Lars Bollen

Guillermo Juarez

H U Hoppe

Mobile Notes: Mobile Devices in Creative Discussions

Introduction

The trendy notion of "mobile learning" has different connotations: On the one hand, it can be understood as "learning on the move" -often referred to as "learning any time anywhere". Of course this interpretation relies on specific kinds of technological enabling, but the definition aims at the general setting of learning activities. Particularly, it includes informal learning settings (cf. [START_REF] Scanlon | Mobile technologies: prospects for their use in learning in informal science settings[END_REF]). A second interpretation sees mobile learning somewhat more pragmatically as learning with mobile devices. This may include formal learning in the classroom. Here, mobile devices may enable a one-to-one orchestration with learning devices (one device per learner, cf. [START_REF] Chan | One-to-one technology-enhanced learning: An opportunity for global research collaboration[END_REF]). This workshop contribution clearly bases on the second view in that it explores the use of mobile devices in classrooms. The one-to-one assumption (see above) raises a couple of questions which need more exploration. Among these are the questions of ownership (who owns the personal device -the institution or the student?) and homogeneity (will one device be used throughout different subject matters?). In this context, we explore device heterogeneity even in one specific setting. This setting includes personal mobile devices (PDAs) as well as a publicly visible interactive screen connected to a PC. Similar mixed device scenarios involving tablet PCs and bigger interactive screens have been studied by Liu and Kao [START_REF] Liu | Handheld Devices with Large Shared Display Groupware: Tools to Facilitate Group Communication irn One-to-One Collaborative Learning Activities[END_REF]. Based on the findings of Liu and Kao, we believe that, as a next step, dedicated classroom scenarios with heterogeneous devices should be designed and orchestrated. Here, "dedicated" means an orientation towards specific types of learning activities in which different types of devices would be integrated based on different functional assignments. Evidently, the functional assignments should reflect the specific strengths and potential complementarities of the different devices. E.g., following Liu and Kao [START_REF] Liu | Handheld Devices with Large Shared Display Groupware: Tools to Facilitate Group Communication irn One-to-One Collaborative Learning Activities[END_REF], public interactive displays complement the lack of shared visual focus with smaller personal devices. In our own previous work, we have explored extending collaborative visual modelling tools to PDAs [START_REF] Pinkwart | Lightweight Extensions of Collaborative Modeling Systems for Synchronous Use on PDA's[END_REF]. In one "light weight" version, we used PDAs to graphically annotate models without replicating the functionality (or "operational semantics") of the modelling environment on the PDAs. The fully fledged modelling environment would run on a PC and would be shown on a large interactive display. On the PDA, an area of the visual modelling space could be selected and annotated using and hand writing and drawing. These annotations were directly transferred to the public display. This type of interoperability could be characterised as synchronous bitmap sharing with graphical annotation. But this usage scenario also suffers from a problem of shared visual focus, due to the necessary switching between the PDA view and the public view. It is also somewhat clumsy to adjust the viewing area on the PDA. This environment decreases the problem of switching focus from the PDA to a shared display area, since here the PDA is only used as an input device; upcoming discussions are still conducted with the help of a large, shared display. In this contribution, we focus on an application in which the function of PDAs is more limited but more focussed: The PDAs are used as as input devices in a quasi-synchronous mode. The usage scenario is a face-to-face brainstorming discussion in which contributions are prepared on the participants' PDAs. When a contribution (either text based or a hand written sketch) is completed, it is sent to a database from which it can be retrieved and transferred to the public display application. The transfer of a contribution to a public display is done either instantaneous (synchronous) or it can be selectively controlled by a discussion moderator, in which case the transmission would be delayed and asynchronously. All public discussions would only refer to the public display area.

Mobile Notes

Building on the conceptual background described in the previous section, we designed and implemented an application to support classroom discussions with mobile devices and electronic whiteboards similar to what has been presented in [START_REF] Bollen | SMS-Based Discussions -Technology Enhanced Collaboration for a Literature Course[END_REF] and [START_REF] Bollen | The Use of Mobile Computing to Support SMS Dialogues and Classroom Discussions in a Literature Course[END_REF]. The environment -we called it "Mobile Notes" -is based on FreeStyler [START_REF] Gaßner | Diskussionen als Szenario zur Ko-Konstruktion von Wissen[END_REF] and Cool Modes [START_REF] Pinkwart | A Plug-In Architecture for Graph Based Collaborative Modelling Systems[END_REF][9], two applications that have been developed at the University Duisburg-Essen. These applications support interactive and collaborative modelling with heterogeneous semantics (e.g., Petri Nets and concept maps) in computer supported classroom scenarios. Mobile Notes' main functionality is to use PDAs to write short textual contributions, to draw sketches or to participate in a vote and submit these products to a central server in order to display and arrange them on an electronic whiteboard and to initiate further discussions and collaboration. Mobile Notes' main components are:

• A computer running FreeStyler with the Mobile Notes plug-in (the server).

• Mobile devices acting as input clients.

• A database as communication interface between the clients and the server.

• An interactive whiteboard used as the common output device for all the clients. This board is based on the FreeStyler application. Figure 1 shows a typical arrangement of these components. Following the concept of functional assignments (see above), the mobile devices is designed not to copy or emulate what is shown on the screen. Being aware of the limitations as small screens and the lack of conventional input devices as full size keyboards, our approach was to develop a custom interface that enables an easy interaction between user and system.

Figure 1. Typical arrangement of Mobile Notes' components

Typical use cases for this architecture are e.g. teacher presentations with comments from the audience, brainstorming scenarions with participants freely collecting ideas and comments or test questions, that are prepared by a teacher / moderator that have to be answered by students. Figure 2 and Figure 3

System Demo

The envisioned system demo consists of an interactive and productive discussion between the workshop participants while using the hardware and software described above. Figure 4 shows a situation simular to what is planned for the system demo. For this purpose, we will bring several (5-7) PDAs and a laptop to the workshop. Participants will be able to use the PDAs in a moderated discussion as input devices. The laptop will be used by the moderator to run a discussion support application (FreeStyler). We will need a video projection to be connected to the moderator's notebook.

  show a handwritten annotation in the PDA and its representation in FreeStyler and a typical FreeStyler workspace during a Mobile Notes session. This screenshot originates from a recent usage of Mobile Notes during the "Big Issues in Mobile Learning Workshop" in Nottingham, UK, in June 2006.

Figure 2 .Figure 3 .

 23 Figure 2. Creating handwritten notes and sketches for use in FreeStyler

Figure 4 .

 4 Figure 4. Using Mobile Notes in a discussion.

Additionally, we will need an open wireless network. If a wireless network is not available, we could bring our own access point.


S Mironov

A S Mel'nikov

A Buzdin

Electromagnetic proximity effect in planar superconductor-ferromagnet structures

The spread of the Cooper pairs into the ferromagnet in proximity coupled superconductor -ferromagnet (SF) structures is shown to cause a strong inverse electromagnetic phenomenon, namely, the long-range transfer of the magnetic field from the ferromagnet to the superconductor. Contrary to the previously investigated inverse proximity effect resulting from the spin polarization of superconducting surface layer, the characteristic length of the above inverse electrodynamic effect is of the order of the London penetration depth, which usually much larger than the superconducting coherence length. The corresponding spontaneous currents appear even in the absence of the stray field of the ferromagnet and are generated by the vector-potential of magnetization near the S/F interface and they should be taken into account at the design of the nanoscale S/F devices. Similarly to the well-known Aharonov-Bohm effect, the discussed phenomenon can be viewed as a manifestation of the role of vector potential in quantum physics.

The proximity phenomena in condensed matter physics are known to include the interface effects which are usually associated with the exchange of electrons between the contacting materials. This exchange is responsible for the mutual transfer of different particle-related qualities through the interface such as superconducting correlations, spin ordering etc. [1][2][3] The goal of the present work is to show that this spread of particle-related qualities in some cases should be supplemented by the longrange spread of the electromagnetic fields. As an example, we demonstrate that such electromagnetic proximity effect can strongly affect the physics of superconductorferromagnet (SF) systems which are widely discussed as building blocks of superconducting spintronics 4,5 .

To elucidate our key observation we can consider an exemplary bilayer system consisting of a superconducting (S) film placed in contact with a ferromagnetic (F) layer with the magnetic moment parallel to the layer plane (see Fig. 1). Considering the S and F subsystems to be isolated we get a perfect example of complete separation of the regions with a nonzero concentration of Cooper pairs and magnetic field. The latter is completely trapped inside the ferromagnet. As we allow the electron transfer between the subsystems the Cooper pairs immediately penetrate the ferromagnet inducing the pair electron correlations there. This process which is usually referred as a standard proximity effect leads to the series of fascinating transport phenomena 1,6 . The inverse proximity effect namely the transfer of the magnetic moment from the ferromagnetic to the superconducting subsystem is also possible and has been recently studied in a number of theoretical and experimental works [7][8][9][10][11][12][13][14][15][16][17][18] . This inverse proximity effect is related to the spin polarization of electrons forming the Cooper pair near the S/F interface and results in the small magnetization of the superconducting surface layer at the depth of the or-der of the Cooper pair size, i.e., the superconducting coherence length ξ 0 ∼ 1 -10 nm. Experimentally, it has been observed with the help of muon spin rotation techniques 13,14 , nuclear magnetic resonance 15 and neutron scattering measurements [16][17][18] .

It is commonly believed that neglecting this short range inverse proximity phenomenon we can assume the magnetic field to remain completely trapped inside the F film even when the electron transfer between the films is possible. This conclusion is based on the obvious observation that an isolated infinite single-domain ferromagnetic film with the in-plane magnetization does not produce the stray magnetic field in the outside. In this paper we show that, contrary to this belief, the direct proximity effect is always responsible for the exciting the supercurrents flowing inside the ferromagnet itself and, thus, for the appearance of the compensating Meissner supercurrents in the superconducting subsystem. The appearance of these currents is accompanied by the generation of the magnetic field in the S film which decays at the distances of the order of the London penetration depth λ which can well exceed the coherence length in type-II superconductors (see Fig. 1). From the experimental point of view, this electromagnetic proximity phenomenon reminds the Aharonov-Bohm effect 19 since the current inside the attached superconductor is induced by the ferromagnetic layer which does not create the magnetic field outside the layer in the absence of such superconducting environment. At the same time, the true physical key point is that the wave function penetrating the ferromagnet is responsible for this effect, when the vector-potential of the F magnetization unavoidably generates current in quantum system described by a common Cooper pair wave function (superconductor and a region near S/F interface). Quite surprisingly this long-range electromagnetic (orbital) contribution to the inverse proximity effect has

F x 0 z S d F M H z -d S 0 J dia 4 M π 0 H z para F x z S M 0 4 M π 0 H z magneƟsm leakage of s (a) (b) (c) FIG.
1. The sketch of the superconductor/ferromagnet bilayer. (a) When the layers are separated from each other the magnetic field exists only inside the F layer. (b) In contrast, when the layers are put in contact the magnetization inside the ferromagnet becomes the source of the long-range magnetic field in the superconductor. In both panels the orange solid (blue dashed) curves schematically represent the profile of the magnetic field when the total current inside the F-layer is diamagnetic (paramagnetic).

(c) The spatial profile of the screening parameter λ -2 (x) inside the ferromagnetic layer in the dirty limit. We take σs = 10σ f .

been overlooked in all previous studies devoted to this subject while the relevant magnetic fields strongly exceed the ones induced by the spin polarization discussed in 8,9 and should dominate in the experiments. Indeed, in 14,16 the induced magnetic field was observed in S/F hybrid systems at distances much larger than all relevant superconducting coherence lengths. The existing theories of proximity effect fail to provide the interpretation of these results, while our approach provides a natural explanation of these phenomena. Thus, the currentcarrying states appear to be unavoidable for the SF systems and should be taken into account in the design of the devices of superconducting spintronics 4,5 . Further consideration illustrating the above qualitative arguments is organized in two steps: (i) first, we consider a simple model assuming a phenomenological form of the relation between the supercurrent j and vector potential A in conditions of the long range inverse proximity effect; (ii) second, we present the results of microscopic calculations of the j(A) relation which support and justify the phenomenological findings. Accounting for the proximity effect we write the Maxwell equation in the form:

rot rotA = 4π c (j s + j m ) , (1)
where j s is the Meissner current and j m = c rotM is the current associated with the magnetization M.

For the sake of definiteness we consider a bilayer consisting of the superconductor of the thickness d s ≫ λ and a ferromagnet of the thickness d f ≪ λ with the uniform magnetization M 0 = M 0 e z . We choose the x-axis perpendicular to the layers with x = 0 at the S/F interface (see Fig. 1). First, we assume the local London relation which is relevant, e.g., for dirty S/F sandwiches:

j s (x) = - c 4π 1 λ 2 (x) A(x), (2)
where the London screening parameter λ -2 becomes dependent on x. The penetration of Cooper pairs into the F film gives rise to the supercurrent there and also slightly modifies the Meissner response of the S layer in the small region of the thickness of the order of the superconducting coherence length ξ from the S/F interface. The latter modification can even vanish if the conductivity of the ferromagnet is much smaller than the normal conductivity of the S layer. So, let us, first, neglect the changes of λ inside the superconductor and assume it to be equal to the constant value λ 0 . Then the solution of Eq. ( 1) in the S layer reads A y (x) = A 0 exp(x/λ 0 ) where A 0 is a constant and inside the F layer A y (x) = A 0 + 4πM 0 x. Here we neglect the spatial variations of the vector potential in the ferromagnet related to the weak London screening since the typical scale of such variation is of the order of λ ≫ d f . Thus, the exact solution of Eq. ( 1) is redundant for most realistic structure and material parameters and we restrict ourselves to the approximate calculation procedure described below. To find A 0 we integrate Eq. ( 1) over the width of the ferromagnet accounting the relation B z = ∂A y /∂x. The magnetic current j m = c rotM integrated over the sample thickness is zero and, thus, the relation between the magnetic field B z (d f ) outside the sample and the field B z (0) = A 0 /λ 0 inside the superconductor close to the S/F interface takes the form

B z (d f ) -B z (0) = A 0 d f 0 dx ′ λ 2 (x ′ ) + 4πM 0 d f 0 x ′ dx ′ λ 2 (x ′ ) .
(3) The first term in the r.h.s. of Eq. ( 3) can be neglected since it is of the order of (d f /λ)B z (0) ≪ B z (0). Note that for the same reason one can neglect the contribution coming from the renormalisation of λ -2 in the region of the width ∼ ξ inside the S layer if the inverse proximity effect is not small. In the absence of the external magnetic field B z (d f ) = 0 and we finally obtain that the magnetic field induced in the superconductor is

B z = -4πM 0 Q exp(x/λ 0 ), (4)
where

Q = d f 0 λ -2 (x ′ )x ′ dx ′ .
Note that in the con- 2. The dependence of the magnetic kernel Q on the ferromagnet thickness for the clean and dirty limits. For the clean limit we take h = 10πT and choose the temperature T in a way that ∆ = 2πT . Also we define the values 2 where σ f is the conductivity of the ferromagnet and ν0 is the density of states at the Fermi level per unit spin projection and per unit volume.

d f / ξ f 0 1 2 3 4 5 6 Q / Q d -1 0 1 F x S H z x H z F S d f /
Q d = π 2 σ f ξ 2 f tanh(∆/2T )/(2 c 2 ) and Qc = e 2 ν0ξ 2 h (vF /c)
ventional regime when the superconducting condensate penetrating the F layer reveals a diamagnetic response with Q > 0 the magnetic field induced in the S layer is anti-parallel to the magnetization M 0 . In the case when d f is of the order of the superconducting coherence length in the ferromagnet ξ f the basic estimate gives Q ∼ (ξ f /λ) 2 ∼ 10 -2 for the S/F structures based, e.g., on thin Nb films. Considering the experiments probing the change in the magnetic moment in SF structures when cooling down through the superconducting critical temperature and taking the typical magnetization corresponding to Fe or Co films 4πM 0 ∼ 10 4 Oe we find B z ∼ 10 2 Oe which is an easily measurable value.

More accurate estimate for the value Q can be obtained within the Usadel formalism for dirty S/F bilayers 20 . In Fig. 1(c) we plot the typical profile of the screening parameter inside the F layer while the behavior of the kernel Q is shown in the right panel of Fig. 2. Remarkably, the value Q oscillates as a function of d f being either positive or negative which corresponds to anti-parallel or parallel directions of B and M 0 for different values of d f . Somewhat similar results for the magnetic response have been previously found for the S/F structures with a low transparent barrier at the interface in 21 , but the magnetic field generated inside the superconductor was neglected.

Interestingly, the value Q and, thus, the magnetic field in the S layer can be substantially increased provided the magnetization in the F layer has a non-collinear structure -this is relevant to the recent experiments 14 . To illustrate the origin of this effect let us assume that the ferromagnet consists of two layers: one layer F 1 occupying the region 0 < x < d 1 has the magnetization M 0 along the z-axis while another layer F 2 of the thickness d 2 has the same magnetization M 0 but directed along the y-axis. We choose d 2 to be much larger than the normal metal coherence length ξ n = D f /T but still much less than λ. The vector potential in the F bilayer has two components: A = (A 0y + 4πM 0 x) e y + A 0z e z for 0 < x < d 1 andA

= (A 0y + 4πM 0 d 1 ) e y + [A 0z -4πM 0 (x -d 1 )] e z for d 1 < x < d 1 + d 2 .
The functions A y0 (x) and A z0 (x) vary over the distances ∼ λ and are almost constants in the F film. Integrating Eq. ( 1) analogous to the case of the S/F bilayer we find the magnetic field induced in the S layer:

B z = -4πM 0 Q z e x/λ0 , B y = -4πM 0 Q y e x/λ0 , (5)
where

Q z = d1 0 x ′ dx ′ λ 2 (x ′ ) , Q y = d1+d2 d1 (x ′ -d 1 )dx ′ λ 2 (x ′ ) . (6)
In Fig. 3 we schematically show the spatial profiles of the spontaneous magnetic field for parallel and perpendicular orientations of the magnetic moments in the fer- romagnetic layers. Remarkably, for perpendicular orientation the estimate for the value Q y based on the Usadel formalism gives Q y ∼ (ξ n /λ)Q z ≫ Q z where ξ n = D f /T ≫ ξ f . As a result, in the experiments where the angle θ between the magnetic moments in two F layers can be tuned one should observe a strong increase in the magnetic moment of the S layer for θ = π/2 as compared to the case θ = 0 (see Fig. 3). Namely this behavior has been recently discovered in experiments with the Au/Nb/ferromagnet structures 14 .

z S F y x H z H y z S F
Note that all described phenomena should become more pronounced provided one deals with the clean S/F and S/F 1 /F 2 sandwiches. In this case the Cooper pairs penetrate over much larger distances into the ferromagnet which strengthen the proximity effect and, thus, the amplitude of the induced magnetic field in the superconductor. In the clean limit the relation j s (A) becomes non-local and we may write it in a very generic form

j s (x) = - c 4π A(x ′ )K(x, x ′ )dx ′ . (7)
Deep inside the superconductor for |x| ≫ ξ N = v F /T the kernel K in this relation has the standard London form characterized by the penetration depth λ 0 . Then the profile of the magnetic field generated in the superconductor is again determined by Eq. ( 4) but the value Q now can be expressed through the nonlocal kernel K (see supplementary material):

Q = d f -x0 dx d f 0 dx ′ x ′ K(x, x ′ ) -λ -2 0 δ(x -x ′ )θ(-x) , (8
) where θ(x) is the Heaviside step function and the integration limit x 0 satisfies the condition ξ N ≪ x 0 ≪ λ 0 .

The calculations of the screening parameter Q are presented in the supplementary material. In Fig. 2 we show the typical dependence of the value Q on the ferromagnet thickness. The interference between different quasiparticle trajectories results in the oscillations of Q(d f ) with the period of the order of the length ξ h = v F /h which is known to characterize clean ferromagnets 1 . The envelope of the oscillating Q increases as

∝ (d f /ξ N ) 2 at small d f /ξ N values [here ξ N = v F /(4πT )] and decays exponentially at d f ≫ ξ N .
Up to now several experimental papers reported the evidences of the long ranged (at distances larger than ξ) magnetic field generation in S/F systems which could be naturally explained by the above theory. In particular, in Ref. 16 the polarized neutron reflectometry was used to study the magnetic field profile in a single V(40 nm)/Fe(1 nm) bilayer. Below T c these measurements indicate the appearance of the spontaneous magnetic field penetrating the S layer at the distance ∼ 20 nm from the S/F interface. This distance definitely exceeds the coherence length ξ 0 ∼ 5 ÷ 10 nm for vanadium. Another manifestation of the long ranged field generation was recently reported in 14 , where the muon spin-rotation experiments in Au/Nb/ferromagnet structure revealed a remote magnetic field in Au at the large distance (more than 50 nm which strongly exceeds the ξ value in Nb films) from the ferromagnet. In 14 the composite F layer was used allowing a non-collinear magnetic configuration. The remote field was much more pronounced in a perpendicular configuration in accordance with the above arguments regarding the generation of the long-range magnetic moment in SFF' systems. All these observations are pretty hard to explain by the standard theory of the inverse proximity effect characterized by a rather short length scale ∼ ξ 0 . In contrast, our results demonstrate the current and magnetic field generation at much larger distances ∼ λ ≫ ξ from the S/F interface. Note that repeating the experiments 14 without the applied external field one may expect the generation of the magnetic field in the Au layer solely by the spontaneous current predicted in our work.

Sure, the muon spin-rotation experiments permit the direct measurement of the spontaneous magnetic field in S/F structures. An alternative way to detect the currents associated with these spontaneous magnetic fields can be based on the use of different types of local transport probes positioned at the outer boundary of the superconductor. Scanning, e.g., the outer surface by the normal metal tip of the scanning tunneling microscope one can measure the changes in the local density of states caused by the Doppler shift of the quasiparticles energy in the presence of the superflow [22][23][24] . For the case of the superconducting probe one can also propose a simple and elegant experimental setup revealing this effect. It consists of the Josephson junction where one of the electrodes has the thickness of the order of λ and is covered by the ferromagnetic layer (see Fig. 4). Certainly, the superconducting probe should be small enough not to perturb the measured magnetic field distribution. The electromagnetic proximity effect should result in the shift of the Fraunhofer dependence of the critical current on the external magnetic field. Note that the proper choice of the magnetic field can compensate this shift and restore the bare critical current (see Fig. 4).

To sum up, we have revealed a very general mecha-nism of the long-range electromagnetic proximity effect in the S/F structures which results in the strong spread of the stray magnetic field into the superconductor from the ferromagnet. The screening currents accompanying this magnetic field spread appear in the region where in the normal state all the stray fields are completely absent. The only nonzero electromagnetic characteristic in this region is the vector potential which is usually an unobservable quantity. In this sense the current generation which accompanies the superconducting transition illustrates the crucial role of the vector potential in quantum physics of the SF structures.

SUPPLEMENTARY MATERIAL

See supplementary material for a microscopical calculation of the electromagnetic response in the dirty and clean limits.

SUPPLEMENTARY MATERIAL FOR "ELECTROMAGNETIC PROXIMITY EFFECT IN PLANAR SUPERCONDUCTOR-FERROMAGNET STRUCTURES"

A. Magnetic response: dirty limit To estimate the value Q in the dirty limit we use the Usadel formalism. The spatial profile λ -2 (x) is determined by the anomalous Green function f (x) = (f s + f t σ)iσ y which is a 2 × 2 matrix in the spin space (σ is the vector of the Pauli matrices):

λ -2 (x) = 16π 2 T σ c 2 ω>0 |f s | 2 -|f t | 2 . (S1)
In Eq. (S3) the summation is performed over the positive Matsubara frequencies ω, σ is the normal state conductivity and inside the F layer the singlet and triplet components f s and f t satisfy the Usadel equations

D∂ 2 x f s = 2ωf s + 2ihf t , D∂ 2 x f t = 2ωf t + 2ihf s , ( S2
)
where D is the diffusion coefficient and h is the exchange field. For simplicity let us assume the conductivity of the F layer σ f to be much smaller than the one of the superconductor (σ s ) which allows to impose the rigid boundary conditions at the S/F interface for the f function: f s = f s0 and f t = 0 where ∆ is the gap function and

f s0 = ∆/ √ ω 2 + ∆ 2 .
Assuming ∆ to be real we can immediately write the solution of Eqs. (S2): f s = Re(F ), f tz = i Im(F ) where F = f s0 cosh [q(x -d f )] / cosh (qd f ) and q 2 = 2(ω + ih)/D f . The functions f s (x) and f tz(x) are oscillating and, therefore, there are regions where |f tz | > |f s | and the local Meissner response is paramagnetic.

To calculate the value Q for the S/F bilayer in the dirty limit it is convenient to rewrite the expression for the screening parameter λ -2 through the function

F = f s + f tz : λ -2 (x) = 16π 2 T σ c 2 ω>0 Re F 2 . (S3)
Note that the frequencies ω enter only the amplitude f s0 which allows to calculate the sum in Eq. (S3) analytically:

ω>0 ∆ 2 ω 2 + ∆ 2 = ∆ 4T tanh ∆ 2T . (S4)
Also the value Q is defined by the following integral:

d f 0 x cosh 2 [q(x -d f )] dx cosh 2 (qd f ) = q 2 d 2 f -1 4q 2 cosh 2 (qd f ) + 1 4q 2 . (S5)
The last term in Eq. (S5) is purely imaginary and it drops out from the value Q. Finally, after algebraic transformations we obtain:

Q = π 2 σ f ξ 2 f ∆ 2 c 2 tanh ∆ 2T Im q 2 d 2 f -1 cosh 2 (qd f ) , (S6)
where ξ f = D f /h.

B. Magnetic response: clean limit

To calculate the magnetic response kernel in the clean limit we use the approach analogous to the one in [1, 2] which is based on the solution of the Eilenberger equation [3]. We consider a clean S/F bilayer with d f ≪ λ and d s ≫ λ. Inside the S layer the Eilenberger equations for the quasiclassical Green functions read

v F Df = -2ωf +2∆g, v F D * f † = 2ωf † -2∆ * g, ( S7
)
where g is the normal Green function, f and f † are the anomalous Green functions satisfying the normalization condition f † f +g 2 = 1, D = (∇ + 2ieA/ c) is the gaugeinvariant momentum operator, ω is the Matsubara frequency, and v F = v x x0 + v y ŷ0 + v z ẑ0 is the vector of the quasiparticle velocity. For simplicity we neglect the spatial variations of the gap potential ∆ inside the S layer which is accurately justified in the vicinity of T c or in the case of strong mismatch between Fermi velocities in the S and F layers which damps the proximity effect. Note, however, that such variations of ∆ should not qualitatively change the final results, so our simple model makes sense even at low temperatures. Inside the F layer we get:

v F Df = -2(ω + ih)f, v F D * f † = 2(ω + ih)f † . (S8)
We focus only on the solutions of Eq. (S8) near the S/F interface in the region of the thickness ∼ ( v F /T ) ≪ λ, where the deviations of the electromagnetic response from the bulk London relation j(A) are most pronounced. So, again, we put A y (x) = A 0 + 4πM 0 x neglecting the variations of A y (x) close to the interface. The analytical solution of the Eilenberger equation for the function g allows us to calculate the superconducting current flowing along S/F interface:

j y (x) = -4πeν 0 T ω>0 Im [g(x)] v y , ( S9
)
where ν 0 is the density of states at the Fermi level per unit spin projection and per unit volume, and the brackets denote averaging over the Fermi surface: ... = (4π) -1 π 0 dθ 2π 0 (...) sin θdϕ. Here θ and ϕ are the spherical angles at the Fermi surface so that v x = v F cos θ and v y = v F sin θ cos ϕ. Further, it is convenient to represent this current as the sum j y (x) = j M + j surf (x), where j M = -(4/3)(e 2 T v 2 F ν 0 A 0 (0)/ c) ω>0 (∆ 2 /Ω 3 ) coincides with the standard Meissner current flowing in the bulk superconductor and j surf (x) is the surface current induced by the magnetization. Finally, integrating j surf (x) over x we find the desired value of the magnetic response kernel Q.

  FIG.2. The dependence of the magnetic kernel Q on the ferromagnet thickness for the clean and dirty limits. For the clean limit we take h = 10πT and choose the temperature T in a way that ∆ = 2πT . Also we define the valuesQ d = π 2 σ f ξ 2 f tanh(∆/2T )/(2 c 2 )and Qc = e 2 ν0ξ 2 h (vF /c) 2 where σ f is the conductivity of the ferromagnet and ν0 is the density of states at the Fermi level per unit spin projection and per unit volume.

FIG. 3 .

 3 FIG.3. The profiles of the spontaneous magnetic field in the superconductor-ferromagnet-ferromagnet trilayers when the magnetic moments in the two F layer are (a) parallel to each other and (b) perpendicular to each other.

FIG. 4 .

 4 FIG. 4. Shift in the Fraunhofer critical current oscillations for the Josephson junction with one electrode covered by the ferromagnetic layer.

ACKNOWLEDGEMENTS

The authors thank G. Karapetrov and Zh. Devizorova for valuable comments. This work was supported by the French ANR SUPERTRONICS and OPTOFLUXONICS, EU COST CA16218 Nanocohybri, Russian Science Foundation under Grant No. 15-12-10020 (S.V.M.), Foundation for the advancement of theoretical physics "BASIS", Russian Presidential Scholarship SP-3938.2018.5 (S.V.M.) and Russian Foundation for Basic Research under Grant No. 18-02-00390 (A.S.M.). A.B. would like to thank the Leverhulme Trust for supporting his stay in Cambridge University.


Wouter Van Joolingen

CIEL: Integrating Approaches for Inquiry Learning

Keywords:

come

Introduction

The world is changing and changing fast. Contemporary students live in a world full of impressions, fast changing viewpoints, ubiquitous information, zapping behaviour, play and joy etc. They use facilities such as MSN, on-line gaming, and SMS. They expect a seamless access to information, mostly in a social context. At the same time we see drastic changes in learning environments, moving them to rich, multi-media, collaborative, experiential, and individualised environments. Though starting from different angles, these two developments seem to come together. Modern learning environments such as WISE [START_REF] Slotta | The web-based inquiry science environment (WISE): Scaffolding knowledge integration in the science classroom[END_REF] or BioLogica [START_REF] Hickey | Integrating curriculum, instruction, assessment, and evaluation in a technology-supported genetics learning environment[END_REF] combine all kinds of opportunities to learn and communicate in a facilitative setting.

In these learning environments, often called powerful learning environments (de Jong & Pieters, in press), specific design measures are taken to foster learning. In summary these design measures a) increase the engagement of learners, b) leverage the situatedness of learning, c) improve the quality of collaboration in learning, d) facilitate students in expressing their knowledge, e) scaffold students in employing key learning processes. Also, within the Kaleidoscope context, and more specifically the labs participating in this ERT proposal, a number of these encompassive learning environments that include one or more of the above mentioned design guidelines on different domains have been created. Examples are: CoolModes [START_REF] Hoppe | Collaborative Learning in Open Distributed Environments -Pedagogical Principles and Computational Methods[END_REF]), SimQuest (van Joolingen & de Jong, 2003), Co-Lab (van Joolingen, de Jong, Lazonder, Savelsbergh, & Manlove, 2005), FLE [START_REF] Leinonen | Fle3: Future Learning Environment[END_REF][START_REF] Mørch | Integrating agents with an open source learning environment[END_REF], JEMUE [START_REF] Baggetun | Pedagogical agents for collaborative telelearning scenarios[END_REF], ARI-LAB [START_REF] Bottino | Technological Advances and Learning Environments[END_REF] and Tangibles [START_REF] Cerulli | Exploring randomness via studying, building and sharing models[END_REF].

The CIEL project aims to achieve a level of integration between these diverse learning environments for inquiry and experiential learning. The two main reasons for this are:

• Combining specializations. Each tool has its merits of supporting specific parts of the collaborative inquiry learning process, including support for collecting data, modeling and specific means of collaboration. Instead of trying to create a single tool that can do it all CIEL strives to enable interoperability between tools. • Support of longer term learning scenarios. Most learning environments support a specific type of activity, such as doing research on the greenhouse effect, or create a model of volcanic activity. This means that the typical duration of use will be limited. CIEL strives to design scenarios that extend over longer time periods, involving multiple activities and tools.

In a typical inquiry learning scenario that takes more than a few lesson periods, learners will perform diverse activities. Each of these activities will represent steps in the process of scientific knowledge building, and the whole scenario may therefore be supported by a variety of tools. CIEL aims to create an architecture in which many tools can interoperate to construct many different inquiry learning scenarios.

Figure 1 presents a possible example of such a scenario for inquiry learning, in which the learners collect data in a lab using a mobile device, and use the data to construct a model of the domain they are investigating using a modeling tool. In the mean time they can debate and construct an argument, using a dedicated argumentation builder such as Belvedere [START_REF] Suthers | Belvedere: Engaging students in critical discussion of science and public policy issues[END_REF]. The whole learning process is integrated around a common repository through which the different tools exchange information (e.g., data or models). On the one hand such a scenario should be semantically integrated [START_REF] Koedinger | Component-based construction of a science learning space[END_REF], meaning that the learning process as a whole, including the transitions from tool to tool take place in a smooth and meaningful way. On the other hand integration should also take place at a technical level, ensuring smooth data exchange between tools.

Repository Data Sets

Experiments Hypothesis

Chem lab Modeling tool

Hypothesis Argument

Argument builder

Figure 1 A multi-tool scenario to be supported by CIEL.

On the semantic level, it is important to support the flow of information and activity in the task context. This can be facilitated and implemented in different ways. One possibility is to map data onto specific objects that can be physically transferred (e.g., data can be captured on a PDA and thus be transferred). Another possibility is to transfer information between places and different environments by real time synchronization. In [START_REF] Milrad | Exploring the use of mobile devices to facilitate educational interoperability around around digitally enhanced experiments[END_REF], several of such interoperability patterns are discussed from both an engineering and an educational design view.

Here the environments can possibly be heterogeneous. This second option requires a tight coupling in terms of processing and time constraints. A loose coupling which implies much less restrictions is often facilitated through a common repository, although a real time synchronization is not supported in a loose coupling model. It needs a common data format which can be 'understood' and generated by the different tools. In CIEL, we have first adopted the loosely coupled model for our technical approach.

CIEL aims to support both levels of integration by bringing together several systems that support collaborative inquiry learning and make them interoperate at the two indicated levels. At the semantic level CIEL works on integration of concepts such as learning processes, learning objects and learning tools. For each of these shared definitions are created that provide meaning to data that are stored. Examples of these definitions include "data set", "experimental setup", "phenomenon", "experiment", and "model". Such definitions allow consistent support of learning processes across learning environment. If two systems treat the same object, for instance an "experiment" in a semantically consistent way, continuity of the learning scenario can be reached. The shared definitions in CIEL are created within a CIEL ontology.

The CIEL ontology also provides a context to the technical level of integration. In order to exchange data between tools, XML definitions are given of the terms in the ontology. This leads to a common data format shared by a set of tools that each can play a part in scenarios for collaborative inquiry and experiential learning. This approach is depicted in Figure 2.

Processes Concepts Tools Objects

Experimentation

The demonstrated system

The system demo consists of a first version of an integrated scenario along the lines described above. For the topic of "sampling" in the domain of statistics, a scenario has been designed and implemented that involves multiple activities and tools. The learner is placed in the role of an employee of a forest agency that is involved in the production of wood. For a certain forest area it must be determined whether the trees are tall and wide enough to harvest the wood. In doing so a sample of trees has to be taken. The task for the learner is to determine the best sampling strategy, so that a reliable decision can be made based upon the sample.

Save these results [ 40 ;50 [ 1 [ 50 ;60 [ 2 [ 60 ;70 [ 0 [ 80 ;90 [ 1 [ 70 ;80 [ 0 [ 30 ;40 [ # Class 2 [ 40 ;50 [ 1 [ 50 ;60 [ 2 [ 60 ;70 [ 0 [ 80 ;90 [ 1 [ 70 ;80 [ 0 [ 30 ; [ 40 ;50 [ 10 [ 50 ;60 [ 187 [ 60 ;70 [ 55 [ 80 ;90 [ 234 [ 70 ;80 [ 0 [ 30 ;40 [ # Class 2 [ 40 ;50 [ 10 [ 50 ;60 [ 187 [ 60 ;70 [ 55 [ 80 ;90 [ 234 [ 70 ;80 [ 0 [ 30 ; The scenario is supported by two simulations, one in Flash and one made in Co-Lab. Both allow creating samples and computing parameters such as the average and standard deviations of each of the samples. The data from the samples are stored as learning objects in an XML format defined by the project and can be imported in Cool Modes and Freestyler, which are modeling tools that allow for modeling a variety of domains, including statistics. The sampling strategy can be modeled in FreeStyler.

2

Figure 3 provides an impression of the demo as well as the underlying architecture. Underlying the scenario are the definitions of datasets and models, as well as a working CIEL broker that allows for storage and retrieval of the objects produced by the learners. Tools can connect to the broker and store the objects, accompanied by automated generation of metadata, that helps with their retrieval.

The demonstrated scenario illustrates a step on the road that is pursued by CIEL towards a wider integration of tools, supporting different modes of collaboration and experimentation.

Figure 2

 2 Figure 2 Structure of the CIEL work, depicted as two layers: theoretical and technical. The contents of the four cells are examples of a wider range of possible content.

Figure 3

 3 Figure3Overview of the demo. The two data generating tools at the top (Co-Lab and an independent flash application share their data with the broker. The data is used in FreeStyler modeling application depicted at the bottom.

Persons who contributed to this demo are: Jakob Sikken, Andre Berloth (University of Twente), Lars Bollen, Björn Hassing, Ulrich Hoppe (University of Duisburg-Essen), Muriel Ney, Cedric d'Ham (MeTAH), Aleksander Krzywinski (University of Bergen), Augusto Chioccariello (CNR) and Jeremy Toussaint (University of Oslo).

Demo requirements

The demo will be run from two brought-in laptops. Participants will be able to work with the applications. We will need a working internet connection from them (wired or wireless) as well as a beamer or large monitor to display the demo.


Luis Radford

Serge Demers

José Guzmán

Michele Cerulli

CALCULATORS, GRAPHS, GESTURES AND THE PRODUCTION OF MEANING

In this paper we report an analysis of a teaching sequence in which Grade 11 students were asked to produce some graphs corresponding to the relationship between time and distance of a cylinder moving up and down an inclined plane. The students were also asked to carry out the experience using a TI 83+ graphic calculator equipped with a sensor, and to discuss and explain the differences between their own graphs and the ones obtained with the calculator. We analyze the students' processes of meaning production in terms of the way diverse semiotic resources such as gestures, graphs, words and artifacts become interwoven during the mathematical activity. Our findings suggest that a complex relationship between gestures and words allow the students to make sense of the time-space graphic expressions.

INTRODUCTION AND THEORETICAL FRAMEWORK

In an artifact-mediated classroom activity, Grade 11 students were asked to investigate and graphically express the relationship between time and distance of a cylinder moving up and down an inclined plane. Strictly speaking, the temporalspatial relationship of the cylinder's motion cannot be seen, even if an experiment is materially carried out. Indeed, in such a case, crude perception merely allows one to see the cylinder going up and down the inclined plane. Although motions of this kind were only systematically mathematized in the early 17 th century, and since then there may be a certain intuition that "traveled distance" and "consumed time" -to use Galileo's words-bear a certain mathematical relationship, the graphic expression of such a relationship (which was not conceived until many years after Galileo's work) is certainly much less intuitive. In fact, the temporal-spatial mathematical relationship of a body's motion is an abstract, conceptual and cultural entity. To render this relationship apparent in the classroom requires a fine understanding of space, time and movement. In particular, the graphical account of motion may require students having recourse to diverse semiotic resources, such as gestures, words, drawings, coordinate systems, artifacts, etc.

Recent research has shown the cognitive import of gestures, words, and artifacts in the production of graphical as well as algebraic symbolic expressions [START_REF] Arzarello | From Body Motion to Algebra through Graphing[END_REF][START_REF] Roth | Scientific investigations, metaphorical gestures, and the emergence of abstract scientific concepts[END_REF][START_REF] Robutti | Approaching graphs with motion experiences[END_REF][START_REF] Radford | The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge[END_REF][START_REF] Radford | Gestures, speech and the sprouting of signs[END_REF]. The reported research, as well as other research carried out in other scientific fields like linguistics and psychology, indicates that, in the students' talking and gesturing activity, words and gestures play a substantial role, even if the specific role of words and gestures may vary according to the adopted theoretical perspective. For instance, in the early 1980s Kendon contended that gestures express underlying cognitive representations as words supposedly do (Kendon 1981, p. 38). Following this line of thought, McNeill suggested that gestures and speech share the same psychogenetic source [START_REF] Mcneill | So You Think Gestures Are Nonverbal?[END_REF], see also [START_REF] Crowder | Gestures at Work in Sense-Making Science Talk[END_REF]. In a more social, interpersonal perspective, gestures and words can be seen as semiotic means that students use to objectify knowledge (i.e. to make things and relations apparent in their universe of discourse). It is within the latter perspective that the analysis of the students' activity will be conducted in this paper. Considering gestures as (a loose type of) signs, our intention is to investigate how gestures, words, and artifactual actions are mobilized by the students in order to objectify and endow with meaning the emerging mathematical content (i.e. the referent) of the sign-graph expressing the conceptual mathematical spatial-temporal relationship of the cylinder motion. More specifically, our goal is to investigate what we want to call "semiotic nodes", that is, pieces of the students' semiotic activity where action, gesture, and word work together to achieve knowledge objectification.

METHODOLOGY Data Collection:

The data presented here come from an ongoing longitudinal classroom-based research program where teaching sequences are elaborated with the teachers. As the research program unfolds, theory, data, and hypothesis are cyclically generated. Usually, in these sequences the students work together in small groups of 3; then the teacher conducts a general discussion allowing the students to expose, confront and discuss their different solutions. In addition to collecting written material, tests and activity sheets, we have three video-cameras filming three groups of students. Subsequently, transcriptions of the video-tapes are produced. These transcriptions allow us to identify salient short passages that are then analyzed in terms of interaction and students' use of semiotic resources. In this paper, we will focus on one of the small groups.

The Teaching Sequence: A two-day mathematical activity based on a hands-on investigation of motion along an inclined plane included different tasks and questions. The instructional design rested on the premise that the mathematical investigation of spatial and temporal relationships in motion problems supposes the cognitive capability of conceptualizing motion from different mathematical reference systems. Bearing this in mind, we will discuss only 3 questions here.

In Question 1, the teacher propelled a cylinder (called cylinder A) upwards, from the bottom of the inclined plane. The students saw the cylinder go up and come down. The students could repeat the experiment as many times as they wished. The teacher then provided the students with an activity sheet and asked them to produce a graph (called graph A) representing the relationship between time and space of cylinder A's motion. The students were given no information concerning the initial point (or point zero) from where the distance should be (qualitatively) measured. We expected the students to locate the point zero on the bottom part of the plane, that is, the point where the cylinder was put in motion (a point that coincides with the body's position). The teacher also asked the students to sketch a second graph (Graph B) in the same coordinate system where graph A was drawn for a hypothetical cylinder (cylinder B) put in motion on an identical inclined plane one second after cylinder A started moving (in other words, cylinder A starts at 0 = A t and cylinder B starts at

1 = B t
). (Cylinder B's motion was hence a "thought experiment").

In Question 2, the students were asked to perform two experiments (motion starting at t=0 and motion starting at t=1 sec) using a TI 83+ calculator and the Calculator Based Ranger (CBR -the motion detector). In the calculator-based experiments, they were instructed to place the CBR at the top of the inclined plane. The students, who had previous basic experience with the graph calculator and motion sensor detector, had to compare their graphs A and B to the ones they obtained with the calculator.

Finally, in Question 3, the students were asked to study the graph shown in the right corner of Figure 1. The graph was accompanied by the following instruction: "A group of students drew the following curve to represent the relationship between time and space when a cylinder is propelled upwards on an inclined plane. This group placed the distance origin around the center of the inclined plane. Is this curve correct? Explain in detail your answer."

RESULTS AND DISCUSSION

Question 1: As expected, the students produced graphs starting at distance D=0 (see Figure 2). Key words with which the students gave meaning to the graphs here were the "initial point" (which was equated to point zero of traveled space), "going up", "maximum point", "going back".

1. Judith: The initial point is at zero and goes up to the maximum point, then, then … 2. Vanessa (interrupting): [it] continues to fall to point zero 3. Judith: (adding) while time runs out.

Question 2: As mentioned previously, in this question the students were asked to put the CBR at the top of the inclined plane. Let us focus here on the discussion concerning the students' comparison between their delayed motion graph (Graph B in Fig. 2) and the calculator's graph (called Graph C and shown in Figure 3).

The students noted several differences, among them the following: (1) Graph C was not perfectly curved in the part after its minimum value, (2) contrary to Graph B, in Graph C the value of the variable D (distance) in the ending points is not the same (i.e. In Lines 1 and 3 Judith makes an "iconic gesture", that is, a gesture that bears a resemblance with its referent. The iconic sign-gesture enacts the falling trajectory of the cylinder. It allows Judith to call her group mates' attention to a specific part of the phenomenon. The iconic gesture affords a segmentation of the phenomenon and operates a choice of what has to be taken into account. Thus, the iconic gesture does not stress speed, time, accurate distance and other elements. What it stresses is the fact that the cylinder went off the table. However, the students mobilized more semiotic resources than gestures. There is, in fact, a coordination of gesture, gaze, and words. Along with gestures, Judith uses locative words and time-related expressions to achieve a coordination of time, space, and movement. This is an example of semiotic node (see Figure 4).

In Line 5, Judith has recourse to an "indexical gesture": pointing with her finger, she indicates two parts of the calculator graph on the screen. In this case, numbers (30 cm and 45 cm) come to play the role of the iconic gesture that has previously shown the cylinder falling off the table. The first number represents the students' estimated distance from the cylinder's maximum point to the bottom of the table.

However, the cylinder never went 15 cm off the table (i.e. 45-30), for it was caught in the air as it fell off. Numbers are not accurate, and the students do not worry -accuracy is not at stake. Difference (3): As the previous excerpt intimates, Judith was able to provide an interpretation for Difference (2), i.e. why Graph C starts and ends at different values of the variable D. Nevertheless, the students' understanding of the relationship between time and distance was still vague. The reference point for the distance remained ambiguous. What the students understood was that the cylinder traveled more distance (absolute distance) in its falling back trajectory than its moving up one. The students kept discussing without success why Graph C does not start at D=0. When the teacher came to see their work, he did not provide an answer. His presence, however, catalyzed the students' ideas, which at the end he reformulated using a metaphor -the "eyes metaphor": We consider the gesture-word systems of Line 5 and Line 7 as two supplementary examples of semiotic nodes. In each case, indeed, a new kind of awareness is made apparent. In Line 5, the semiotic node serves to make sense of the fact that f D is greater than 0 D in Graph C. Epistemologically speaking, this semiotic node has a sense-making constructive dimension. In Line 7, the semiotic node brings to a higher degree of awareness the importance of the position of the spatial origin. It provides the students with a way to better interpret graph motions and to understand what has experimentally happened. Let us now turn to Question 3.

Question 3:

The students remarked that, in the graph, some values of the distance axis "D" are negative. They argued that negative distances are impossible.

1. Judith: No because your distance can't become negative […] It moves away from you or it comes close to you but (inaudible). 2. Carla: Well on our graph it does both. 3. Judith: It is because it doesn't go beyond the point? (the word « point » is accompanied by a gesture of both hands indicating an imaginary point in front of the body) Let's say that this is zero, zero is here (she turns her body to the right and places her right hand at the bottom of the right part of her desk to indicate the zero point; there is a coordination of the gesture and the deictic word "here". She is imagining a distance axis having an origin at the bottom of her desk, where her hand is) and it doesn't go negative because it doesn't go beyond (she moves her left hand from the top to the bottom of her desk and her left hand goes beyond her right hand that is still signifying the origin. She is implying that, in this reference system, points to the right of this zero point -i.e. points falling beyond the deskare negative). 4. Vanessa: I don't know if it is because of that, but what you say makes sense. 5. Carla: (Carla is not convinced. She interprets the bottom edge of Judith's desk as the horizontal axis of time in their graphs. She says:) Yeah, but those are the seconds (after a relatively long pause of approximately 2.5 seconds she waves her hand and draws in the air a concave graphic similar to Graph C while saying:) On the graph it goes like this … that's the seconds (she gestures a horizontal line) … it goes up and comes back down (she makes again a concave graphic similar to Graph C), the distance ... (she makes a vague gesture in the air that tries to locate a position for the distance; she falls silent for a relatively long pause of approximately 3 seconds while she and the other girls think)

6. Judith: Like your distance starts at zero (zero is again emphasized using a gesture that indicates a point on the desk close to her. Of course, this assertion is true if the position of the CBR coincides with the body's position).

[…] 7. Carla: Like the first [Graph A, i.e. the case where the distance is measured from the bottom of the inclined plane]… the closer it goes to the CBR it will be negative because we started here (indexical gesture pointing to the actual bottom of the inclined plane) … [this] started at the top (the word 'top' refers now to the initial point of the concave graph C that she reproduces here with a right hand gesture), the lower we were on the x axis … or whatever … the lower we were, the closer we were to the CBR. If we had gone beyond 0.5 m (i.e. the approximate maximum distance that the cylinder could travel from its maximal position on the inclined plane to the bottom of the table) it would be negative.

8. Judith: Exactly, it doesn't go beyond the point.

This excerpt stresses the students' difficulties in conceptualizing the difference between the spatial origin of the cylinder motion and the mathematical spatial origin. While the first one was perceptually seen, the second, in contrast, requires the students taking into account a theoretical perspective. As Line 1 makes plain, body position provides a powerful perspective ("it moves away from you or it comes close to you"). But this perspective has to be shifted in order to make apparent (or objectify) the phenomenon from other perspectives. Despite the success of the "eyes metaphor" in the previous question, the students could not elaborate a conceptual idea for the point zero distance. In Line 3, Judith mentions the word "point" and accompanies it with an indexical gesture. The concrete point on the bottom of the desk becomes the origin. "Zero is here", she says, and keeps her right hand there. Her left arm (initially extended) starts traveling -like the cylinder-from a far position towards the bottom of the desk. And while she is saying that "it doesn't go negative because it doesn't go beyond" her left hand does go beyond the supposed point zero. Here the complex system of iconic and indexical gestures contradicts what is uttered. In a sense, Judith is providing us with the enactment of a gesturaland-word-proof by contradiction. And Vanessa finds it meaningful (Line 4). In Line 5, Carla, talking to herself as much as to the other girls, makes an iconic gesture. This time the content of the iconic gesture is not the motion of the traveling cylinder but the calculator-produced graph. Carla's iconic gesture hence has a different referent from Judith's in Line 3. However, in referring to Graph C, the CBR (i.e. the distance origin) should be located at the top of the inclined plane. In the following line (Line 6), Judith says that the distance starts at the bottom. We see then the students talking about two different origins. The misunderstanding is not clarified. On the contrary, in Line 9, Carla refers to Graph A, switching thereby the origin albeit seemingly without being aware of it. This confusion allows her to interpret Judith's argument and, in the end, consensus is wrongly reached. Line 3 exhibits another example of semiotic node. Line 5 does not. In the latter, the gesture-word system has a heuristic role but it does not produce any novelty in terms of knowledge objectification or meaning production.

CONCLUDING REMARKS

The analysis of the students' semiotic activity carried out in this paper sheds further light on the students' conceptual strategies in understanding motion problems. In our analysis, we paid particular attention to the word and gesture system. Our theoretical construct of semiotic node allowed us to locate specific points in the students' semiotic activity where gestures and words achieve a coordination of time, space, and movement leading to the social objectification of abstract mathematical spatial-temporal relationships. The fact that the detected semiotic nodes were strongly oriented to the objectification of the mathematical space origin and the actual motion of the cylinder may explain, to some extent, the students' failure in securing a good mathematical understanding of the problem at hand. Indeed, in these semiotic nodes, time was rarely mathematized. In the students' discussions, time appeared mostly as marking the starting and ending points of the cylinder motion or else it was considered in a very rough qualitative way (as in Line 3 of the students' dialogue related to Question 1 or as in the first example of semiotic node; see Fig. 4). It is true that, in Figure 2, the beginning of Graph B correctly shows the characteristic type of delayed motions, but Graph B ends at the same time as the non-delayed motion Graph A! It may be true, as [START_REF] Koyré | Études d'histoire de la pensée scientifique[END_REF] remarks, that it is more difficult to think in terms of time than in terms of space. A suggestion for teaching would be to encourage students to pay due attention to the time variable and to incorporate it in a more sustained way in the analysis of spatial-time relationships.

Figure 2 .

 2 Figure 2. Graphs A and B

  0 D D f > ), and (3) Graph B starts at D=0 and its shape is different from Graph C. Difference (1): This difference was explained by a slight turn of the cylinder when it was rolling upwards on the inclined plane. Difference (2): This difference was more difficult to understand. After discussing different ideas Judith said: 1. Judith: … (looking at the inclined plane)

Figure 3 .

 3 Figure 3. Calculator's Graph C

  1. Carla: (talking to the teacher and pointing to the initial point of Graph C) We don't understand why it didn't start at zero […] 2. Vanessa: It's because it started the other way around, right? […] 3. Teacher: Ah! I don't understand […] we have always rolled the cylinder from the bottom to the top… (he makes a gesture as if he is rolling up the cylinder) 4. Carla: (talking at the same time as the teacher) Is it because you're further from the thing [i.e. the CBR]?

Figure 4 .

 4 Figure 4 . Example of a "semiotic node" where word and gesture achieve a coordination of time, space, and movement.

  You started on the table [i.e. the table that served as the inclined plane for the experiment], right? (Vanessa : Yes) And when it was rolling it fell off the table (with a similar gesture her arm is bent again and goes beyond her desk, as the falling cylinder did during the final part of its motion when it fell off the inclined plane and was caught by the student)… I don't know… 4. Vanessa: It has nothing to do with that. 5. Judith: It does have something to do with that […] That's the curve, right? Here (she points to the horizontal segment of the left part of Graph C on the calculator screen) suppose this is when you started on the table and when you finished (she points now to the horizontal segment of the right part of Graph C), you've finished further, that's further. […] Let's say that your distance here would be 30, and 45, that's the error! […] Now why it started there (initial point of Graph C) … I don't have any clue…

This thing there [the cylinder], does it go further? (the other two girls turn to see the inclined plane which was behind the students' desks) … like this … (she makes a gesture with her right arm; the gesture starts with her arm extended in front of her body and moves back, miming the cylinder motion in its coming back down trajectory) does it measure the …? Oh! 2. Vanessa: What? 3. Judith:

  You started on the table … and when it was rolling it fell off the table Carla while the teacher is still talking to Vanessa) That's true … Ah! Yeah! I get it! It is like we watched the cylinder leave and arrive like this (she puts her hands on the bottom of the desk) when it was at the bottom of the table … but now (she makes a complex gesture: with her left hand placed far from her she signifies the position of the CBR and with her right arm extended and then bending it she mimes the movement of the cylinder coming back to the bottom of the table) … it's the thing [i.e. the CBR] that is at the top! 6. Vanessa: (Understanding the other girls) Ah! Well we weren't looking from the point of view of the thing, it's because of that! O.K. 7. Teacher: O.K. Well there, the point of view … your eyes (he points to his eyes) … it's the CBR. For one of the graphs [Graph B] your eyes were at the bottom [of the inclined plane] (he puts his right hand in front of him and close to his body to signify closeness) and for one of the graphs [Graph C] your eyes were on … (he makes a gesture putting his hand in front of him and far from his body to signify the top of the inclined plane) […] 8. Judith: (Understanding) O.K. It's the same thing as that but from a different point of view.

	5. Judith: (Understanding
			iconic gesture
	pronoun	locative word	pronoun	locative word
		time-related expression

Acknowledgment

This paper is a result of a research program funded by the Social Sciences and Humanities Research Council of Canada (SSHRC/CRSH).