A Amor

A Fernandes
email: amancio.fernandes@sorbonne-universite.fr

J Pouget

Snap-through of elastic bistable beam under contactless magnetic actuation

Keywords: Bistable mechanism, buckling, snap-through, Laplace force, bifurcation

We propose the modeling of contactless switching of a bistable slender beam using Laplace force actuation. The model beam is based on the elastica approach which allows large amplitudes of the cross-section rotation and large elastic transformations. Furthermore, the extensibility of the elastic beam is accounted for and it plays a crucial role in the switching process. The study is devoted particularly to the actuation mechanism of the bistable beam uniformly loaded by a density of lineic force which is permanently perpendicular to the beam deformation. Such actuation can be produced by the Laplace force due to an electric current travelling along the beam placed in a magnetic induction. The mechanism of the bistable switching is analyzed in detail using a variational formulation and stable and unstable equilibria are described. We investigate numerically the post-buckling behavior to capture the exact buckling modes that follow the path in the unstable region. The numerical simulations are also used to obtain the bistable response in terms of actuating force (or electric current amplitude), beam end-shortening and mid-point displacement of the beam. We also perform a finite element computation and provide a validation of the results obtained by solving the set of equations of the boundary value problem.

The second part of the work is focussed on the experimental validation of the switching mechanism of the bistable beam presented in the analytical part. We design an experimental set-up for the fine measurement of the mid-point displacement of the bistable beam as a function of the electric current travelling the beam. The region of bistable instability is revealed by experimental adjustment of the bifurcation point associated with the actuating force. All the results extracted from experimental tests are compared to those coming from the modeling investigations, which ascertains with good accuracy the approach of the proposed model for bistable beam.

Introduction

Consider an elastic slender beam. One end of the beam is fixed, the other one undergoes a small end-shortening caused by a compressive force applied in the direction of the beam axis. As long as the force is weak, less than a certain critical value, the beam is in compression as predicted by linear elasticity. As soon as the force exceeds a critical value, the beam is no longer straight, and undergoes a transverse deformation or deflection [START_REF] Bazant | Stability of structures[END_REF][START_REF] Bigoni | Nonlinear solid mechanics : bifurcation theory and material instability[END_REF][START_REF] Dym | Stability theory and its applications to structural mechanics[END_REF][START_REF] Thomson | A general theory of elastic stability[END_REF]. The amplitude of deflection increases with end-shortening as well as with compressive force. We are faced with a bifurcation phenomenon, resulting in the buckling of the beam (Timoshenko and Gere , 1963). The later can deflect either upwards or downwards.

The elastic structure that we are describing is a bistable system and both positions of the beam (upwards or downwards) are stable states. However, the most interesting aspect of this bistable system is to be able to switch from one stable state to another regardless of how the transition is completed. Although a great emphasis has been paid to beam buckling and several studies are available in the literature, the mechanism of switching between stable positions remains an active topic of research, especially because a linear model is inadequate and recourse to a nonlinear model becomes inescapable. Furthermore, some questions are raised such as how does the bistable system switch between stable positions and what are the buckling modes which are involved in the process? Actuating a bistable beam to force it to switch can be accomplished using various means. A first idea is to load laterally the beam with a punctual force at any point of the beam, or with a punctual moment applied to the beam in the direction perpendicular to plane deformation [START_REF] Cleary | Modeling and experimental validation of actuating a bistable buckled beam via moment input[END_REF]. However, the bistable beam response under an applied punctual force depends on the way of applying the force, more precisely, on whether the force is vertically fixed onto the midpoint of the elastic beam or if the force stays on a central line between beam ends while the bistable snaps. Loading a bistable beam with a vertical force which is always fixed onto the beam has been reported in some studies, among them Pippard [START_REF] Pippard | The elastic arch and its modes of instability[END_REF] and Patricio [START_REF] Patricio | An elastica problem : instability of an elastic arch[END_REF]. Nevertheless, this situation is not convenient for experimental considerations. An extensive study of the actuation of a bistable beam by punctual force along a fixed central line in space (not necessarily on the midline) has been presented in [START_REF] Camescasse | Bistable buckled beam: Elastica modeling and analysis of static actuation[END_REF] and experimental validation of the model has been discussed in [START_REF] Camescasse | Bistable buckled beam and force actuation : Experimental validations[END_REF]. A comparative study has been proposed by [START_REF] Hung | Snapping of an elastica under various loading mechanism[END_REF] according to the way of applying the force onto the beam force while the latter is switched. Then Cheng and Hung examine three cases : (i) the force is attached onto the beam mid-point, (ii) the force is applied along a fixed central line in space and (iii) the force is applied by means of a device such that the force is always perpendicular to the deformed beam and its point of application is allowed to slide on the beam, while staying along the central line in space.

Switching or snapping-through of bistable systems can be obtained by means of active materials such as piezoelectric ceramics, shape memory alloys, electroactive polymers or magnetostrictive materials to cite the most popular. In this situation, the active elements bonded onto the elastic beam are able to induce local strains or stresses onto the buckled beam to control the switching process [START_REF] Maurini | Distributed piezoelectric actuation of a bistable buckled beam[END_REF].

In each the above-mentioned situations, the actuation is in contact with the bistable system and the influence of the actuating mechanism must be accounted for. The present study proposes an innovative means of actuating a bistable system by a uniform density of lineic force, which is able to trigger the snap-through of the buckled beam as soon as the density of force is great enough to make the bistable pass through the zone of instability to reach the other stable position. Here, the loading mechanism is such that the lineic density of force is permanently fixed onto the beam.

Moreover, when the beam is deformed while switching, the lineic force stays perpendicular to the beam deformation. Such a loading mechanism can be properly produced by the Laplace force [START_REF] Griffiths | Introduction to electrodynamics[END_REF][START_REF] Jackson | Classical electrodynamics[END_REF]. More precisely, this is the force applied to an electrically conductive wire traversed by a volume density of electric current placed in a magnetic induction field.

The elementary force is then perpendicular to the plane formed by the magnetic induction vector and vector of the electric current density or, in other words, the tangent vector to the deformed beam.

One of the main purposes of the present study is to investigate the bistable buckled beam on the basis of the elastica model, especially because a rather small disturbance of the actuating work can produce displacement and cross-section rotation of the beam of relatively large amplitudes.

Bistable buckled beam modeling is subject of numerous research studies which can be divided in different approaches according to the starting hypothesis. Elastica model has been extended to elastic arches loaded at its center by Pippard [START_REF] Pippard | The elastic arch and its modes of instability[END_REF]. Vangbo [START_REF] Vangbo | An analytical analysis of a compressed bistable buckled beam[END_REF] investigated the switching response of a bistable beam including the compressive effect by using an expansion of the solution to the beam equation on the buckling modes. In the framework of modal formulation, the detailed behavior of the stable and unstable branches of the bistability response under loading has been proposed by (Cazottes et al. , 2010). These authors have examined on the basis of a similar modeling, that is, buckling modal formulation, the actuation of a bistable buckled beam subject to a moment [START_REF] Cazottes | Design of actuation for bistable structures using smart materials[END_REF]. Beside the elastica beam theory, some weakly nonlinear approches are often considered. In particular, a theory accounting for extensibility of the beam leads to geometric nonlinearity combining bending and stretching deformations. This kind of approach has been developed by [START_REF] Nayfeh | Exact solution and stability of post-buckling configurations of beams[END_REF] for a buckled beam and by Pinto and Gonçalves [START_REF] Pinto | Active non-linear control of buckling and vibrations of a flexible buckled beam[END_REF] for a simply supported shallow elastic arch. The weakly nonlinear approach has been considered also for both static and dynamic snap-through of bistable structures by [START_REF] Chandra | A numerical investigation of snap-through in a shallow arch-like model[END_REF]. The present work is devoted to the bistable beam but we also mention some extensions of bistability effect and even tristability to elastic thin plates or shells. Such studies have been conducted by Schultz et al. [START_REF] Schultz | Snap-through of ansymmetric laminates using piezocomposite actuators[END_REF] for bistability control of composite plates using piezoelectric actuators, or by [START_REF] Vidoli | Tristability of thin orthotropic shells with uniform initial curvature[END_REF] for tristability of thin orthotropic shells, or by [START_REF] Fernandes | Multiparameter actuations for shape control of biatable composite plates[END_REF] for the study of shape control of bistable composite plates.

One of the most attractive development of bistable systems (or multistable systems by extension)

is their ability to be miniaturized from millimeter scale to dimensions of the order of the micron.

Such bistable systems become particularly interesting to design microelectromechanical devices (MEMS). At the microscopic scale, the actuation making use of Lorentz force is quite attractive.

According to the Lorentz theory of electrons, the force on an electric punctual charge consists of two parts. The first part is a force produced by an electric field on an electric charge (electrostatic force). The second part is the force applied to a moving electric charge placed in a magnetic induction. When the material deals with a continuum of density (per unit of volume or mass) of electric charge, the Lorentz force becomes a kind of effective Lorentz force. Subsequent use of Lorentz force leads to the notion of a so-called ponderomotive force or electromagnetic force. More precisely, the latter force is deduced by applying the classical Lorentzian (space) averaging procedure to an assembly of electric punctual charge contained within a microelement of volume ∆V [START_REF] Maugin | Continuum mechanics of electromagnetic solids[END_REF]. The macroscopic quantities thus obtained by this spatial averaging procedure are considered to be sufficiently smooth functions of space and time. Now, going back to our problem, if the charges are travelling in a conductive wire placed in a magnetic induction, the force acting on the wire is called the Laplace force [START_REF] Griffiths | Introduction to electrodynamics[END_REF][START_REF] Jackson | Classical electrodynamics[END_REF]. Obviously, we then have a contactless actuation of the bistable system. Moreover, such electromagnetic actuations are easily controlled using electronic circuits.

Electromechanical system technology presents important advantages including, small size, light weight, limited energy consumption and integrated electronic components. Electrostatic force has been exploited to excite sensors at their resonant frequencies, based on bistable concept (Herrera-of electric potential between a semi-infinite rigid electrode and the initially curved beam has been analyzed for applications to MEMS [START_REF] Das | Pull-in and snap-through instabilities in transient deformations of microelectromechanical systems[END_REF][START_REF] Pane | Investigation on bistability and fabrication of bistable prestressed curved beam[END_REF][START_REF] Williams | Modeling of initially curved beam structures for design of multistable MEMS[END_REF]. [START_REF] Park | Pre-shaped buckled-beam actuators : theory and experiments[END_REF] present a study of the actuation of pre-shaped beam controlled by electromagnetic force and experimental validation of the snap-through mechanism of the bistable micro-beam is also conducted. Some other interesting works concerning micro-systems based on bistable beam have been conducted mainly by [START_REF] Buchaillot | Post-buckling dynamic behavior of self-assembled 3D microstructures[END_REF], [START_REF] Krylov | Dynamic stability of electrostatically actuated initially curved shallow microbeams[END_REF] or [START_REF] Krylov | Bistability of curved microbeams actuated by fringing electrostatic fields[END_REF].

Actuating a bistable beam by means of active materials is an interesting development in order to control the stability of the bistable beam during the snap-through process by using electric field, gradient of temperature or magnetic induction. Such materials induce local strains leading to trigger switching the bistable beam as soon as the strain is large enough to enter the unstable zone.

Electrostatically actuated bistable systems are well adapted for controlling the bistability and they are perfectly designed for MEMS [START_REF] Krylov | Bistability of curved microbeams actuated by fringing electrostatic fields[END_REF][START_REF] Medina | Bistability criterion for electrostatically actuated initially curved micro plate[END_REF]. Deformation of a bistable beam bonded with a piezoelectric element is induced by elongation or compression in the piezoelectric layer by means of the application of an appropriate electric field. Aimmanee and Tichakorn [START_REF] Aimmanee | Piezoelectrically induced snap-through buckling in a buckled beam bonded with a segmented actuator[END_REF] have proposed such an actuation of a bistable beam using smart materials. A study of distributed multiparameter actuation for the buckling and snap-through control of a pinned elastic beam was reported in [START_REF] Maurini | Distributed piezoelectric actuation of a bistable buckled beam[END_REF]. There, the authors propose an elastic beam sandwiched by two pairs of piezoelectric actuators driven by two differences of electric potential. An extension to the bistable shell made of magnetic rubber composite using the magnetic actuation of the shell cap has been conducted by [START_REF] Seffen | Eversion of bistable shells under magnetic actuation: a model of nonlinear shapes[END_REF].

The present work is based on an extensible elastica beam model. We describe the beam deformation by two strain measures which are the extension strain along the beam axis and the beam curvature.

The beam kinematic is presented in the next section (Section 2). Both descriptions in the fixed frame and local frame attached to the beam deformation are given in that section. We then derive the equations of the buckled beam from the virtual work principle. Section 3 is devoted to this task. Moreover, the boundary conditions -clamped-clamped beam -are specified in the same section. Section 4 deals with the numerical study of the set of nonlinear boundary value problems. The numerical problem thus stated relies on the model equations based on the shooting method associated with a numerical procedure to capture the unknown shooting parameters. In the same section we compare the solutions derived from the model equations to those obtained with the help of the finite element method. One of the main results coming from the numerical investigations is the bistable response, that is, the actuating driving force (or the electric current) as a function of the beam mid-point displacement. The results are discussed according to the chosen parameters (geometry of the beam, the end-shortening). A part of the work is devoted to the experimental validation of the proposed model (Section 5). We report the investigation of the experimental measurements in Section 5 and discuss the results thus obtained and compare them to those deduced from the model and numerical simulations.

2 Description of the bistable system

Kinematics

We consider an elastic beam clamped at each of its ends and having length L 0 at rest. The left end is fixed while the right end is subject to an end-shortening of small amount ∆L. The cross-section of the beam is homogeneous and rectangular with a width b and thickness h. A fixed Cartesian reference frame {A; e 1 , e 2 , e 3 } is attached to the structure in its initial configuration R 0 . The reference frame is such that the unit vector e 1 lies in the direction of the beam axis. Due to the end-shortening, the distance between clamps becomes L = L 0 -∆L. The elastic beam is initially straight and free of stress (see Fig. 1.a for a sketch). The beam undergoes a buckling caused by the imposed end-shortening at the right end which is the current configuration. A compressive force P is associated to the end-shortening and it can be viewed as the conjugate variable of the end-shortening ∆L. This force will be determined once the problem is completely solved. The beam in the reference configuration is described by the set

R 0 = {OG 0 = q 0 (s) = s e 1 , with s ∈ [0, L 0 ]} (1)
where s is the abscissa along the beam axis and q 0 (s) is the position vector of the material point G 0 . The deformed beam in the current configuration is a smooth curve (see Fig. given by We denote by τ the unit vector tangent to the curve describing the deformed beam at the point G(s). The tangent vector is defined by

OG = q(s) = x(s) e 1 + y(s) e 2 , s ∈ [0, L 0 ] . (2
τ = dq ds , ( 3
)
where s is the curvilinear coordinate measured along the deformed beam. In addition, we introduce Λ = ds ds , the ratio of the length of differential line element of the beam in the deformed configuration to that of the undeformed configuration. The parameter Λ measures the extensibility of the beam.

The normal unit vector n is perpendicular to the tangent vector τ . Accordingly, the set of orthogonal unit vectors {τ , n, e 3 } forms the local frame attached to the deformed beam at the curvilinear abscissa s. These vectors can be written in the reference frame {A; e 1 , e 2 , e 3 } using the following transformation τ (s) = cos θ(s) e 1 + sin θ(s) e 2 , (4a)

n(s) =-sin θ(s) e 1 + cos θ(s) e 2 , (4b)
where the angle of rotation θ is θ = (e 1 , τ ). The reader is referred to Fig. 2 for the sketch of different vectorial variables used in the forthcoming sections. The gradient of the beam transformation characterized by the vector q(s) can be easily obtained from Eqn.

T (s) n N (s) τ R (s) F (s) n M (s) e 1 e 2
(3) accounting for the extensibility Λ of the beam and it can be written down as

q = Λτ , (5)
where the prime is the derivative with respect to the curvilinear abscissa s.

The extensional strain ε and the beam curvature κ are chosen as the two strain measures. They are given by

ε = Λ -1 , (6)
κ = θ, s = dτ ds • n = 1 Λ θ, s . (7)
The vector position in the current configuration of the beam is written in the local frame {G; τ , n, e 3 } given by

q(s) = q T (s) τ + q N (s) n , (8)
which can be compared to that of given by Eqn. (2). Consequently, we have the choice to compute the beam transformation either in the reference frame {A; e 1 , e 2 , e 3 } or in the local frame {G; τ , n, e 3 } by sake of convenience. The formulation from one frame to the other can be obtained by using Eqn. ( 4). The geometric relations Eqn. ( 5) become

q T -θ q N = Λ , (9a)
q N + θ q T = 0 . (9b)
In the forthcoming sections, especially for the variational formulation, the use of variables defined in the local frame will be most useful.

Constitutive equations of the beam

The beam cross-section is rectangular and homogeneous and it undergoes internal resultants in force and moment which is defined as distributor of the internal actions given by the set {R(s), M(s)}.

The force and moment read

R(s) = R x (s) e 1 + R y (s) e 2 = N (s) τ + T (s) n , (10a)
M(s) = M (s) e 3 . ( 10b
)
The quantity N is the axial (perpendicular to the cross-section) force along the beam axis, T is the shear force and M is the bending moment with respect to the e 3 direction. The resultants in force and moment acting on the beam cross-section are functions of the strain measures defined in the above section. The elastic behavior of the beam is assumed to be linear with respect to the strain measures. Accordingly, the constitutive equations are stated as

N (s) = EA ε(s) , (11a)
M (s) = EIκ(s) , ( 11b
)
where E is the Young modulus, A the area of the cross-section (in the present case A = b h) and I is the moment of inertia of the cross-section with respect to the e 3 direction. In addition, EA and EI are the axial beam and bending stiffnesses, respectively.

3 Variational formulation for the buckling bistable beam

Principle of virtual work

To obtain the beam equations, we adopt the principle of virtual work. The principle for the present model of beam and configuration is stated as follows (in the static case)

δS = δW i + δW e = 0 . (12)
The different contributions to the principle are W i for the work of internal forces, W e for the work of the applied actions.

The virtual work of internal forces of an elastic beam takes on the general form (the details of the algebraic manipulation can be found in [START_REF] Camescasse | Bistable buckled beam: Elastica modeling and analysis of static actuation[END_REF])

δW i = - L0 0 {R(s) • δq (s) + M(s) • δp (s) -(q (s) × R(s)) δp(s)} ds (13)
where "×" denotes the vector product. We recall that the vector q(s) is the position vector of the material point G (s) (see Eqn. (2)) and p = θ e 3 is the vector rotation of the beam cross-section.

The bistable beam is subject to a lineic density of force applied into the beam staying in (e 1 , e 2 )plane. The virtual work of this density of force reads

δW e = L0 0 F(s) • δq(s)ds (14)
where δq(s) is the virtual displacement of the material point G(s) belonging to the deformed beam.

Dimensionless equations

For the sake of consistency and in order to highlight some key parameters, we introduce dimensionless variables and parameters, defined by

• lengths (S, Q, ∆ ) = (s, q, ∆L) /L 0 , (15a)
• Forces and moments (N, T, P, M) = (N/F 0 , T /F 0 , P/F 0 , M/M 0 ) ,

• Density of forces

F = FL 0 /F 0 , (15c)
• Energy

E = E tot /E 0 , (15d)
where the reference parameters have been defined as

F 0 = EAk, M 0 = EI/L 0 and E 0 = F 0 L 0 .
Moreover, the following key parameter has been introduced

k = I AL 2 0 (16)
which characterizes the ratio of the compression energy over the bending energy. The parameter k ∝ (h/L 0 ) 2 where L 0 /h is the slenderness ratio of the beam. Now, with the new dimensionless variables the position vector takes the form We now propose a new form for the variational equation Eqn. ( 12). However, the variational equation is written with the variables defined in the local frame. It reads as

Q(S) = Q X (S) e 1 + Q Y (S) e 2 = Q T (S) τ + Q N (S) n (
δS[Q, θ] = 1 0 {(N -θ T) (δQ T -Q N δθ) + (θ N + T -F) (δQ N + Q T δθ)}dS + 1 0 {M + (1 + ε) T}δθdS = 0 . (18)
It has been assumed here that F = F n, that is, every element of force density is perpendicular to the deformed bistable beam. Obviously, in the algebraic manipulations leading to Eqn. (18), we accounted for the boundary conditions at the end of the beam. In particular, for clamped boundary conditions we have δθ = 0 and δQ = 0, because the displacement and the rotation of the beam at its ends are given.

Equations for the bistable buckled beam

The variational equation Eqn. ( 18) must be satisfied for any arbitrary variation δQ and δθ which meets the boundary conditions at S = 0 and S = 1 (clamped-clamped beam). We deduce the equations of the bistable beam as

N -θ T = 0 , (19a) θ N + T -F = 0 , (19b)
M + (1 + ε) T = 0 . ( 19c
)
Along with the above equations the associated boundary conditions read as

Q (0) = 0, Q (1) = X B e 1 , θ (0) = 0, θ (1) = 0 . (20)
In the second boundary equation Eqn. ( 20), X B denotes the abscissa of the right end of the beam due to the end-shortening of the clamp B. With the view of completing the above differential equations, it is convenient to write the geometrical equations Eqns. ( 9) in non-dimensional notation

Q T -θ Q N = 1 + ε , (21a)
Q N + θ Q T = 0 . (21b)
A similar transformation holds for equations Eqns. ( 5). Their form using the non-dimensional notation introduced in Eqn. ( 15) can be obtained easily. Now, we have almost all the ingredients to solve the problem of a bistable buckled beam subject to a density of lineic force. It is worth noting that each elementary lineic force is always perpendicular to the deformed beam while the beam is snapping.

At first sight, we have a set of nonlinear differential equations with given boundary conditions.

This kind of problem is not trivial because first of all, there exist unstable equilibrium regions spanned while the beam switches from one stable state to the other one. In addition, the classical method cannot capture the configuration of the beam in the unstable region. The second difficulty is that at each step of the process method, the configuration of the lineic density force is modified because the density force is always normal to the deformed beam which is an unknown of the problem. Therefore, a more sophisticated algorithm to solve the problem becomes necessary.

4 Numerical analysis and results

Problem statement

This section is devoted to the numerical investigation of the nonlinear boundary value problem governing the snap-through of the bistable buckled beam. The equations to be solved were obtained in the previous Section. More precisely, they are the equations of the beam equilibrium Eqns. ( 19) and the geometric relationships Eqns. ( 21).

We recall that the axial strain and bending moment satisfy the constitutive equations Eqns (11) written in dimensionless form as

ε = k N, M = θ . ( 22
θ (0) = 0 , (23a)
Q T (0) = 0 , (23b)
Q N (0) = 0 . (23c)
• At the B end (S = 1) Figure 3 shows the sketch of the clamped-clamped buckled beam for a given density of lineic force while switching. The detail of the lineic density produced by Laplace force is also shown, especially the induction field and the vector of electric current density par unit of area. In classical electrodynamics, the Laplace force acting on a small part of elementary volume dv of the beam is given by [START_REF] Jackson | Classical electrodynamics[END_REF])

θ (1) = 0 , (24a)
Q T (1) = X B , (24b)
Q N (1) = 0 . ( 24c
dF = j × B dv ( 25
)
where j is the density of electric current per volume unit such that dv = Ad (A = b h being the beam cross-section area). On introducing the electric current per unit of length i τ = A j where τ is the unit vector tangent to the deformed beam, the Laplace force acting on a piece of the beam of length d is then

dF = i d τ × B . ( 26
)
The beam is placed between two magnets producing the magnetic induction B assumed to be uniform in the volume between the two magnets. For a given magnetic induction B, the only free parameter is the electric current. It will be the control parameter.

Algorithm for solving the switching bistable beam problem

To solve the set of nonlinear equations Eqns. ( 19) using an algorithm based on a Runge-Kutta method, we need the initial values for N, θ, M and T at S = 0. At first sight, the values N (0) = N A , M (0) = M A and T (0) = T A which are the clamping efforts are not known. Accordingly, a numerical shooting algorithm must be used (see [START_REF] Ascher | Numerical solution of boundary value problem for ordinary differential equations[END_REF] for the implementation method).

The input parameters of the method are given by boundary condition Eqns. ( 23) and we start with an initial guess for the set of parameters {N A , T A , M A = θ A }. The objective parameters are given by Eqns. ( 24). Moreover, we do not know the buckling force P = N A since only the end-shortening of the right end of the beam is given. The buckling force P is therefore obtained by solving the buckled beam problem with loading set to zero (F = 0).

The boundary value problem to be solved is not trivial since we are faced with unstable branches of solution while the beam is switching from one stable state to the other one. To avoid the divergence of the algorithm and discrepancy of the final results, we control the vertical displacement of the material point of the beam at the curvilinear abscissa S C such that

Q X (S C ) = (1 -∆ ) /2.
The idea is to increment the mid-point by a very small amount at each value Q Y (S C ) = Y C that corresponds to an unique deformed beam, while a same density of lineic actuating force can produce, a priori, different beam deformations.

A second numerical method is proposed to ascertain the first one, based on the finite element method. More precisely, we take the advantage of the FEniCS numerical libraries [START_REF] Logg | Automated solution of differential equations by the finite element method : The FEniCS book[END_REF].

The numerical code deals with the minimization of the total potential energy associated to the bistable beam and its actuating energy. The problem at hand is to find the equilibrium configurations, as with the first method. The difficulty is that for a given loading, different beam deformations can be obtained. We must have recourse to a more sophisticated algorithm using a so-called deflated continuation method by varying iteratively the loading parameter [START_REF] Farrel | The computation of disconnected bifurcation diagrams[END_REF] in order to capture the successive bifurcation points and the associated beam equilibria that can be either stable or unstable.

Results and comments

The forthcoming results have been obtained for a thin elastic beam made of amagnetic stainless steel A316L with a Young modulus of E = 207 GPa, length of 60 mm, width of 5 mm and thickness of 0.05 mm. The average magnetic induction between both magnets is about 0.3 Tesla. One of the first numerical results is the beam response under the density of lineic force produced by the Laplace force. Using the numerical method we compute the actuating force or, more conventently, the electric current, as function of the mid-point displacement of the beam. large (large end-shortening of the order of 3% or great buckling deflection). At first sight, the curves are quite similar in shape. The present results can be compared to those obtained with a punctual force actuation [START_REF] Camescasse | Bistable buckled beam: Elastica modeling and analysis of static actuation[END_REF]. The part of the force-displacement diagram which is almost a line corresponds to the unstable region. Note that all the curves pass through zero (y m = 0, i = 0), and are symmetric with respect to the origin. Moreover, all the curves display a negative slope between extrema, which means that the bistable system possesses a negative stiffness on this part of the branch, which corresponds to instability. An important comment from the present results is that both numerical methods give quite similar results, which confirms the robustness of the modeling of the bistable buckled beam based on elastica theory. The results are applied not only for weak buckling deflection but also for rather large deflection in comparison to the beam slenderness. The concept of Laplace force is not so easy to handle. A question which could be raised is what would be the bistable response if the Laplace force is replaced by vertically fixed force? This point is discussed in Appendix A.

The second result extracted from the numerical simulations is the compressive force P applied at the right end of the beam in order to maintain a given end-shortening as function of the displacement computed at the beam mid-point. The results for three different values of the end-shortening are reported on Fig. 5.

We observe that, for each curve, the compressive force for null loading corresponds to the buckling critical force depending on the chosen end-shortening. As soon as the electric current is switched on and it begins to increase, the compressive force increases rapidly as well until it reaches a plateau.

As in the previous results the comparison of both numerical approaches gives very close results, which confirms, once more, the reliability of the present modeling based on elastica beam theory.

Instructive results can be obtained by computing the total potential energy. We recall that the numerical using FE method is based on the minimization of the total potential energy of the system. The latter reads as

E = 1 0 1 2 Mκ + 1 2k Nε dS . ( 27
)
The first term on the right hand side in Eqn. ( 27) represents the density of bending energy and the second term is the energy density of compression. The potential energy is computed using the results from the numerical simulations of the model equations (see Eqns. ( 19)) for each configuration of the deformed beam while the beam is snapping. Figure 6 shows the variations of the total potential energy as function of the beam deflection at the beam mid-point from the upper stable position to lower one and vice versa. The FE results have been superposed to the results of the first method in Fig. 6. It worth noticing that the potential energy possesses two minima corresponding to the stable positions of the bistable beam. The relative error between both numerical results is less than 0.4%. One of the advantages of the elastica model and the path-following technique used to solve the nonlinear boundary value problem is that we are able to capture with enough accuracy the beam deformation when passing through the zone of instability of the bistable system. However, for rather small end-shortening, the beam cross-section rotation and displacements have small amplitudes. Therefore, the equations for the bistable beam can be linearized. More precisely, the set of equations Eqns. (19 and 21) are linearized with zero actuating loading (F = 0) around the critical buckling force P c for P > P c . Trivial algebraic manipulations lead to the following equation for the beam deflection

Q (4) Y + P (1 -kP) Q Y = 0 . ( 28
)
Here we recover the classical Euler equation for a buckling beam; however, in Eqn. ( 28) the buckling force P is altered by the factor (1 -kP) accounting for the compressibility of the beam in the prebuckling regime. We point out that the deflection of the beam Q Y has been chosen instead of the rotation θ because the deflection must satisfy four boundary conditions at the ends of the beam, which not the case for the rotation. Eqn. ( 28) possesses two kinds of solution (symmetric and asymmetric modes) satisfying the boundary conditions for clamped-clamped beam, namely

Q Y (0) = 0, Q Y (0) = 0, Q Y (1) = 0, Q Y (1) = 0.
Appendix B provides the buckling mode solutions to Eqn. ( 28) along with the associated boundary conditions. Going back to the solution deduced from the numerical simulations of the boundary value problem, we can write the solution as an expansion of the buckling modes

Q Y (S) = J j=1 A j Y j (S) (29)
where Y j (S) are the buckling modes (see Appendix B for their exact forms). The factors A j correspond to the contribution of the modes. The first derivatives (and also the second derivatives) form an orthogonal vectorial base. Therefore, by multiplying the derivative of the expansion Eqn. ( 29) with respect to S by the j's buckling mode, we arrive at

A j = Q Y (S) , Y j (S) (30)
where •, • is the scalar product in the functional space L 2 [0, 1].

The result of the computation is shown in Fig. 7 for the first five modes. We observe that the first mode is predominant in the vicinity of the stable positions, and that its influence decreases rapidly when the bistable is switching as the second buckling mode takes over from the first one.

When the bistable passes through a zero deflection at the mid-point of the beam, the second mode prevails, and the bistable is in the region of instability. Fig. 7 gives interesting information about the buckling mode contribution, especially the third mode, which is an even mode. It plays a significant role in the switching process since 30 per cent are involved in the snapping scenario.

The other modes, the fourth and fifth are not really significant, as their contributions are less then 4%. This result demonstrates that if we want to use a reduced model, the first three buckling modes should be enough to describe the beam switching in the framework of small end-shortenings.

Fig. 8 summarizes the beam deflection at the different step of the snapping process.

On using the path-following procedure with the control parameter defined by vertical coordinate of the crossing point of the beam with the vertical line fixed at the mid-point between both clamps, we are able to capture, step by step, the beam deformation. More precisely, when the bistable is switching, the crossing point moves along the vertically fixed line. To each crossing point corresponds a unique beam deformation. Accordingly, the switching scenario can be seen as follows : (a) when the actuating force is zero the bistable beam deformation corresponds to the first buckling mode; (b) when the electric current increases, the beam mid-point increases as well; however, for a strong enough lineic force, the beam starts snapping, accordingly the beam

-4 -3 -2 -1 0 1 2 3 4 y m [mm] 0 20 40 60 80 100 A i [%] A1 A2 A3 A4 A5
Figure 7: Contributions of the first five buckling modes in the snap-through process (in percent) for an end-shortening of 1% of the total length. profile is mainly a combination of the first three buckling modes with the mode contribution given in Fig. 7; (c) the bistable beam is passing through the domain of instability, the electric current must be decreased, the beam mid-point passes through zero, the second buckling mode is therefore predominant; (d) the beam mid-point continues to increase and we recover the profile of the beam at point (b), but symmetrically inverted and finally; the beam ends its running at the upper stable position (e) and we recover the first buckling mode, but upside down.

4 3 2 1 0 1 2 3 4 y m [mm] 3 2 1 0 1 2 3 i [A]

Experimental validations

This section reports an effort to bridge the modeling approach presented in the above sections with experimental observations. The emphasis is placed on the validation of the bistable response in electric current (lineic density of force) as function of the mid-point vertical displacement of the beam subject to electromagnetic Laplace force. Comparisons of the proposed experimental results to those derived from the bistable beam based on elastica theory will be discussed. A thin elastic beam having the same dimensions and material properties as used in the analytical and numerical studies was fabricated for the experiment (see Section 4.3).

Description of the experimental set up and process of measures

The thin elastic beam is manufactured from a sheet of amagnetic stainless steel A316L using electro-eroding process. Several test beams have been cut with an accuracy of less than 1/100 mm.

Particular attention is paid to obtaining a high quality planar beam with a minimum of residual stress. The beam is thus subject to metallurgical process consisting in heating at 500 0 C in an oven for few hours and maintained absolutely planar and then in cooling gradually to the ambiant temperature.

The thin beam is placed between two clamps which maintain the beam ends very tightly. Fig. 9 shows the photos of the complete experimental set-up. The test set-up is equipped with a micrometric translator at one of the beam ends. The translator allows us to control the displacement of the beam ends (the right end) while the other is fixed. Using travel translator stage we are able to monitor the end-shortening of the beam with 10 µm adjustment (see Fig. 9). Moreover, a digital micrometer with an accuracy of 1 µm is used in order to control the displacement of the platform of the moving clamp.

Experimental measurement and discussion of the results

Bifurcation diagram. One of the first experimental results concerns the bifurcation diagram, namely, the beam deflection at its center as function of the end-shortening when actuation is set to zero. On using the travel translator we impose the desired end-shortening at the right end of the beam. The end-shortening is read directly on the digital comparator. The beam deflection at its center is measured thanks to the laser profilometer and the value is transmitted to the computer. Accordingly, by varying the axial translation we obtain first, the value of the end-shortening and then, the corresponding deflection of the beam. Figure 11 shows the bifurcation diagram. The experimental data are depicted by small red points and they are superimposed to the result deduced from the numerical simulations (the curve in solid line). We notice a strong fit between the model and experimental data. Obviously, because of the high slenderness ratio of the beam L h = 12000, the pre-buckling zone is so tiny that it is not observable.

Discussion and concluding remarks

The main objective is to develop and analyze a model for bistable buckled beam based on an elastica approach of beam, including extensibility. On using the complete model that is, by considering rotations and displacements of large amplitudes, we examine in detail the switching process of the bistable beam. The beam actuation is produced by a lineic density of uniform force. This means that the force configuration follows the beam deformation variation while the bistable transits from one stable position to the other one. As a matter of fact such an actuating force can be produced by an electromagnetic Laplace force. The latter is due to the electric current travelling in the beam placed in a magnetic induction.

The most pertinent results describing the switching process of the bistable buckled beam are obtained first with the use of semi-analytical solution by solving the nonlinear boundary value problem. At last, experimental validation of the bistable beam actuated by an electromagnetic Laplace force is proposed and confirms with a high quality of accuracy the present model, which demonstrates that this apparently simple bistable system actuated by a contactless process is rich enough to provide particularly interesting comments :

• The bistable modeling in the elastica theory framework allows us to write down the governing beam equation for transformation of large amplitudes. The equations of the nonlinear boundary value problem are deduced from a variational formulation. Moreover, the equations of the bistable beam are written in the local referential frame attached to the actual configuration of the beam (deformed beam).

• Important results concern the response of the bistable beam to the Laplace force. The responses have been computed for several end-shortening values and they describe the evolution of the electric current (through the computed applied lineic force) as function of the vertical displacement of the beam mid-point. The response exhibits an N-shaped curve. The numerical algorithm is efficient enough, which allows us to capture the beam configurations at each step of the vertical increment, even for unstable branches. A second numerical investigation based on the FE method and path following procedure is also reported. The results of the second numerical method confirms perfectly the first numerical results, especially the robustness of the beam modeling based on elastica beam theory.

Among the results which can be extracted from both numerical investigations, we have examined (i) the bistable response, the electric current as function of the displacement of the beam mid-point, (ii) the buckling loading applied to one of the beam ends to maintain the given end-shortening, (iii) the total potential energy exhibiting minima at the each stable position of the buckled beam (upward and downward) and (iv) the analysis of buckling modes and their precise contribution to the snapping process.

• Finally, a section is devoted to experimental tests which have ascertained the validity of the model. The comparisons between the experimental tests and numerical results are quite good, with very small discrepancies. In short, a good correspondance is observed between the bistable response modeling prediction and experimental results, which establishes a reliable validation of the present model.

All the results presented in this study are good reasons to undertake complementary and forthcoming works, on the nonlinear dynamics of the bistable beam in the post-buckling regime when applying time-dependent electric current through the beam. Experimental investigations will have to be considered to confirm the model predictions. The model along with the obtained results opens new views to design innovative devices in the area of micro-robotics, medical engineering, etc.

Appendix A Comparison with bistable switching by means of a lineic density of vertically fixed force.

A question that we might ask ourselves is what would the bistable beam response be if the Laplace force was replaced by a lineic density of force with fixed vertical direction. Would there be a noticeable difference? To understand the problem better, we performed a numerical simulation of the bistable response subject to a lineic fixed force. The result is illustrated in Fig. 13. The curves are superposed for the same end-shortening of 3 % and for the same order of lineic density of force. A difference is clearly observed, as the maximum of lineic density of follower force actuation is little bit greater than that of the vertically fixed force. In addition, the relative difference between maxima have been computed and presented in Fig. 14. The relative difference is defined by where F V is for the vertical force and F L referred to the Laplace force. We notice that δF increases with the end-shortening. For instance, an end-shortening of the order of 5% gives rise to a relative difference between both kinds of actuation of about 5.4%.

Accordingly, we cannot reasonably use the lineic density of vertically fixed force as an approximation of the Laplace force.

Appendix B Buckling modes for a clamped-clamped beam

The buckling modes are obtained by solving Eqn. ( 28 In Eqns (B.1) and (B.3) A j is an amplitude that we have at hand to be adjusted in order that the mode is normalized.

  1.b). The point G 0 in the initial configuration is then transformed into the material point G of the deformed beam

)Figure 1 :

 1 Figure 1: Clamped-clamped elastic beam: (a) Beam in the reference configuration at rest (no load) (b) Beam in its buckled configuration.

Figure 2 :

 2 Figure 2: A part of the beam in the deformed configuration: (i) Internal actions (force and moment) at the material point G (s), (ii) Fixed and local frames.

  17) which is the position vector of any material point of the beam at the curvilinear abscissa S written either in the fixed frame or in the local frame. The different coordinates in Eqn. (17) satisfy the dimensionless definitions introduced in Eqn. (15).

)

  Note that at the ends A and B of the beam we have τ = e 1 and n = e 2 . In the set of equations Eqns. (19), Eqns. (21) have been included to compute the local displacement Q N and Q T as function of θ and N and boundary conditions concern those displacements. The set of equations that meets the boundary conditions (clamped-clamped beam) reads as • At the A end (S = 0)

Figure 3 :

 3 Figure3: Clamped-clamped buckled elastic beam: the Laplace force acting on the deformed beam and detail of an element of beam undergoing Laplace force produced by a volume density of moving electric charges placed in a magnetic induction.

Figure 4 :

 4 Fig.4. A classical N-shaped curve is obtained for three different end-shortenings: (i) weak (small end-shortening 0.3% of the beam length), (ii) intermediate (end-shortening of 1%) and (iii) rather

Figure 5 :

 5 Figure 5: Bistable response: or buckling loading as function of the beam midpoint displacement for three different values of the beam end-shortening (in solid line: the elastica model, small triangles and diamonds: FE results).

Figure 6 :

 6 Figure 6: Bistable response: Total potential energy as function of the beam mid-point displacement for three different values of the beam end-shortening (in solid line: the elastica model, small triangle and diamonds: FE results).

Figure 8 :

 8 Figure 8: Switching scenario of the bistable beam, (a) beam at the lower stable position, (b) region of instability, combination of buckling modes, (c) second buckling mode, (d) beam profile symmetrically inverted with respect to that at (b), (e) beam at the upper stable position.

Figure 9 :Figure 10 :

 910 Figure 9: Photos of the experimental set-up: (a) the global view of the different apparatus, (b) the detail showing the beam placed between both magnets and (c) the measurement of the mid-point beam displacement owing the laser profilometer.

Figure 11 :

 11 Figure 11: Diagram of bifurcation: beam center deflection as function of the end-shortening. Experimental results: red spot, model: blue solid line

Figure 12 :

 12 Figure12: Bistable response: experimental results superposed onto the numerical simulations of the bistable beam model.

Figure 13 :Figure 14 :

 1314 Figure 13: Lineic density of force as function of the beam mid-point displacement. Comparison between bistable actuations by means of lineic density of Laplace force and vertically fixed force. Laplace force (in solid line) and vertically fixed force (dashed line).

  ), accounting for the boundary conditions for a clamped-clamped beam. The solution splits into two kinds of modes.(i) The symmetric buckling modes read asY j (S) = A j (1cos (α j S)) (B.1)withα j = 2jπ (B.2)Moreover, we have set α 2 = P (1 -kP) in Eqn. (B.1).(ii) The antisymmetric modes are given byY j (S) = A j 1cos (α j S) -2 α j (α j Ssin (α j S)) (B.3)where now the values of α j are the solutions to the implicit equation

Acknowledgements

The research work reported in the paper has been supported by the research project BISCELTECH funded by Fonds Unique Interministériel (FUI-APP21).


Ola Helenius
email: ola.helenius@ncm.gu.se

Designing stakeholder scaffolding for implementing a teaching model at scale

Keywords: Professional development, design research, large scale, arithmetic, implementation

The challenge of implementing pedagogical innovations at scale is discussed by looking at the set of stakeholders. A large-scale multi-municipality implementation of a highly specific research-based teaching model for elementary school arithmetic in Sweden is used as an exemplary case. To create a lens through which the stakeholder structure can be discussed, the concept of stakeholder scaffolding is introduced. The main message is that what is to become a stakeholder in an implementation program should not only be decided through making an inventory of possible stakeholders and deciding which ones to engage. It is also a matter of designing new groups with given roles and responsibilities in the implementation program.

Introduction

In organization theory, the concept of stakeholder refers to "any group or individual who can affect or is affected by the achievement of the firm's objectives'' (Freeman, 2010, p. 25). When the stakeholder term is used in connection with implementation research in the field of education it is rarely explicitly defined. Certainly, when an educational innovation is implemented, there is not often a firm with particular objectives at the center. Instead, it makes sense to view many implementation efforts as coordinated by networks of parties. Therefore, I will look at multi-stakeholder network theory [START_REF] Roloff | Learning from multi-stakeholder networks: Issue-focussed stakeholder management[END_REF] for guidance and, define a stakeholder as any group or individual who can affect or is affected by the approach to the educational issue addressed by the network of collaborating parties.

By stakeholder scaffolding, I shall mean the intended interaction patterns between identified or constructed stakeholder groups and how these intended interaction patterns are designed to support the implementation process of a pedagogical innovation. In this sense, the stakeholder scaffolding is an aspect of the program theory of the implementation project. I use the theoretical construct program theory in the same way as Chen, that is, "as a set of explicit or implicit assumptions by stakeholders about what action is required to solve a social, educational or health problem and why the problem will respond to this action" (Chen, 2012, p. 17). In Chen's conceptualization, a program theory has two components, the change model and the action model. The change model specifies how activities in the program are supposed to affect certain determinants in the system that are to be changed in order to reach specified goals or outcomes. The action model is "a systematic plan for arranging staff, resources, settings, and support organizations to reach a target group and deliver intervention services" (Chen, 2012, p. 18). Stakeholder scaffolding, as I will describe the concept, is hence a part of the action model. The purpose of the concept of stakeholder scaffolding is to provide a lens that helps researchers and program designers to focus on the specific roles different stakeholder groups can assume in relation to the action model.

The aim of the present paper is to discuss stakeholder scaffolding in relation to the scaling of educational interventions. The theoretical basis for my discussion builds on [START_REF] Coburn | Rethinking scale: Moving beyond numbers to deep and lasting change[END_REF] characterization of scale as a four-dimensional construct concerning depth, sustainability, spread, and change of ownership. Further, the issue of scaling will be discussed from a design perspective where I do not only discuss which potential stakeholders a particular educational system might already involve, but also how to deliberately design stakeholder groups with the purpose of achieving scalability.

The discussion will be carried out by exemplifying the stakeholder scaffolding in a large-scale multimunicipality implementation of a highly specific research-based teaching model for elementary school arithmetic in Sweden. This means that this is not an empirical article but descriptive and theoretical. Because of space limitations, I will discuss Coburns construct directly in relation to how the implementers have tried to take the four components of scale into account in the implementation program design. For readers that prefer an introduction to [START_REF] Coburn | Rethinking scale: Moving beyond numbers to deep and lasting change[END_REF] conceptualization of scale, the original article is a very worthwhile read as are many of the shorter summaries (e.g., section 5 in [START_REF] Roesken-Winter | Evidence-based CPD: Scaling up sustainable interventions[END_REF]. I will now go on to describe the implementation program, before moving on to discussing it relative to Coburn's four scaling dimensions.

Thinking, reasoning, and reckoning (TRR) 1-3

The pedagogical program I will describe has two roots. On one hand, it builds on a specific teaching model communicated through descriptions of teaching sequences in the form of very detailed teacher guides. On the other hand, it builds on a professional development (PD) program that is a collaboration between the National Center for Mathematics education (NCM) in Sweden, a multitude of municipalities, and Sveriges Kommuner och Regioner (SKR). SKR is a national organization of municipalities and regions that stands free from the national political system. In what follows, the Swedish acronym TRR will be used to designate both the teaching model and the collaborative program for implementing it. Before describing the structure of TRR as an implementation program, I will briefly describe the TRR teaching model. TRR for grades 1-3 comprises four themes for each grade where each theme is a collection of five teaching cycles. A cycle covers three lessons. The teacher guides include sequenced descriptions of which representations to use, which terminology and expressions that are supposed to be used by the teacher, which questions to use when probing pupils reasoning, suggestions for how to sequence qualitatively different types of pupil reasoning and so on. Each cycle focuses on a particular mathematical phenomenon from one of the conceptual fields number, additive structures, or multiplicative structures [START_REF] Säfström | Implementing alternative models for introducing multiplication[END_REF]. A cycle has six phases: chanting, introductory whole class discussion, pair work, follow-up discussion, individual documentation, and follow-up discussion of the documentations. For each phase, the social arrangements and expectations of teachers' and students' roles are different. The TRR teaching model builds on a randomized control trial evaluated design research project for grade 0 in Sweden [START_REF] Sterner | Reasoning about Representations: Effects of an Early Math Intervention[END_REF][START_REF] Sterner | Number by reasoning and representations-the design and theory of an intervention program for preschool class in Sweden[END_REF].

The core of TRR is that participating teachers use the TRR cycles as a basis in their teaching. Each year, about half of the teaching time will then be directed by the TRR model. The rest of the time teachers can plan as they please. Participating teachers are formed in professional development (PD) groups of at least three teachers, teaching the same grade and teaching the same cycle at the same time. Before the first cycle in a new theme, between cycles and after the last cycle of the theme, the PD groups meet and discuss the upcoming cycle and the experience of teaching the previous cycle guided by structured PD-questions from a booklet under the guidance of a local facilitator appointed by the municipality.

I will now move on to discuss TRR from a top-down perspective. The initiative is joint between NCM and SKR and there is a joint steering group but no clear ownership of the project as a whole. TRR is run in (so far) yearly rounds, each engaging around five municipalities working together in so-called working networks of normally five municipalities. After a municipality has agreed to participate and signed a formal agreement with SKR, they appoint a process leader, which is normally a mathematics teacher that already has certain special leadership assignments in the municipality and a networking group consisting of the highest responsible politician for school matters in the municipality, the highest non-political leadership, two principals, two teachers, and the process leader. What I have described so far can be called the institutional scaffolding of TRR and is illustrated in Figure 1. The temporal scaffolding, which is how activities are distributed over time, is illustrated in Figure 2. In Sweden, a school year begins in the autumn. Therefore, in the temporal description of TRR what we describe as, for example, year 1 (Y1) is the year that starts in the autumn when the teachers of an attending municipality start to use the TRR teaching model at scale as well as the associated PDprogram. Y2 is hence the second year of TRR usage and Y0 is the preparatory year before.

The first thing that happens after the municipality has signed the contract to participate is to appoint the process leader and the network group. All process leaders in the current round then meet for a process leader conference (PLC). This is a first or a rather tight series of PLCs that during the pandemic has been held almost every month. Slightly after, the network conferences (NWC) are repeated two times a year. To ensure knowledge transfer between the different rounds, around half of the PLC and NWCs in Y0 and into Y1 are joint with the previous round and later joint with the next round. Both PLCs and NWCs are two-day sessions.

An important decision that municipalities have to take early is at what scale to run the program. We have examples where municipalities have joined with all their teachers and schools in all grades 1-3, some that join with all grades in a few schools, some that join with all teachers in grade 1, and one municipality that only joined with one school. An early decision on the scale is important because the number of teachers and how they are distributed between grades and schools determine the number of local facilitators and we want the local facilitators that are to work with PD-groups when Y1 starts to start preparing, by being introduced to TRR in Y0 and start to use some cycles in the classes they currently teach. Municipalities are then responsible to book introductory sessions with NCM for the teachers to start in Y1. Some want two such sessions and some stick with one. In subsequent years, the individual municipalities can also decide to book Q&A sessions with NCM for both facilitators and teachers. These are normally local events. In the temporal diagram in Figure 2, I have also indicated where pupil testing can be, but that can vary between municipalities depending on their scaling plan and how to set up a control group design.

Formally, the project runs three years for each municipality, but in practice, it has so far continued with less and less PLCs and NWCs. Municipalities in Y3 and beyond have so far continued to book introductory sessions for new teachers and Q&A sessions for working teachers with NCM.

Four dimensions of scale and how it applies to TRR

I shall now discuss the stakeholder scaffolding in TRR described above, from the perspective of scale as [START_REF] Coburn | Rethinking scale: Moving beyond numbers to deep and lasting change[END_REF] four-factor construct: depth, sustainability, spread, and shift in ownership. It should be noted that the stakeholder groups I describe below are intended to be examples of possible roles for stakeholders in relation to scaling. I make no claims that the particular stakeholder scaffolding of TRR is particularly good or that it is always well functioning.

Depth

Coburn's conceptualization of depth concerns "change that goes beyond surface structures or procedures [...] to alter teachers' beliefs, norms of social interaction, and pedagogical principles as enacted in the curriculum" (Coburn, 2003, p. 4). The focus in Coburn's description is on students' and teachers' roles in the classroom, like for example the patterns of communicational interaction. As described, TRR gives very strong guidance for teachers to follow particular interaction patterns, as well as particular ways of treating the mathematical content. Obviously, not all TRR intentions are followed. The implementers recognize that small adaptations from teachers affect how the content is portrayed to the pupils [START_REF] Koljonen | The documentational approach of didactics as a gauge of fidelity of a teaching model implementation[END_REF]. Therefore, highly prescriptive teaching designs still need deliberate stakeholder scaffolding to support depth. In TRR, the local PD-groups are one such structure intended to create a support for interpreting the intentions of each cycle as well as following up the classroom implementation and also induce a form of peer pressure. The local facilitators are a stakeholder structure designed to support the PD groups. Forming the facilitators as a group with a particular role is a key element in the stakeholder scaffolding. It is our experience from previous large-scale PD-programs [START_REF] Helenius | A stakeholder analysis of the development of a large-scale professional development program[END_REF] then letting one group of teachers, the facilitators, work ahead and giving them some additional PD as well as officially designating them as a facilitator group tends to shift of identity from a receiver of the PD to a deliverer of the PD which enforces loyalty to the program intentions and supports them in their role relative the PD groups.

Sustainability

Sustainability concerns the chances that deep changes stay in effect in classrooms, schools, and municipalities where they have been established. Referring to a wealth of literature, Coburn describes sustainability as dependent on four factors, which all reinforce a professional community of colleagues in the school that supports depth of change of which we will deal with two here. The first of these is the need for knowledgeable and supportive school leadership. In TRR, all levels of school leadership from principals and up are specifically engaged. The principal level is represented in the network group, which is the most complex of the designed stakeholder groups. It is also a common arrangement that the PL is co-opted into the municipality principal group, typically led by the municipality school leader, which further allows the process leader to influence and engage principals. The second factor I will mention is the connection with other schools and teachers engaged with a similar reform. Since TRR is a municipality driven there is plenty of such interaction within municipalities.

Spread

Spread may be the most obvious factor of scaling, as it involves the fundamental factor of replicating the implementation of some reform or innovation with more students, teachers, schools, or districts. Coburn makes a point of considering spread also within different levels. Spread within classrooms could for example involve pedagogical innovations proposed in one subject spreading to a teacher's teaching in other subjects. Spread within a district or municipality could involve certain norms related to a pedagogical teaching innovation spreading to local school policy [START_REF] Coburn | Rethinking scale: Moving beyond numbers to deep and lasting change[END_REF].

The stakeholder scaffolding of TRR has several elements designed to support spread to other schools and classrooms. The scaling that occurs to new municipalities with new rounds obviously leads to scaling, and so far all participating municipalities have decided to continue from Y1 to Y2, and beyond and have included more schools and teachers except in the cases where all schools were already engaged.

Shift in Ownership

The aspect of scale that Coburn perhaps puts the most emphasis is on a shift in ownership. It is worth noticing that the metaphor of shift of ownership presupposes an initial owner, an item to be owned, and a new owner. In Coburn's description, it is some reform program that is the item and the initial owner is some actor external to teachers, schools, and districts. With such a shift, knowledge of the reform can become local and a theory-based practice can arise so that the reform can become selfgenerative [START_REF] Coburn | Rethinking scale: Moving beyond numbers to deep and lasting change[END_REF]. With TRR, the stakeholder scaffolding sets up a quite advanced structure for shared ownership right from the start, that gradually can shift towards municipalities and teachers. Municipalities have to themselves decide the scale with which to enter the program, depending on how much they believe in the program and in their own ability to administer it and later decide how to scale it up. Recall, the arrangement that teachers should meet regularly in PD-groups and be able to teach the same cycles in parallel puts quite high administrative demands on all levels of leadership. It is questions like this that are discussed in the early PLCs and NWCs. Because municipalities are responsible for scaling also makes it important that teachers remain positive and even when feeling like the teaching is very difficult keep feeling that they have support from the organization. It is so far our experience that facilitators, but also many teachers, identify with the program and help to spread the word to teachers in the municipality that are yet not engaged. A final note on ownership is the several of the municipalities, without consulting SKR or NCM, have produced their own professional videos on TRR, to share with parents, the general public, and other teachers.

Discussion

Designing models for implementing pedagogical innovations at scale is a vital concern in contemporary implementation research. In the present paper, I took a closer look at the stakeholder issue related to scale and coined the concept of stakeholder scaffolding, which I presented as a part of the action model of the program theory [START_REF] Chen | Theory-driven evaluation: Conceptual framework, application and advancement[END_REF]. Chen already identifies particular types of stakeholders when schematically describing an action model, the implementing organization, peer organizations and community partners, and the program implementers. It is worth noting that Chen is discussing a wide section of innovations spanning over the social, educational, and health sectors. This means that the specificity when discussing the roles of the stakeholder groups he mentions is low. In the case of TRR, it is for example not very easy to separate what is the implementing organizations from the peer organizations. But a point Chen is making when pointing out different types of stakeholders as part of the action model is that types and roles of the stakeholders should be planned in advance and be part of the action model. Going back to the description I gave drawing on [START_REF] Roloff | Learning from multi-stakeholder networks: Issue-focussed stakeholder management[END_REF], it states that a stakeholder can be any group or individual who can affect or is affected by the approach to the educational issue addressed by the network of collaborating parties. A main message in the present article is that what is to become a stakeholder in an implementation program should not only be decided by reviewing groups that can be affected by the program and can affect it, like researchers, policymakers, principals and teachers. Instead, the view I am advocating is that stakeholder groups can be deliberately created as crossover groups between such levels as part of the design of a program.

In TRR, one such designed stakeholder group is the group of process leaders in each round. By identifying such a group and giving them opportunities to work together, their roles in the respective municipalities will be strengthened. The process leader group is formed through the multitude of process leader conferences, which also ensures that the process leaders are the most informed persons which further strengthens their authority and agency within municipalities. The arrangement with process leaders coworking with the peer group from a previous round, and eventually, with the peer group from the successor round further strengthens their agency. Other examples of designed stakeholder groups are the network groups in the municipalities with the role of giving coherent support for TRR across all institutional levels in the municipality.

In summary, TRR F-3 is an example of a teaching model that is very demanding for teachers that are to use it in their classrooms. Despite the highly scripted form of the TRR teacher guides, we believe it is virtually impossible for an individual teacher to operationalize TRR in their classroom. It can only be done in deliberate collaborative settings with a high degree of organizational support. In short, it can only be done by ensuring that all of Coburns four aspects of scale are accounted for in the implementation program. Designing and implementing a well-thought-out stakeholder scaffolding is hence an important part of the program design. When researching effects of large-scale implementation programs, I see it is a good idea to put special focus on if the stakeholder scaffolding worked as intended or not.

Figure 1 :

 1 Figure 1: The institutional stakeholder scaffolding of TRR

Acknowledgement

This work was financed by the Swedish Research Council Grant 721-2014-2468


Tuula Koljonen
email: tuula.koljonen@liu.se

Ola Helenius
email: ola.helenius@ncm.gu.se

Linda Marie Ahl
email: linda.ahl@ncm.gu.se

The documentational approach to didactics as a gauge of fidelity of a teaching model implementation

Keywords: Documentational approach to didactics, fidelity, implementation, mathematics, education

We examine the feasibility of using the Documentational Approach to Didactics to gauge the fidelity and character of teachers' implementation of scripted teaching sequences for the teaching of arithmetic in primary school. By analyzing a video-recorded lesson, we found that the teacher adapted the scripted teaching sequence by merging it with other resources such as self-made worksheets and an interactive whiteboard. The teacher also made adaptations to the prescribed organization of the teaching. Through the analysis, we gain insight into the nature of the adaptions and find that Documental Approach to Didactics can be used as a lens to distinguish between surface adaptions, that do not interfere with the program theory and adaptions that violate core ideas of the innovation. The results provide a background for a discussion of pro-fidelity versus pro-adaption point of view in implementation research.

Introduction

The extent to which an educational innovation implemented in classrooms reaches its desired outcomes depends on how teachers adapt the innovation in their practice. The extent to which teachers faithfully follow the instructions and intentions of an intervention can be discussed in terms of fidelity. The rationale for measuring fidelity and striving for high fidelity is to allow analytical separation of the quality of an educational innovation from the quality of the implementation of it, as well as to provide a reference for scale-up or replication research involving the intervention. [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF] call the view that more fidelity is better the pro-fidelity perspective. They argue that to understand the nature of implementations of pedagogical innovations rather than just the effects of them, the pro-fidelity perspective ought to be complemented with a pro-adaptation perspective encompassing a research focus on "why adaptations are made, where implementation challenges come from, and how the nature of an adaptation matter for outcomes " (p. 200).

It has been noted by researchers that innovations that are general, non-specific, and involve little prescribed structure might need more adaptation [START_REF] Cohen | Relations between policy and practice: A commentary[END_REF]) and hence would benefit from a pro-adaptation research design [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF]. Highly structured innovations that are specifically prescribed could on the other hand benefit from a pro-fidelity research design [START_REF] O'donnell | Defining, conceptualizing, and measuring fidelity of implementation and its relationship to outcomes in K-12 curriculum intervention research[END_REF]. The study presented here reports a case from a context where the innovation is a highly prescribed curriculum based on a specific teaching model, but as we will show, a proadaptation perspective still has merits.

The context of the study is a large-scale intervention, spanning 14 municipalities and over a thousand teachers, where the outcome on the student level is being evaluated with a control group, pre-posttest design. Yet, despite an intricate implementation design involving collegial collaboration to stimulate high fidelity, the project team was aware that adaptations would be made, and that fidelity would vary. To evaluate the intervention and gain worthwhile feedback, we were therefore in search of a theory base for researching teachers' adaptations of resources. This methodological paper reports on the use of the Documentational Approach to Didactics, DAD, [START_REF] Trouche | The documentational approach to didactics[END_REF] for this purpose. DAD has previously been used to characterize teachers' use of and blending digital online resources. However, we see the potential to use DAD to characterize teachers' implementation of a highly prescriptive curriculum resource. Therefore, the present paper aims to examine the feasibility of using DAD to characterize the fidelity and adaptation of a highly prescriptive curriculum resource.

Theory

The theoretical framework DAD can be used for studying teachers' goal-oriented work through their interactions with resources used in their teaching [START_REF] Trouche | The documentational approach to didactics[END_REF]). An objective with DAD is to achieve a comprehensive understanding of teachers' work through elaborating what it can mean "to use" a resource. To understand the relevance of the term documentational, it helps to know the etymology of the word document. The word originates from Old French where it, in the late 15th century came to mean "teaching, instruction". The concept is also linked to the Latin "documentum" which refers to "lesson" and "to teach". DAD builds on French didactics' theory and the tradition of the instrumental approach. The instrumental approach takes as a point of departure, normally, a digital artifact that is to be used by teachers or pupils. The fundamental observation is that when an artifact becomes incorporated in an individual's handling and thinking scheme, there are two processes involved: instrumentation and instrumentalization. The instrumentation process is when the affordances of the artifact shape the persons' practice. The instrumentalization process is when an individual uses her knowledge and dispositions to use the artifact for fulfilling her needs and goals. Combined, the instrumentation and instrumentalization processes interact to create the instrumental genesis, the transformation where the artifact becomes an instrument for the individual [START_REF] Artigue | Revisiting the French didactic tradition through technological lenses[END_REF]. The instrumental genesis, in turn, involves an interaction in which the user and the potentialities of the artifact (the physical tool) are reciprocally constituted in action [START_REF] Artigue | Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work[END_REF].

In the documentational approach, the role of the artifact is replaced by the role of one or several resources that a teacher might use or be influenced by in her teaching. The concept of resource encompasses any artifact with the potential to promote mediating teaching activities, including textbooks, curricular guidelines, digital resources, student worksheets, or work with a colleague. In DAD the teacher's role is to be a designer who, guided by a goal, transforms resources into documents in the process of interacting with a resource or a set of resources. This process is called documental genesis, in line with how instrumental genesis is described above [START_REF] Trouche | The documentational approach to didactics[END_REF]. This process ends with a document that is the product of a teacher's resource design work in each situation. The totality of a teacher's developed documents is called a document system.

A document is thus a combination of when a resource is appropriated and reshaped by a teacher, in a way that reflects their professional experience with the use of resources. Document = Resources + schemes of usage. A teacher's resource system is developed and formed by all the resources the teacher uses for a given activity [START_REF] Trouche | The documentational approach to didactics[END_REF]. The resource systems will be different for different teachers, even though they use the same resources since the resulting documents are dependent on the teacher's characters and prior knowledge. Thus, the process of documental genesis is ongoing since teachers' utilization schemes, how teachers adapt and are adapted by the resources they use change with experience [START_REF] Vergnaud | A comprehensive theory of representation for mathematics education[END_REF].

Method

We have compared the enacted actions of a teacher involved in a large-scale implementation program, with the scripted actions that carry the core ideas of the program, the program determinants. To set the stage for this study, we will elaborate on the implementation program by explicating elements of its program theory. When we use the concept program theory, we refer to Chen ( 2012), where we refer readers for further elaboration. Briefly, a program theory is an explication of a program's goals and how the program is supposed to achieve its effects. The change model of a program theory is the description of the "leverage or mechanism upon which [the program] can develop a treatment or intervention to meet a need. That leverage or mechanism is variously called the determinant or the intervening variable" (Chen, 2012, p. 18).

The teaching model builds on a randomized control trial evaluated design research project for grade 0 in Sweden [START_REF] Sterner | Reasoning about representations: Effects of an early math intervention[END_REF][START_REF] Sterner | Number by reasoning and representations-the design and theory of an intervention program for preschool class in Sweden[END_REF]. The program has a comprehensive organization and involves municipality collaboration, teacher professional development, and local facilitator and leadership organization (Helenius, n.d.). In what follows, we will designate both the teaching model and the program for implementing it by its Swedish acronym TRR. In its current form, TRR is implemented in a program run in 14 municipalities in Sweden with over 1000 teachers. The TRR teaching is arranged in cycles composed of six phases, each comprising a particular pedagogical arrangement (see [START_REF] Sterner | Reasoning about representations: Effects of an early math intervention[END_REF] for a description). Participating teachers complete 20 cycles that cover half of the mathematics teaching over a school year. A cycle normally covers three lessons and is communicated by a resource book, with detailed descriptions of how to carry out teaching. The detailed descriptions carry the program determinants and view as the desirable outcome in instruction. Of the large selection of possible determinants to distinguish in TRR, we present four of relevance in the sequence of teaching we will analyze.

Determinants of TRR resource book

Determinant 1 (D1). Building on the pupil's productions and reasoning. While the tasks, representations, and language to use are explicitly described in the TRR resource book, the main role of the teacher is to get the pupils to reason about their work and engage pupils in reasoning about each other's work. To aid the teacher, the material contains advice on how to sequence student presentations, depending on different approaches taken to the task and on particular types of questions or prompts to use when engaging pupils to talk about their work.

Determinant 2 (D2). The problem-solving strategy. In TRR, we endorse a particular strategy for approaching word problems: Formulate in your own words, use one of the endorsed representations for the situation type represented by the problem, write a mathematical expression, and find out the answer. A normative choice in TRR is that we prefer if pupils write the mathematical expression as close to the mathematical situation that is represented in the problem as possible. In this case, we prefer the situation: The rabbit jumps five times farther than the bear to be represented as rabbit's jump = 5 • bear jump, regardless of if the task continues: The bear jumps 3 meters, how far does the rabbit jump? or The rabbit jumps 15 meters, how far does the bear jump? Determinant 3 (D3). The multiplicative model. In TRR we use three different models to build the multiplication concept on, none of which is repeated addition. One of them is rectangular arrays, initially built by cubes and later cut out from square paper. In the second teaching cycle on multiplication, the class constructs what is referred to as the big multiplication table which is a 10by-10 matrix of rectangles, where the entry (i, j) displays the i times j rectangle with the product i • j written on it [START_REF] Säfström | Implementing alternative models for introducing multiplication[END_REF].

Determinant 4 (D4). The chanting phase of the teaching cycle. The first phase in a teaching cycle is always chanting. The chanting signals that a mathematics lesson starts and connects to the mathematics to be treated in the teaching cycle. The chanting phase is not supposed to contain teacher explanations. But despite that, it might contain mathematical information not apparent to the pupils. In the cycle under scrutiny here, the chanting comprises variants of If 3 times something is 12 then something is 4. The resource book instructs the teacher to have the big multiplication table visible and pointing at the 3-row, while uttering three, moving the hand along the row and stopping at the rectangle with 12 on it when uttering twelve, and then moving the hand to the top column and stop at 4 when uttering four. The chanting is hence supposed to be accompanied by a gestural indication of how to solve an open multiplication problem of the type a • _ = b by using the big multiplication table.

The data for the study presented here comes from a 49 minutes iPad video recording of one lesson with 8-year-old children (grade 2). The teacher has a formal teacher education for grades 1-7 and 24 years of teaching experience. At the time of the data collection, he had been involved in the TRR program for three years but had not taught the cycle discussed here before. The parts showing the teacher's use of resources were transcribed. Only the excerpts that are presented in this paper were translated into English. In collaboration, all three authors conducted a thematic analysis of the teacher's actions. With the theoretical lens of DAD, we describe the teacher's document system, that is his schemes of usage of the two resources, the TRR resource book and the whiteboard. Then we compare his document with the determinants D1-D4 from TRR's change model.

Results

Two resources, the TRR resource book and an interactive whiteboard with an attached document camera, are in play in our excerpts. D1-D4 are references to the TRRs determinants presented in the method.

As per TRR resource-book instructions, the chanting of the type: If 1 times something is 2 then something is 2 is displayed on the whiteboard, which in the teacher's case is an interactive whiteboard. For each chanting, the teacher draws the corresponding rectangle next to the chant, e.g., if 3 times something is 12, then something is 4, the 3 by 4 rectangle with 12 printed on it is drawn. Next to the rectangle, the teacher has also written the open multiplication statement 3 • _ =12 on the interactive whiteboard.

The sequence starts 01:26 minutes into the lesson and is close to two minutes long. In line with the teaching model, the teacher leads a collective chanting phase (D4).

Teacher:

We now start with the chanting, so read with me. If 1 times something is 2 then something is 2 (points at the chanting text). So, 1 times something is equal to 2 (points at the open statement) then it is 2 of course (points at the rectangle) /.../ If 7 times something is 42 then something is 6 (points at the chanting text). Here (points at the rectangle) we can see sixes, 7 pieces, 6 pieces of sevens, and 7 pieces of sixes, so ... hm, we stop here, and we move on to the presentations.

The teacher uses the whiteboard through instrumentalization, where using the whiteboard is a part of his utilization scheme. The use of the resource, the interactive whiteboard, forces instructions in the TRR resource book to be instrumented through the interactive whiteboard, leading to a transfer of the instructions. In this transfer, the chanting instructions themselves pass through unchanged. But, the inclusion of the arithmetic expression in the form of an open multiplication problem is an addition that does not conflict with the determinants. The omitting of the use of the large multiplication table (D3, D4), in favor of a single rectangle image, circumvents the idea of the 10-by-10 matrix as a support for pupils' reasoning. As a result, the gestural indication ( D4) of how open multiplication problems can be solved using the big multiplication table (D3) will make the pupils less prepared for treating the word problem in the cycle, according to the preferred strategy (D2).

The second excerpt, starting at 03:10, displays a whole class discussion aiming to build on pupils' reasoning (D1). The teacher starts by asking a pair of pupils to come forward to discuss their work, from the lesson before. The pupils do not bring any work with them, because the teacher has beforehand created a slide for the whiteboard. He simultaneously displays the problem formulation at the top, and the pupils work, followed by three representations: the number line, a rectangular array, as well as the instructions from the TRR resource book (D2, D3). The teacher says: The teacher moves on to leading questions, to guide the pupils:

Teacher: Hm, aha...help...if Pi has 5 (points at the number line), so how many did Pi have?

The pupils that have approached the whiteboard to describe how they worked with the problem (D1, D2) turn their attention to the whiteboard and say 5.

Teacher: Yes, and how do you know? It says here (pointing at the task formulation on the whiteboard)

The initiative has transferred from the pupils that should have been asked to describe their work in their own words, leading to a loss of fidelity to D1 and D2.

What follows next is that the teacher takes over the initiative and explanations more and more. The teacher uses the picture of the pupils' work on the interactive whiteboard, which enables him to draw and write on the picture when explaining. When it is time for writing mathematical expressions, the teacher leads the discussion, even more. At this stage, the teacher also involves the rest of the class (D1). The excerpt below is 10:43 minutes into the lesson The teacher successfully led the pupils to reason about the properties of multiplication with the rectangles (D3). However, the fidelity to using pupils' productions and reasoning (D1) is low.

High fidelity can be traced in the teacher's document, the instrumentalization of the whiteboard and the choice to place the pupils' solutions projected on the board instead of having pupils posing their solutions on the static board. As a consequence, the teacher saves time, and the class can easily participate in the discussion since the written solutions are visible to everyone in the class (D1).

Through the instrumentation of TRR, the teacher includes the rectangular array (D3) and open multiplication statements (D2) into pupils' worksheets and presentations.

Discussion

To examine a method for gauging fidelity, we have reported a single case study of a teacher with several years of experience in implementing TRR -a pedagogical innovation in the form of scripted teaching sequences. We examined the teaching by focusing on the concept of the document system from the theory Documental Approach to Didactics, DAD. The analysis showed adaptions of the highly prescriptive curriculum resource in the teacher's documents. Some adaptions retained fidelity to the determinates in the program theory's change model, while others led to less fidelity relative to the determinants. The two resources, the TRR resource book, and the interactive whiteboard shaped the document system. It is expected that when a teacher already has an established document including a particular resource (here: whiteboard) and a new resource (here: TRR resource book) is to be incorporated and joined with the previous resources, an adaptation takes place. This adaptation results in a new document including the use of both resources.

The teachers' utilization scheme involving the interactive whiteboard shaped the teaching. The teacher's instrumentalization of the interactive whiteboard on one hand strengthened the instrumentation of certain TRR determinants, for example enabling swift and easy presentation of pupil's work (D1). On the other hand, the usage of the whiteboard drives a tendency to reshape TRR instructions away from the instructions in the TRR resource book. One example is when the reference to the big multiplication table is replaced by the reference to individual rectangles that symbolize a particular multiplication. In principle, the transformation of the TRR chanting instructions (D4) to the whiteboard format could have been done as faithful instrumentation but ended up involving an adaptation of the TRR instruction, essentially breaking D4 as well as D2, by not presenting the big multiplication table (D3) as a resource in the problem-solving strategy (D2). The instrumentalization of the whiteboard can be said to affect the instrumentalization of the TRR resource book so that the resulting document system breaks some of the TRR determinants. The other example from our result section can be understood in similar ways.

A pure pro-fidelity stance could perhaps indicate a need to complement the printed TRR instructions with ready-made interactive whiteboard resources. Some teachers have asked us, developers, for such supporting resources. Despite our choice to present TRR in a highly prescriptive form, we are however worried that more specific resources could lure the teachers into a surface adaption of TRR, omitting the determinants. Instead, in line with [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF], we think a pro-adaptation perspective is vital. The DAD framework provides a good basis for examining the nature of adaptations. In the presented case, we could for example conclude that some adaptions made were related to the mixing of resources. Since we know it is a relatively common practice to use interactive whiteboards, we can suspect that similar adaptions might be made by other teachers too. As developers, this is useful information since it might for example call for revisions of the teacher guides concerning how careful we describe the core features of certain figures or practices in the material.

A downside of the DAD-based analysis reported in this paper is that it is too extensive to carry out on a large scale. But even examining some distinct cases can inform us on why and how adaptations are made, as in this case. Through DAD analyses we can gain insight into the nature of the adaptions. We get a tool to distinguish between surface adaptions that do not interfere with the determinates in the change model from the program theory [START_REF] Chen | Theory-driven evaluation: Conceptual framework, application and advancement[END_REF] and adaptions that violate core ideas of the innovation. The insights from the analysis give important feedback for future development of the professional development program that goes with the implementation, as well as the structured teaching instructions it builds on.

  what you have done? Tell us what you have done, and start with the number line. So, can you tell us a little about it? Presenting pupil: We know that Pi has 30 cucumbers [the teachers encourage them with mmm] and if Pi has 5 then Nitty has ehh… Nitty had...I do not know...you… The TRR resource book encourages open questions. Can you describe in your own words what you have done? But, the teacher's call to start with the number line deprives the pupils of the initiative. The pupils end up having difficulties telling what they have done, and how they reached their answers.

  expression fits this one (points at a rectangle, 3 • 6). Now I want you in the audience to think as well. What fits that rectangle? What is the multiplication for this (points at the rectangular array)? You too are audience. Now you are an active audience (addressing a pupil) and so you have to think. Ehh Teacher:(points at the class) ... shshh ... (the teacher asks a pupil in the class for an answer) is that? (writes on the board 3• 6 =)Pupil in class: It's 3 pieces of sixes Teacher:Because it is 3 pieces of sixes, sure. And how much is 3 pieces of sixes? (asks the presenting pupil)Presenting pupil: 18Teacher: You remembered, great! Thank you for your presentation.