake of ease, we refer these lines to three graphs labelled with numbers 1, 2 and 3). The three graphs added for Bob's movement have some similarity. They all show that taking into account information about opposite direction and capture it visually in the diagram is not an issue for the students. Each added graph is made up of four pieces, which embed the opposite way of moving with respect to Rob: first getting close, then returning to the start (first a decreasing piece, then an increasing piece). Not even slope is an issue: the double speed of movement is double slope in the three diagrams. However, the duration of Bob's movement is problematic for the students. In fact, while there is correspondence between ways of moving there is no embodiment of duration: there is no correspondence between time intervals in which both robots either move or stand still. The lengths of the horizontal pieces are different from each graph to the other, and the constraint for Bob to move at a double speed with respect to Rob is no longer preserved. Thus, the problematics of duration evolved along various accidental threads for the students, driven by their perceptual and bodily engagement with the task. These broke with causal connections and direct determination, opening up to speculative and inventive investments and to a generative movement, implicating the perturbation of spatio-temporal relationships. For example, in the case of graphs 1 and 3 (Figures 3a and3c), some encountered the event for which Bob already stands still while Rob is still moving and, later, Bob moves towards the starting position while Rob is still standing still. Some groups introduced the new event in which the second robot stops just after fifteen seconds, in the very middle of the experience with WiiGraph, and ideally disappears from the view of the sensor, so that the second graph might accidentally stop in the middle of the diagram (Figure 3a). Almost all the students engaged with the kinaesthetic question of Rob and Bob always covering the same space, no matter the time spent, as shown in the three diagrams. These threads are actualized through the groups' written explanations, then during class discussion. Types of explanation are the following:

Graph 1: "The line we represented is half of Rob's line. The lines are steeper because speed is doubled and Bob moved faster than Rob and in the opposite way. The graph ends at 15 because Bob, moving at a double speed, stopped at half of 30."

Graph 2: "The two configurations are different from each other, indeed slopes also change since times change: Rob is slower. So, covering the same space in different time, there will be a higher steepness."

Graph 3: "Bob goes at a double speed with respect to Rob, so it finishes "the lap" before Rob. The rest of the way it stood still and at the end it met Rob. Speed changes between the ways of Bob and Rob, indeed Bob has to cover the same space backwards using half of the time."

The logical equivalence between double speed as double distance in the same time and as the same distance in half of the time is lost, and the problematic of covering a fixed space in less time drives students' perception and visualisation in the diagramming of the missing graph. Graph 1 (Figure 3a) is the most coherent in respect to the axiomatic way of reasoning about double speed but at once the most incoherent in relation to kinaesthetic actions with the technology. Briefly speaking, it is nothing but a temporal shrinking of the given graph. Instead, graph 3 (Figure 3c) is in line with the usage of WiiGraph, because it embraces all the thirty seconds of the modelling process. The same occurs in the case of graph 2 (Figure 3b), which is particular though, since it struggles to depict the simultaneity of the two robots' movements. In the discussion, different students actualize in different ways the problematics that sustain the mathematical events that occurred in solving the task. Below, Lorenzo, Luigi, Giulio and Oliver bring forth in the discourse the issues of duration and simultaneity of movements, of moving at a double speed and of covering the same space, issues that are entangled in their diagrammatic and written activity. Lorenzo speaks about graph 1 (Figure 3a), Luigi and Giulio refer to graph 2 (Figure 3b), while Oliver argues about graph 3 (Figure 3c).

Lorenzo: 'Cause, moving at a double speed, distance remained constant, even though it was the opposite, but maybe, if Rob performed a movement in 10 seconds, Bob performed it in 5 seconds because speed was double.

Luigi: For me, hem, the graph took up the same time because, moving simultaneously, maybe, at the time they were moving, it took less time for one than for the other one to cover the same space, to move in the same space, but then one stood still until the other one also did move again, so both graphs last for 30 seconds. (. Giulio: Yes, for me he's right about the first piece, but then if you stand still, so speed is null, zero times two is always zero and so speed has to be equal in the positions in which it stands still, for me.

Discourse with the researcher unfolds the event-nature of unexpected threads traversed in solving the task. We see how the students inscribe themselves into the temporality of imaginary situations with the robots. The ways of perceiving this temporality are different for different (groups of) students: some imagine that one robot has to "wait for" the other to know what to do (Giulio, Luigi); for others, coordination is not needed (Oliver) or considered (Lorenzo). Time is duration and simultaneity of movements: both aspects become problematic for learners. Both are crucial in making sense with WiiGraph of time as the independent variable in space-time functions.

Conclusive remarks

In this paper we have discussed some unexpected graphical solutions to a given diagramming task based on modelling motion through the use of Wii graphing technology. We have focused not on how these solutions were incorrect with respect to the expected diagram, but on ways in which they brought forth new possibilities for the students to encounter spatio-temporal relationships. In so doing, we looked at visual, proprioceptive and kinaesthetic aspects of experiencing the technology as that which sustained the occurrence of new mathematical events in the classroom, bringing into being problematic perturbations of the given situation, like shrunk graphs as well as not coordinated movements and fixed paths, which break with the conventional visualization and activity of the representational system in use. The written, the diagrammatic, the discursive and the bodily, as the groups attempted to grapple with the task (to make sense of it), have to be seen as that which animated the task without ever exhausting it or fully determining it. The temporality of the events speaks directly to the material contingency of learning: the students are dynamically affected by the diagrammatic activity while telling stories of motion related to graphing technology.

Figure 1 :

 1 Figure 1: The graphical system in Line and two students moving

Figure 2 :

 2 Figure 2: (a) The given graph, (b) The expected solution to the task given by one group

  Figure 3: Unexpected solutions -(a) graph 1, (b) graph 2 and (c) graph 3

  Bob] does things in half of the time, it's true, however it's not that he can know what it [Rob] will do later, hem, 'cause we do know it, but if they move simultaneously...

	Researcher: Did you say that time is halved?
	Giulio: Yes, 'cause it [Oliver: Bob does our graph with the movements with which it's been set up, and then, in
		the end, it's not that it waits for Rob, it goes on just as it likes and wants, then at a
		certain point, when movements are finished, it stops and the line keeps straight for
		the rest of the time.
	Researcher: Are you saying that time is halved?
	Oliver:	Yes, because of the double speed. (...)
	Lorenzo:	Indeed, for explaining a little the graph, I said that if there's a distance to cover, and
		that distance is 50 kilometres and you go at a speed of 50 kilometres per hour, it
		takes you 1 hour to cover that distance. Instead, if you go at 100 kilometres per hour
		it takes you 30 [minutes].
		..)
	Giulio:	For me, it [the graph] finished at 30, 'cause it's not that Rob [Bob] could know
		Bob's [Rob's] movement in advance, so it [the graph] cannot finish at 15, it [Bob]
		has to wait for it [Rob] to perform the same but opposite movement, 'cause we did
		see Bob's graph (miming it in the air) but if they move simultaneously, it means
		that one cannot anticipate the movements, so it cannot finish at 15 seconds. (...)

Giulio:

We depicted slope at 2,5 seconds but then we stood still until 20 seconds, 'cause anyway it's true that time is halved, but Bob doesn't know what Rob will do later, so it has to wait for it.

Proceedings of CERME10