Iskander Abroug
email: iskander.abroug@unicaen.fr

Nizar Abcha

Armelle Jarno

François Marin

Physical Modeling Of Extreme Waves Propagating From The Open Sea To The Coastal Zone

Keywords: Extreme waves, Solitary waves, Shoaling, Energy damping, Breaking waves, Runup

. A new formula is proposed for the estimation of runup height.

Figure 1. Sketch of the experimental setup

A 0 = Amplitude of solitary wave 3 meter form the wave maker, m. A 1 , A 2 = Wave heights for water depths h 1 and h 2 , m. A b = Breaking height, m. A s = Solitary wave amplitude, m A t = Amplitude of solitary wave at the toe of the beach, m. C gr = = Group velocity corresponding to the main solitary wave, m.s -1 E = Energy on a unit length in the direction transversal in the direction of wave propagation, J.m -1 g = Acceleration of gravity, m.s -2 h 0 = Stillwater depth, m h b = Breaking depth, m. L 0 = Wave length scale, m. R = Runup height, m. S 0 = Slope parameter. T = Duration of impulse, s β = 0.04= Slope, rad. ε=A 0 /h 0 = Non linearity parameter (1). ε'=A t /h 0 = Non linearity parameter (2). ɳ = free surface elevation, m <ɳ> = mean water level, m ρ = Water density, 1g.cm -3

Introduction

Giant waves or extreme waves correspond to waves of very high amplitude. These waves may be accompanied by deep troughs that may appear before or after the highest peak. This type of wave has been a mythical phenomenon in the marine environment for centuries. Extreme waves occur in basins of different depths with and without current. One of the firsts who observed this phenomenon was the captain and naval officer of an expedition in 1826 Dumont d'Urville, who reported about 30 meters high waves. These waves may be due to a large variety of physical mechanisms [START_REF] Kharif | Physical mechanisms of the rogue wave phenomenon[END_REF]. Among others, solitary waves may be considered. The first description of these waves was made by John Russel (1834) who observed a boat trained rapidly by a pair of horses in a channel at Hermiston. When the boat was suddenly stopped, Russel noted that a wave having the shape of a lonely high altitude wave was propagating along the channel without any evident change in shape. (J.S. [START_REF] Russel | Report on waves[END_REF]. Boussinesq (1871) derived a mathematical expression for the solitary wave profile generated on a horizontal, rectangular channel. Solitary waves are traditionally used as tsunami waves approximation because of the hydrodynamic similarity between them (e.g. Shih-Chun [START_REF] Hsiao | Tsunami-like solitary waves impinging and overtopping an impermeable seawall: Experiment and RANS modeling[END_REF]). Some laboratory studies have considered the shoaling [START_REF] Camfield | Shoaling of solitary waves on small slopes[END_REF]; [START_REF] Kishi | The shoaling, breaking and runup of the solitary wave on impermeable rough slopes[END_REF]) and the runup of solitary waves (e.g. [START_REF] Synolakis | The runup of solitary waves[END_REF]; [START_REF] Li | Solitary wave runup on plane slopes[END_REF]; [START_REF] Zelt | The runup of nonbreaking and breaking solitary wave[END_REF]). Gradual damping was investigated by [START_REF] Keulegan | Gradual damping of solitary waves[END_REF] and [START_REF] Daily | The solitary wave: its celerity, profile, internal velocities and amplitude attenuation in a horizontal smooth channel[END_REF]. [START_REF] Synolakis | Evolution of Maximum Amplitude of Solitary waves on Plane Beaches[END_REF] investigated the amplitude evolution of breaking solitary waves on a plane beach. They found that the complete amplitude evolution can be divided into four regions (especially in the case of smooth slopes). Synolakis (1991) confirmed mathematically that, the solitary wave amplitude before breaking, can be perfectly modelled by Green's law.

Waves with abnormally high amplitudes do not occur only on high seas, but also in the coastal zone. A lot of studies have considered these waves in the open ocean; however, they have received little attention in the coastal zone. The physical processes involved in the transformation of these waves from the open sea to the coastal zone are poorly understood, despite the significant impact these waves may have on the coasts, such as overtopping, material damages, and erosion [START_REF] Strusińska-Correia | Nonlinear Behavior of Tsunami-Like Solitary wave over Submerged Impermeable Structures of Finite Width[END_REF], [START_REF] Marin | Laboratory study of sand bedforms induced by solitary waves in shallow water[END_REF]). The aim of the present work is to bring a contribution to the study of these processes from physical modeling. The paper is organized as follows. Section 2 describes the experimental set-up. Results are presented in Section 3. The shoaling, wave energy, wave breaking, and runup are considered. Section 4 is devoted to discussion and conclusions.

Experimental set-up

Experiments are carried out in a wave flume of the coastal and continental morphodynamcis Laboratory at Caen University, France. The flume is 20m length and 0.8m width. An Edinburgh design Ltd piston-type paddle is used to generate waves. Waves propagate through uniform water depth to a 1/25 sloping smooth beach, at the other The flume side walls are made of glass. The beach toe is located 9,5m from the wave maker. Flume bottom is elevated 1.15m from the ground to facilitate the visualizations from the side glasses. Free surface elevations are measured by resistive waves gauges. Mainly, three wave gauges are used. Gauges 1 and 2 are used to plot the space-time diagram STD (more details in the next section) and gauge 3 is used for the measure of the free surface elevation at the toe of the beach. Waves of different spectrum and amplitudes are generated in the flume under conditions of deep water and shallow water. To generate surface waves in the flume, a computer controlled wavemaker is used. A schematic sketch of this experimental setup is shown in Figure 1. In order to confirm the repeatability of present tests, two surface elevation measurements at the same location are conducted. Differences were less than 2%. However, tests are less repeatable once the wave breaks down due to the entrapped compressed air present during the breaking process. The governing parameters include the still water depth h 0 , the local water depth h, the initial solitary wave amplitude A 0, the local solitary wave amplitude A, the free surface elevation ɳ, the wave amplitude at breaking A b, the corresponding water depth h b and run-up height R. A high-speed camera is used for tracking the wave features while propagating in the breaking and swash zones. The camera is operating with a 30 Hz acquisition frequency.

Results

a. Wave profile

Figure 2 shows a comparison between the experimental and theoretical profiles of a solitary wave. A nonlinear solution obtained by Boussinesq approximation has been used for the theoretical profile, for ε=A 0 /h 0 =0.24. A very good agreement is obtained. We can note that the experimental profile exhibits a very weak dispersive tail.

In order to generate solitary waves, [START_REF] Goring | Tsunami-the propagation of long waves onto a shelf[END_REF] used derived Boussinesq's first order solution (1)

(2)

Where A s is the solitary wave amplitude, V s the theoretical wave velocity, A 0 , and A t the amplitudes of solitary wave, 3 meters from the wave maker and at the toe of the beach, respectively. A satisfactory agreement is obtained between (2) and experimental wave velocities, as shown in Figure 3.

b. Shoaling and energy

Shoaling

The free surface elevations were recorded at different locations along the flume with resistive wave gauges (gauges 1 and 2, Figure 1). Free surface instantaneous components can be overlaid chronologically at regular time intervals in order to plot space-time diagrams. Gauge 1 is fixed 3m from the wave maker and used as temporal reference and gauge 2 is moved 10 cm after each acquisition. The information obtained along the flume from the resistive probes are compiled in a matrix describing the evolution of free surface elevation along the flume.

An example of spatio-temporal diagram is shown in Figure 4. In this diagram, the evolution of the amplitude of the wave, and the width of the solitary wave along the channel are well described. A very weak dispersive tail is observed behind the main solitary wave.

The solitary wave has a symmetric profile along the flat bottom. When it reaches the slope, the wave height increases and the shape becomes more and more asymmetric (Figure 5). [START_REF] Synolakis | Evolution of Maximum Amplitude of Solitary waves on Plane Beaches[END_REF] mentioned that the amplitude evolution of a solitary wave on a sloping beach can be divided into four zones for plunging and spilling breaking waves. They called the zones gradual shoaling and rapid shoaling before the breaking point, and rapid decay and gradual decay inside the surf zone. In particular, in each zone the maximum local wave elevation can be adequately described by a simple power-law-type formula as a function of spatial location with a normalization of breaking water depth for convenience [START_REF] Synolakis | Evolution of Maximum Amplitude of Solitary waves on Plane Beaches[END_REF].

In this study the measured amplitudes amplifications are compared to the following shoaling relationship based on linear long waves (Green's law):

(3)

where A 1 and A 2 are wave amplitude for the water depths h 1 and h 2 , respectively. Figure 6 shows that Green's law describes satisfactorily the variation of the amplitude along the slope.

When the crest velocity exceeds the solitary wave celerity, the wave breaks. By zooming in on the STD, position and height of breaking point can be identified. In practice it is better to zoom in as much as possible and detect the brutal change in amplitude (Figure 4). The STD is a way to get a good overview of what is happening along the flume, before and after shoaling. Otherwise, the video visualization gives a good resolution of the local water surface elevation (30 fps). Therefore, when investigating breaking height and distance, the camera is a better solution. Table 1 summarizes comparisons between STD and video visualization results. It appears that breaking heights are slightly different when STD is used. This is due to the air entrainment phenomenon generated by breaking waves which can have an influence on the signal given by the gauge at the breaking point (x b ).

Energy

The waves energy is investigated on the flat bottom and on the slope. Equation 4 (Tian el al., 2010) is used to calculate the energy of the solitary wave on the flat bottom for a given position:

(4)

where is the group velocity corresponding to the main solitary wave, g the acceleration of gravity,  the water density, T the duration of impulse,  and <  > the free surface displacement and mean water level, respectively.

Table 1. Height and distance of breaking; comparisons between STD and camera measurements

Figure 8 shows that the amplitude dissipation factor ( ,which describes amplitude evolution along 6.5m on the flat bottom, decreases linearly for increasing values of A 0 /h 0 from 17% for the lowest solitary wave amplitude up to 4% for the highest wave amplitude. This has also been experimentally obtained by [START_REF] Ippen | The shoaling and breaking of the solitary wave[END_REF].

When the solitary wave reaches the slope and before breaking, the wave celerity decreases, the wave height increases and the total energy is approximately conserved. (Ying li and Fredric Raichlen 2002).

The energy spectral density (Figure 7) shows that the spectral density (f excitation =0.25Hz) of the main solitary wave (A 0 /h 0 =0.28) is conserved during the wave propagation until wave breaking which occurs approximately for x=15m. A second band of energy is identified as the spectral density of the wave generated by the piston pullback. In term of intensity this band and small bands of higher frequencies (1.2 Hz and 1.4 Hz) are less strong than the main band. Therefore, '' super harmonics'' waves do not have a significant impact on the shoaling phenomenon and on the runup of the main solitary wave. Moreover, higher frequency waves travel slower than lower frequency waves and reach the shore after the solitary wave [START_REF] Orszaghova | Importance of second-order wave generation for focused wave group run-up and overtopping[END_REF]. The band for frequencies around 0.85 Hz is slightly inclined when compared to the main spectrum. When propagated along the flume, this band tends to get closer to the solitary wave spectrum. This inclination is due to the instable form of the wave generated by the pullback of the piston. Otherwise, it is also shown that a band around 0.15Hz, which corresponds to low frequency waves, appears especially when the bed slope starts. These subharmonic waves propagate along the flume. They are amplified when reaching the slope (shoaling zone). However, in term of spectral density intensity the energy is approximately three times lower than the energy of the main solitary wave. Therefore, we consider that subharmonic wave effect is not significant.

Breaking [START_REF] Grilli | Breaking criterion and characteristics for solitary waves on slopes[END_REF] and [START_REF] Synolakis | The runup of solitary waves[END_REF] defined a breaking criterion. They proposed two thresholds, ( 5) and ( 6) are used to predict if the incident solitary wave is a breaking wave or not.

(5) ( 6)

Where β = 0.04 is the beach slope, and ɛ'=A t /h 0 is a nonlinearity parameter. For the present study, the nonlinearity parameter is in the range 0.10<'<0.27. Therefore, all waves are breaking waves for present tests. [START_REF] Grilli | Breaking criterion and characteristics for solitary waves on slopes[END_REF] and [START_REF] Yasuda | BIM simulation on deformation up to breaking of solitary waves over uneven to bottoms[END_REF] introduced empirical parameters for slope parameter (Equations 7 and 8):

(7) (8) S 0 is the slope parameter, and L 0 is a characteristic horizontal length scale for the initial wave [START_REF] Grilli | Breaking criterion and characteristics for solitary waves on slopes[END_REF] for more details about L 0 ). According to [START_REF] Grilli | Breaking criterion and characteristics for solitary waves on slopes[END_REF], S 0 in equation 7 has a 99.9% correlation with ξ s introduced by [START_REF] Yasuda | BIM simulation on deformation up to breaking of solitary waves over uneven to bottoms[END_REF]. S 0 and ɛ' are used to determine breaking type:

-S 0 < 0,025: Surging breaking -0,025 < S 0 < 0,3: Plunging breaking -3 < S 0 < 3,7: Spilling breaking According to these criteria, breaking in this study can be considered as plunging breaking for all solitary wave amplitudes (0.06 ≤ S 0 ≤ 0.09). It is validated by direct observations. Using a speed camera of high resolution, we are able to follow the evolution of the free surface in space, and to identify the position of the breaking point. Figure 9 illustrates snapshots of breaking process for a solitary wave with A 0 /h 0 =0.28. The vertical tangent to the free surface corresponds to the spatial position of the breaking wave

The variation of the dimensionless breaking height with the dimensionless amplitude of the wave at the beginning of the slope is depicted in Figure 10. The relative breaking height is found to decrease for increasing values of '= A t /h 0 , as suggested by these authors. ( 9) [START_REF] Grilli | Breaking criterion and characteristics for solitary waves on slopes[END_REF] have derived (9) using a least square method for a wide range of slopes (between β = 1/100 and β = 1/8). Present data has a 99.25% correlation with equation ( 9). In Figure 11, the variation of the dimensionless water depth at the breaking point h b /h 0 is plotted as a function of A t /h 0 . [START_REF] Grilli | Breaking criterion and characteristics for solitary waves on slopes[END_REF] and [START_REF] Hafsteinsson | Solitary wave run-up: wave breaking and bore propagation[END_REF] suggested respectively ( 10) and ( 11) for water depths determination.

(10)

Present data has 99.33% and 98.71% correlation with ( 10) and ( 11) respectively. Therefore ( 10) is slightly better when investigating breaking depths.

c. Runup

Measured runup heights are compared with literature on Figure 12. This figure shows that, for the same ɛ', runup heights are higher for steeper slopes. Present results (β = 1/25) clearly show that the runup height increases for increasing values of ε'=A t /h 0 . This results from the large energy of large solitary wave amplitudes. [START_REF] Synolakis | The runup of solitary waves[END_REF] suggested the following empirical formula: (12

)
Where R is the maximal run-up amplitude. This formula is exclusively for the slope that Synolakis used (β = 1:19.85). Numerous studies have suggested several empirical formulas similar to that of Synolakis, in the form , a and b being parameters depending on the slope. Equation 13 and 14 predict well maximum runup amplitude for low waves (up to ɛ' =0.15)), whereas they over estimate it for ɛ' higher than 0.2 (Figure 13). It was mentioned in [START_REF] Hsiao | On the evolution and run-up of breaking solitary waves on a mild sloping beach[END_REF] that Equation 14 contains no physical meaning, but it gives reasonable maximum runup predictions for a wide range of slopes. It is the case when equation 16 is compared to Briggs et al. (1993) who have worked with a 1/30 slope. Agreements are less good when ɛ' becomes higher than 0.20. From the present data, an empirical formula is suggested for the estimation of runup height, for a 1:25 slope: (15)

Figures 14 and15 show that waves with high amplitude are characterized by high values of the time for runup saturation defined as the time needed by the wave to reach maximum runup. Runup velocity is determined using high speed camera. A video recording with 30Hz acquisition frequency is used for each solitary wave. Furthermore, each solitary wave is identified by only one runup saturation height. Briggs et al. (1993) have plotted the maximal runup amplitude versus an energy-based parameter using runup speed to reach the associated runup saturation. They suggested a linear relationship that can be used to predict the runup saturation for a known runup velocity.

(16)

In Figure ( 16), the maximal run-up amplitude versus runup velocities is plotted. Runup speed for 1:25 slope is greater than for a 1:30 slope. This is not true for small wave amplitudes (mainly nonbreaking waves). This is due to the energy dissipation which is more important for low amplitudes. Variations in wave characteristics for solitary wave in shallower are discussed in this paper. The behavior of the solitary wave above a flat bottom and a sloping bottom is given. We find that the energetic dissipation of the solitary wave on the flat bottom is related to the non-linearity parameter '. In term of energetic intensity, low and high frequency components due to the waves generation mode are negligible when compared to the component of the excitation frequency of solitary waves. It is found that experimental results are in good agreement with the literature. The present study on the propagation of individual solitary waves representing the main component of a tsunami waveform should be extended to include other experimental generation mode. Future research needs to be focused on laboratory investigations of waveforms composed of group focused waves. The challenge is to compare these two types of generation.

List of figure

Figure 2 .Figure

 2 Figure 2. Comparison between theoretical and experimental solitary wave form

Figure 6 .

 6 Figure 6. Amplitude evolution on the slope for '=A t /h 0 =0.26 and comparisons with Green's Law

Figure 8 .Figure 7 .

 87 Figure 8. Variation of the amplitude dissipation factor with =A 0 /h 0 on the flat bottom (from 3m up to 9.5m)

  Figure 11. Snapshots of a plunging breaking for a solitary wave. =0.28

Figure 13 .

 13 Figure 13. Maximal run-up amplitude comparisons with equation 13 and equation 14 Hughes (2004) employed the momentum flux parameter to predict maximum runup: (13) Where , is the wave momentum flux parameter (This parameter is easily determined in Hughes (2004)). He noted that this equation gives reasonable predictions as far as the slopes are between 1/10 and 1/30 for ɛ < 0.25. Hsiao et al in 2008 suggested a simple predictive model based on the nonlinear least-square algorithm: (14)

  Figure 14. Runup evolution versus dimensionlesstime for 4 different waves

Figure 1 .Figure 6 .

 16 Figure 1. Sketch of the experimental setup ............ 2 Figure 2. Comparison between theoretical and experimental solitary wave form ............................ 3 Figure 4. Zoom on the main solitary wave; A 0 /h 0 =0.28 (13 m up to 16.7 m from the wave maker) .................................................................... 3 Figure 5. Evolution of the solitary wave (A 0 /h 0 =0.28) with time: instantaneous signals at different distances from the wave maker; a: 6m; b: 8m; c: 10m; e: 12m; f: 14m .................................... 3 Figure 3. Comparison between theoretical (Eq. 2) and experimental solitary wave velocities ............. 3 Figure 6. Amplitude evolution on the slope for '=At/h0=0.26 and comparisons with Green's Law 4 Figure 7. Zoom on the spectrogram of complete solitary wave signal (6 m up to 16.7 m from the wave maker); A0/h=0.28 (a.u: arbitrary units) ....... 5 Figure 8. Variation of the amplitude dissipation factor with =A 0 /h 0 on the flat bottom (from 3m up to 9.5m) .................................................................. 5 Figure 10. Variation of the breaking height with ' 6 Figure 11. Breaking distance evolution with ' ..... 6 Figure 9. Snapshots of a plunging breaking for a solitary wave. =0.28 ............................................. 6 Figure 12. Normalized runup variation with ' ...... 6 Figure 13. Maximal run-up amplitude comparisons with equation 13 and equation 14 .......................... 7 Figure 14. Runup evolution versus dimensionless time for 4 different waves ................. 7 Figure 15. Dimensionless time of runup saturation versus ............................................ 7 Figure 16. Runup velocity evolution ...................... 8


Iskander Abroug

Nizar Abcha
email: nizar.abcha@unicaen.fr

Armelle Jarno

François Marin

Physical modelling of extreme waves Physical modelling of extreme waves: Gaussian wave groups and solitary waves in the nearshore zone

Keywords: / Transfer region local breaking height, As [m]: Solitary wave amplitude, C [m, s -1 ]: Solitary wave celerity, fc [Hz]: Center frequency, Physical modelling of extreme waves 3 fp [Hz]: Peak frequency, fs [Hz]: Characteristic frequency; Eq, 3, g [m, s -2 ]: Gravitational acceleration, h [m]: Water depth, hb [m]: local breaking depth, k [rad, m -1 ]: Wave number, S0: Global wave steepness (at X=4m from the wave maker) Frequency spectrum, Xb [m]: Breaking position, η (x, t) [m]: Free surface elevation, Δf [Hz]: Frequency stepping, φ [rad]: Phase at focusing

Laboratory-scale waves of two distinct types were generated in a twodimensional wave flume to model the spatial evolution of the frequency spectrum in the nearshore zone. The two generated types are Gaussian wave groups of different spectra widths and solitary waves. In the experiment, the time series of free surface elevation along the flume are obtained along a distance of 10m using wave gauges. For each Gaussian wave train, four regions were defined, namely peak, transfer, high frequency and low frequency region referred to as E1, E2, E3 and E4. The repartition was based on the maximum frequency spectrum situated in E1 at X=4m. Focusing is a complex process; this is mainly due to nonlinear energy transfer between different frequency ranges. It was concluded that the energy keeps stable in E1 and spreads toward E3 before breaking. The frequency spectrum in E2 has a decreasing trend during the wave train propagation and this is more appreciable for stronger wave breaking. After the breaking, the frequency spectrum in E2 stops decreasing. It was found also that frequency spectrum increases during the focusing process in E4. Concerning solitary waves, it was found that the frequency spectrum of the main solitary wave decreases as the wave approaches the breaking zone.

Key words: Gaussian wave train/ frequency spectrum/ solitary wave/ peak frequency region

Introduction

Giant waves correspond to waves of very high amplitude, appearing mainly on high seas [START_REF] Kharif | Physical mechanisms of the rogue wave phenomenon[END_REF]. These waves may be accompanied by deep troughs that may appear before or after the highest peak. The appearance of unusually high waves in the nearshore zone is often associated with the transformation of these waves propagating from the open ocean to the shoreline. The physical processes involved in the transformation of these waves from the open sea to the coastal zone are poorly understood, despite the significant impact these waves may have on the coasts, such as overtopping, material damages, and erosion [START_REF] Strusińska-Correia | Nonlinear behavior of Tsunami-Like solitary wave over submerged impermeable structures of finite width[END_REF]. These waves can be investigated experimentally using a large variety of physical mechanisms including solitary waves and group focused waves [START_REF] Kharif | Physical mechanisms of the rogue wave phenomenon[END_REF]. The first description of solitary waves was made by John Russel [START_REF] Russel | Report on waves[END_REF]. These waves are traditionally used as tsunami approximations because of their hydrodynamic similarity [START_REF] Hsiao | Tsunami-like solitary waves impinging and overtopping an impermeable seawall: Experiment and RANS modeling[END_REF]. Group focused waves are often cited in the framework of freak wave formation and have been widely used to study the statistics of rogue waves and their deviation from linear theory [START_REF] Mori | On the extreme statistics of long-crested deep-water waves: Theory and experiments in[END_REF]. The generation mechanism is defined by the fast spreading of long waves and slow propagation of short waves [START_REF] Touboul | Nonlinear focusing wave group on current[END_REF]. The spatial evolution of frequency spectrum of waves in coastal zones is important in assessing danger to life and to marine constructions. However, data are sparse and the phenomenon is hard to investigate, especially in the inner surf zone.

Several studies have analysed the spatial evolution of the frequency spectrum of wave trains using the method of energy focusing [START_REF] Johannessen | a laboratory study of the focusing and transient and directionally spread surface water waves[END_REF][START_REF] Banner | Wave breaking onset and strength for twodimensional deep-water wave groups[END_REF]. Most of these studies were conducted at deep-water conditions. The evolution of the frequency spectrum of group focused waves in intermediate and shallow water depth have received little attention. Rapp and Melville [START_REF] Rapp | Laboratory measurements of deep-water breaking waves in[END_REF] and Tian et al. [START_REF] Tian | Frequency spectra evolution of two dimensional focusing wave groups in finite water depth water[END_REF] found that energy loss due to breaking concerns mostly high frequency regions. By making simple comparisons of measurements upstream and downstream the breaking in deep-water depth, Yao and Wu [START_REF] Yao | Energy dissipation of unsteady wave breaking on currents[END_REF] found that the frequency spectrum increases during the wave train propagation in low frequency regions (E4).

Recently, Merkoune et al. [START_REF] Merkoune | Focusing wave group on a current of finite depth[END_REF] reported an experimental study on energy dissipation of non-breaking focused wave groups in the presence of currents. They found that losses are around 25% of the total energy. These losses are caused by viscous dissipation and contact line. Energy calculations are simply based on free surface elevation measurements and not on detailed frequency spectra, even though the focusing process is far from linear superposition. Yao and Wu [START_REF] Yao | Energy dissipation of unsteady wave breaking on currents[END_REF] found that during the propagation, the frequency spectrum in low frequency regions increases. This gain is attributed to losses in high frequency regions, more precisely in first harmonic region.

The paper is organised as follows. Section 2 describes the experimental setup and wave parameters. Section 3 is devoted to the results for Gaussian wave trains and solitary waves. The discussion is presented in section 4.

Materiel and methods

Experimental setup

The experiments were carried out in a wave flume of the coastal and continental morphodynamics Laboratory at Caen University, France. The flume is 20m long and 0.8m wide. An Edinburgh designs Ltd piston-type paddle was used to generate waves propagating through 0.3 m depth to a sloping bottom of an angle β=0.04 rad situated at X=9.5m downstream the wave maker (Fig. 1). The flume bottom is elevated 1.15m from the ground to ease side visualisations. The governing parameters include the initial solitary wave amplitude A0 (S0 is the input steepness for Gaussian wave trains), the free surface elevation η (x, t), breaking location Xb and frequency range for Gaussian wave trains. The free surface elevations were recorded at different locations along the flume with resistive wave gauges. The first gauge is fixed and used as a temporal reference, the second gauge is moved 20 cm after each test (Fig. 1). Free surface elevations are overlaid chronologically at regular time intervals in order to obtain space-time diagrams. The fast Fourier transformation (called FFT) is computed for each signal measured by the wave gauges using Matlab [START_REF] Baldock | A laboratory study of non-linear surface waves on water[END_REF][START_REF] Gibson | The evolution of large ocean wave: The role of local and rapid spectral changes[END_REF]. Additional video tests have been conducted to track the breaking process with a camera, which was set to record at a frequency of 100Hz. Each experiment requires a time interval of 5 minutes to ensure that the water surface is calm. For group focused waves, the input focusing point was fixed on X=12m in order to have a focusing on the slope. It is important to clarify that the breaking on the slope will prevent complete focusing. Thus, the focusing is not experimentally perfect and so only breaking locations are mentioned in Table 1. Figure 2 presents time histories of the measured free surface elevation of a Gaussian wave train (GW2) at five locations.

Wave parameters

Gaussian wave trains are generated using dispersive focusing technique. In this experiment, we adopted the Gaussian distribution to generate breaking waves in shallow water using wave energy focusing. According to linear theory, the free surface elevation can be written as follows:

𝜂(𝑥, 𝑡) = ∑ 𝑎 𝑖 cos (𝑘 𝑖 𝑥 -𝜔 𝑖 𝑡 + 𝜑 𝑖 ) 𝑛 𝑖=1 (1)
A wave train contains a range of individual linear sine wave components, each with amplitude ai, angular frequency ωi and wave number ki. The phase angle of each component φi was adjusted so that the crests of the individual components coincide at a selected location and time. We assume that the discrete frequency fi (ωi/2π) is uniformly distributed between f1 and f2. Therefore, the frequency width and center frequency are defined by:

Δf=f2-f1; fc= 0.5*(f1+f2) (2)
Four different bandwidths (i.e. Δf/fc) were set in order to investigate the variation of this parameter. The spectrally weighted frequency (fs) is chosen as the characteristic frequency of the wave train and is calculated using equation (3) [START_REF] Tian | Frequency spectra evolution of two dimensional focusing wave groups in finite water depth water[END_REF].

𝑓 𝑠 = ∑ [𝑓 𝑖 𝑎 𝑖 2 𝑛 𝑖=1 (𝛥𝑓) 𝑖 ] ∑ [𝑎 𝑖 2 𝑖 𝑖=1 (𝛥𝑓) 𝑖 ] (3)
Here, fi is the frequency of the i th component of the wave packet and Δfi is the difference between components, which is constant. Unlike solitary waves, wave group's nonlinearity cannot be characterised by a single amplitude, and therefore global wave steepness is used. Equation (4) given in [START_REF] Tian | Frequency spectra evolution of two dimensional focusing wave groups in finite water depth water[END_REF] was chosen to calculate this parameter through spectrum analysis of wave gauge data at X=4m.

𝑆 0 = 𝑘 𝑠 ∑ 𝑎 𝑖 𝑛 𝑖=1 (4)
Here, ks is the spectrally weighted wavenumber related to fs and ∑ 𝑎 𝑖 𝑛 𝑖=1 is the surface elevation at the focusing point according to linear wave theory. With working depth h=0.3m, the value of kch is calculated using the finite depth dispersion relation. Hence, wave trains start propagating in intermediate water depth (i.e. kch=0.93) to coastal zones. Each measurement is decomposed into 90 Fourier components and the correspondent surface elevation is expressed analytically as a summation of sinusoidal waves in order to determine different amplitudes (ai) and frequencies (fi) (Equation ( 5)) [START_REF] Tian | energy dissipation in two-dimensional unsteady plunging breakers and an eddy viscosity model[END_REF].

𝐹(𝑓) = ∫ 𝜂(𝑡)𝑒 -2𝜋𝑖𝑓𝑡 𝑑𝑡 𝑇 0 (5)
Here, F(f) is the Fourier transform of η(t) in a given location, T is the signal duration and is equal to 35s in our study (the time stepping is 0.02s). Careful observation of the free surface elevation shows that the signal duration is sufficient since it includes all meaningful wave components. The wave frequency spectrum is calculated as𝑆(𝑓) = 2|𝐹(𝑓)| 2 /𝑁 where N is the number of points corresponding to the sampling time. For present tests, N=35/0.02=1750 points Boussinesq derived a mathematical expression for the solitary wave profile generated on a horizontal, rectangular channel [START_REF] Boussinesq | Théorie de l'intumescence liquide, appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire[END_REF]. Equation [START_REF] Touboul | Nonlinear focusing wave group on current[END_REF] indicates the nonlinear solution obtained by Boussinesq approximation [START_REF] Goring | Tsunami-the propagation of long waves onto a shelf[END_REF].

ɳ(𝜉) = 𝐴 𝑠 𝑐𝑜𝑠ℎ -2 (√ 3𝐴 𝑠 4ℎ 3 (𝑥 -𝑉𝑡)) (6)
Here, As is the solitary wave amplitude, V the theoretical wave velocity, x is the horizontal distance and t is the time.

C is the wave celerity given by:

𝐶 = √𝑔(ℎ + 𝐴 𝑠 ) (7)
Figure 3 shows a comparison between the experimental time history of the free surface elevation and the time history predicted by Equation ( 6) at X=4m downstream the wave maker. One can observe an excellent agreement between the experimental and the theoretical solution. In the generation of solitary waves, small trailing waves due to the piston pullback are observed behind the main solitary wave.

. The frequency regions

To distinguish between different frequency regions, a repartition has been performed on the initial spectrum S0(f) at X=4m. E1, E2, E3 and E4 are respectively peak, transfer, high frequency and low frequency regions. These four different frequency regions are illustrated in Figure 4.

Figure 5 shows spectrograms of the seven wave trains. These plots represent the frequency spectrum evolution during the wave train propagation. As mentioned above, different distributions have been performed on all spectra since their shapes differ. It should be noted that the repartition was based on the maximum frequency spectrum situated in E1 at X=4m. The lower limit for E4 is 0.6fp and the upper limit corresponds to 10% of the maximum frequency spectrum. The peak frequency region limits are different from one spectrum to another (Fig. 4). The lower and the upper limits have been selected in order to guarantee that all significant frequency spectra around the peak frequency are considered. The transition to E2 is characterised by a decrease of 60% of the maximum frequency spectrum. The upper limit of E2 coincides with 20% of the maximum frequency spectrum. Subsequently, E3 is situated beyond E2. The upper limit was chosen to be equal to 2.5fp. The spectral energy beyond this region is less than 3% of the total frequency spectrum and can be neglected.

Spatial evolution of the frequency spectrum

In this section, all energy forms are normalized by E0, which is the integral of the wave frequency spectrum S(f) between cut-off frequencies 0.6fp and 2.5fp Energy variations are quantified as a function of space by integrating the wave frequency spectrum obtained at every 20cm from X=4m up to X=14m.

The initial frequency spectrum in E1 is between 45% and 60% of the total frequency spectrum. Figure 6 shows that the frequency spectrum in E1 increases slightly on the flat bottom (4m<X <9.5m) for all studied wave trains. Thus, it increases by 8% of its initial level for GW2. After breaking, the spectral energy in E1 decreases for the case of GW2 and GW7 and eventually reaches around 90% of their initial levels. This decrease is primarily caused by dissipation and nonlinear energy transfer to higher frequency regions [START_REF] Tian | Frequency spectra evolution of two dimensional focusing wave groups in finite water depth water[END_REF].

The frequency spectrum in the transfer region (E2) decreases gradually and significantly upstream breaking. This behaviour is similar for all steepness and for all bandwidths of the studied wave trains. After the breaking, the frequency spectrum in E2 stops decreasing and begins to recover. For the two Gaussian wave trains having the same frequency bandwidth (GW1 and GW2), the frequency spectrum decrease in E2 is more obvious for GW2 when compared to GW1. This is more clear for wider wave trains (GW5 and GW6). The frequency spectrum in E2 for GW5 reaches around 85% of its initial level after breaking and for GW6 it reaches around 50% of its initial level after breaking. The frequency spectrum loss in E2 for the two Gaussian wave trains with approximately the same steepness (WG2 and GW6) is approximately the same.

The frequency spectrum in high frequency region (E3) increases gradually during the wave train propagation. For instance, it reaches 30% of its initial level for GW2 at the toe of the slope. The observed energy transfer from the first harmonic to higher harmonics is consistent with earlier studies in [START_REF] Tian | Frequency spectra evolution of two dimensional focusing wave groups in finite water depth water[END_REF][START_REF] Baldock | A laboratory study of non-linear surface waves on water[END_REF]. Therefore, the energy transfer is similar to that in deep water regions. When the wave train breaks, frequency spectrum stops increasing in this region and starts to decrease. This is more obvious for GW2 and GW7. This can be explained by the reversibility of the nonlinear energy transfer between E2 and E3 [START_REF] Tian | energy dissipation in two-dimensional unsteady plunging breakers and an eddy viscosity model[END_REF].

In lower frequency region (E4), the spectral energy increases during the wave train propagation for the seven studied wave trains. This is consistent with [START_REF] Rapp | Laboratory measurements of deep-water breaking waves in[END_REF][START_REF] Tian | Frequency spectra evolution of two dimensional focusing wave groups in finite water depth water[END_REF]. A possible nonlinear energy transfer from E2 and E1 to E4 could offer an explanation

Characteristic frequency

The characteristic frequency is a good predictor of energy transfer and frequency spectrum changes during the focused wave group propagation [START_REF] Liang | Laboratory study on the evolution of waves parameters due to wave breaking in deep water[END_REF]. According to Equation (3), fs is sensitive to energy transfers as long as all the meaningful frequency components are included in its determination. Fourier frequency components in the range of [0.6fp, 2.5fp] Hz are included in the calculation of fs. The characteristic frequency is determined at X=4m and is normalized by the initial characteristic frequency (fs0).

Figure 7 shows the spatial evolution of spectrally weighted frequency. Two vertical solid lines are plotted in the same figure in order to indicate the breaking regions for different wave trains. It is found that fs is stable from X=4m to X=5.5m. On the flat bottom, the characteristic frequency decreases gradually for all Gaussian wave trains, which is more obvious for wide spectrum waves. This decrease is explained by energy dissipation in E1 and E2. Just prior to breaking, fs increases slightly and this is mainly due to spreading in high frequencies and to spectrum broadening. When the wave train breaks, fs decreases, indicating a decreasing trend in high frequency regions. The characteristic frequency almost returns to its initial value after breaking. However, it does not recover to its initial value for the two weakest wave trains (GW1 and GW3) since they break at X>13.5m.

Solitary wave

Figure 8 presents the spectrogram for SW4, on which the narrow band at f=0.28Hz (Es) corresponds to the spatial evolution of the frequency spectrum of the solitary wave. One can see a second band at approximately 0.75 Hz, which corresponds to the frequency spectrum of the wave generated by the piston pullback. In terms of frequency spectrum, this band and the following bands of higher frequencies (1 Hz and 1.25 Hz) are less intense than the main band (0.28 Hz). Higher frequency waves travel slower than lower frequency waves and reach the shore after the solitary wave [START_REF] Orszaghova | Importance of second-order wave generation for focused wave group run-up and overtopping[END_REF]. Therefore, high frequency waves do not have a significant impact on the spatial evolution of the main solitary wave. Furthermore, the band at approximately 0.15Hz corresponds to low frequency waves. These subharmonic waves are present along the flume and are amplified by shoaling when arriving on the slope. However, they are approximately three times less energetic than the main solitary wave. Hence, we consider that the subharmonic wave effect is not significant. The frequency spectrum of the main solitary wave is mainly situated between 0.7fp and 1.5fp. Figure 9 shows that the frequency spectrum in Es decreases gradually on the flat bottom (4m<X<9.5). It is important to clarify at this point that the decrease rate slightly depends on the initial solitary wave steepness (A0). The frequency spectrum reaches around 97% of its initial level for SW1 and around 93% of its initial level for SW4 at X=9.5m from the wave maker. This decrease is intensified when the solitary wave reaches the slope and is explained by the shoaling phenomenon, which has an impact on the solitary wave shape. This can be seen clearly by the narrowing of the solitary wave spectrum in Figure 8. For instance, this energetic decline reaches around 40% of its initial level for SW2. We can conclude from Figure 9 that the frequency spectrum loss is slightly impacted by the initial solitary wave steepness and the decreasing rate on the sloping bottom is fairly similar for the four solitary waves.

Discussion

This paper describes the spatial evolution of Gaussian wave groups and solitary waves along a flat and sloping bottom using a series of physical experiments. Different breaking intensities, center frequencies and frequency widths are considered for Gaussian wave trains. The spatial evolution of frequency spectrum upstream and downstream the wave breaking is studied for each test, thanks to 50 measurements carried out by wave gauges. On the one hand, four energy regions are defined with the aim of monitoring the spatial evolution of frequency spectrum upstream and downstream the breaking for Gaussian wave trains. The repartition was based on the maximum frequency spectrum at X=4m. It was found that the energetic behavior is relatively similar for the seven generated wave trains and this confirms the limits that were defined. On the other hand, solitary waves are only described by one frequency region.

Based on our experimental data, the energy dissipation for solitary waves on the flat bottom (4m<X <9.5m) is small. For the steepest solitary wave, the energy dissipation is around 10%. These losses are caused by viscous dissipation and contact-line damping. Concerning Gaussian wave trains, the total energy dissipation is insignificant on the flat bottom and is around 8% for the strongest wave train GW7. The total energy keeps stable along the 5.5m for the weakest wave trains and this is consistent with earlier studies in [START_REF] Liang | Laboratory study on the evolution of waves parameters due to wave breaking in deep water[END_REF]. It should be mentioned that in our experiments losses of energy are smaller than in [START_REF] Merkoune | Focusing wave group on a current of finite depth[END_REF]. This difference found in [START_REF] Merkoune | Focusing wave group on a current of finite depth[END_REF] can be explained by the larger influence of viscose boundary layers at sidewalls and on the bottom since our measurements are performed at larger flume width.

On the sloping bottom (9.5m<X <Xb), the shoaling happens. The energy behaviour differs for the two wave types. In fact, the main solitary wave is marked by a strong energy decrease. For Gaussian wave trains; however, the energetic increase in higher harmonic regions (E3) and the energetic decrease in the transfer region (E2) are intensified which indicates a stronger energy transfer than that on the flat bottom.

Along the wave train propagation, the energy decreases in higher frequencies of the controlling frequencies range and increases slightly in the peak region.When the wave trains breaks, the frequency spectrum increase in E3 ceases and begins to decrease. This is consistent with earlier studies [START_REF] Baldock | A laboratory study of non-linear surface waves on water[END_REF][START_REF] Liang | Laboratory study on the evolution of waves parameters due to wave breaking in deep water[END_REF]. Thus, the behaviour of spectral energy in E2 and E3 in intermediate and shallow water depth is close to that in deep water depth.

For the near future, we plan to perform new tests involving other spectra such as Pierson-Moskowitz and JONSWAP spectrum, in order to extend our experimental results to more realistic spectra.

List of figures

Figure 1 :

 1 Figure 1: Experimental setup

Figure 2 :

 2 Figure 2: Free surface elevation as measured by wave gauges of GW2

Figure 3 :

 3 Figure 3: The measured water surface elevation compared to the Boussinesq solitary wave of the same maximum height

Figure 4 :Figure 5 :

 45 Figure 4: Typical repartition of the initial spectrum S0(f) for GW1

Figure 6 :

 6 Figure 6: Spatial evolution of frequency spectrum for (a): GW1, (b): GW2, (c): GW3, (d): GW4, (e): GW5, (f): GW6, (g): GW7. ■ frequency spectrum in E1; • frequency spectrum in E2; ★ frequency spectrum in E3; ♦ frequency spectrum in E4

Figure 7 :

 7 Figure 7: Characteristic frequency spatial evolution

Figure 8 :

 8 Figure 8: Spectrogram of SW2 as obtained by wave gauges through FFT. The colorbar indicates the wave frequency spectrum S(f) (arbitrary units) from X=4m to X=14m

Figure 9 :

 9 Figure 9: ■ spatial evolution of frequency spectrum for SW1; • SW2; ★ SW3; ♦ SW4

Figure 1 :

 1 Figure 1: Experimental setup ................................................................................................... 5

Figure 2 : 5 Figure 3 :

 253 Figure 2: Free surface elevation as measured by wave gauges of GW2 ..................... 5

Figure 4 :

 4 Figure 4: Typical repartition of the initial spectrum S0(f) for GW1............................. 6

Figure 5 : 6 Figure 6 :

 566 Figure 5: Spectrograms as obtained by wave gauges through FFT (a) GW1, (b):

Figure 7 : 6 Figure 8 :

 768 Figure 7: Characteristic frequency spatial evolution ........................................................ 6

Figure 9 :

 9 Figure 9: ■ spatial evolution of frequency spectrum for SW1; • SW2; ★ SW3; ♦

Table 1 :

 1 Gaussian wave trains parameters

		Control frequency
	Wave train	range (Hz)	fc(Hz) Δf/fc	S0	fp(Hz)	Xb(m)
	GW1	[0.6..0.9]	0.75	0.4	0.10	0.63	14.3
	GW2	[0.6..0.9]	0.75	0.4	0.17	0.63	[13.1..13.61]
	GW3	[0.55..0.95]	0.75	0.53 0.12	0.67	13.6
	GW4	[0.5..1]	0.75	0.67 0.13	0.54	13.42
	GW5	[0.4..1.1]	0.75	0.93 0.11	0.46	13.81
	GW6	[0.4..1.1]	0.75	0.93 0.17	0.46	[12.9...13.4]
	GW7	[0.7..1.2]	0.95	0.53 0.15	0.77	[13.11..13.8]

Table 2 :

 2 Solitary waves parameters

	Solitary wave	A0/h A'/h	Ab/hb	hb/h	Xb(m)
	SW1	0.22 0.18	1.56	0.21	15.45
	SW2	0.26 0.21	1.47	0.24	15.21
	SW3	0.28 0.24	1.43	0.27	15.04
	SW4	0.3	0.26	1.35	0.3	14.75

Table 1

 1 indicates the different parameters corresponding to the seven Gaussian wave trains and Table2presents the parameters for the four solitary waves generated during the physical experiments.

Acknowledgments

The authors acknowledge financial support from the Normandy.


Ljerka Jukić Matić

Dubravka Glasnović Gracin
email: dubravka.glasnovic@ufzg.hr

The Influence of Teacher Guides on Teachers' Practice: A Longitudinal Case Study

Keywords: Longitudinal case study, teacher guide, textbook mobilization

The issues concerning the use of curriculum resources such as teacher guides and textbooks are important because they can promote a teacher's pedagogical design capacity, or a teacher's ability to use personal resources to adapt the curriculum materials and to achieve productive and beneficial instructional episodes in the classroom [START_REF] Davis | Designing educative curriculum materials to promote teacher learning[END_REF].

From the idea of teaching as design, [START_REF] Brown | The teacher-tool relationship: Theorizing the design and use of curriculum materials[END_REF] built a Design Capacity for Enactment Framework (DCE) which represents the idea that the curriculum resources and a teacher's personal resources both affect the designing and enacting of classroom instruction. Viewed together, the factors of the curriculum resources and teachers' resources provide a basis for understanding teachers' use of curriculum resources in terms of offloading, adapting, and improvising. These three concepts are different types of mobilization of curriculum resources and are best understood as a spectrum of agency or responsibility for guiding instruction [START_REF] Brown | The teacher-tool relationship: Theorizing the design and use of curriculum materials[END_REF]. The voice of the curriculum resource refers to how the authors or designers of the materials are represented and how they communicate with the teacher [START_REF] Remillard | Modes of engagement: Understanding teachers' transactions with mathematics curriculum resources[END_REF]. Remillard (ibid) distinguishes between talking through the teacher and speaking to the teacher. Most curriculum resources place primary emphasis on what the teacher should do and this is denoted as talking through teachers. When the resource communicates with teachers about central ideas in the curriculum, this denotes speaking to the teacher. Some authors have argued that speaking to teachers is one way that curriculum resources can be designed to be educative for teachers [START_REF] Davis | Designing educative curriculum materials to promote teacher learning[END_REF]. In this way, a curriculum resource offers transparent and direct guidance related to the reasons and purposes underlying task selections (design rationale) or in anticipating student responses [START_REF] Stein | The role of mathematics curriculum materials in large-scale urban reform: An analysis of demands and opportunities for teacher learning[END_REF].

Previous results on the utilization of resources in mathematics instruction in Croatia [START_REF] Glasnović Gracin | Requirements in mathematics textbooks and PISA assessment[END_REF] show that the textbook is the primary resource in teachers' lesson preparation; the next most frequently used resource is the accompanying teacher guide. In 2014, new school textbooks were approved by the Croatian Ministry of Education for the subsequent four-year school cycle. Since the planned curriculum had not been changed, the new editions of the most frequently used mathematics textbook series did not undergo significant changes. However, in line with recommendations from the Ministry of Education, the teacher guides that accompany the textbooks were revised to include, for example, opportunities for discovery learning. This situation, whereby the teacher guides were revised in 2014, while the planned curriculum and the textbook content remained the same, prompted us to conduct an investigation into the influence of teacher guides on teachers' practice. The study encompassed the following questions: 1) What are the differences in the structure, textbook mobilization and voice between the two editions of the teacher guide? 2) How does the teacher guide influence the use of the textbook in the observed mathematics lessons? Do the changes in the content of the new edition of the teacher guide cause a change in teacher practice?

The study was conducted in two time spots, in 2013 and 2017. It encompassed analysis of the two editions of the teacher guide for grade six (published before and after 2014), classroom observations, and interviews with two experienced mathematics teachers in Croatia. Both of the teachers chosen for this study used the same textbook series in 2017 as they did in 2013. The analysis of the transcribed interviews, observation notes and teacher guide content focused on the different mobilizations of resources (offloading, adapting, improvising). The interviews also provided an opportunity for the teachers to explain why a particular type of resource mobilization was used.

The results show the relative stability of teaching practice, but suggest, however, that the teacher guide can have an influence on the instruction. The analysis of the teacher guides showed that both editions have the same structure. The voice of the guides can be described as talking through the teacher, but the design of the activities in the new edition can be defined as transparent. Moreover, the new edition contains additional activities and approaches such as discovery learning. While for most lessons the old teacher guide encourages offloading, the new guide additionally suggests numerous activities that do not use the textbook as a mediator, which promotes adapting and improvising with curriculum resources. In terms of the teachers' practice, the results show that the new edition of the teacher guide inspired the participants to design their own inquiry-based learning activities and that offloading is utilized less often. This indicates that the teacher guide can have an educative impact on teachers, supporting their use of active teaching methods in the classroom.


Berie Getie Kassa
email: beriekassa@gmail.com

Liping Ding
email: liping.ding@ntnu.no

The use of variation theory in a problem-based task design study

Keywords: Variation theory, patterns of variation and invariance, problem-based task design

The aim of this pilot study is to test the usefulness of variation theory as a theoretical framework in guiding the researcher to design the main study of developing students' deep learning in mathematics. Both Marton's variation theory of learning and Gu's theory of teaching with variation were used in the design-based pilot study. Two teachers and 106 Grade four students from an elementary school in a large north-west city in Ethiopia participated in the pilot study. The pilot study data included transcripts of audio-recording of preparatory meetings (the first author with the two teachers) and students' interview, video-recording of lesson observation, and pre and post-tests. The pilot study topic was of learning concepts of area and perimeter of rectangles. We find that the theoretical framework of variation helped us to address the systematic use of patterns of variation and invariance in the problem-based task design and to analyze the three patterns of variation in the problem-based tasks implementation in the classroom learning.

Introduction

School mathematics curriculum has been significantly reformed from kindergarten to upper secondary school in Ethiopia. Consequently, school mathematics is now viewed from a dynamic and experimental perspective rather than only as a body of strict and logical knowledge already in existence [START_REF] Moe | Ethiopian Fourth National Learning Assessment (EFNLA) of Grade 4 & 8 Students, Data Analytic Report[END_REF]. However, the national learning assessments have constantly shown that students have achieved poor results in all areas, and their achievement in mathematics is the least [START_REF] Moe | Ethiopian Fourth National Learning Assessment (EFNLA) of Grade 4 & 8 Students, Data Analytic Report[END_REF]. For example, the fourth national learning assessment (FNLA) result has shown that the overall test performance of grade 8 students in 2013 was far below the national minimum achievement standard of 50%, and their achievement in mathematics is the lowest (the composite average became 25.65%), where 50% of the students scored at or below a score of 22.5% similar in both sampled grades 4 and 8 [START_REF] Moe | Ethiopian Fourth National Learning Assessment (EFNLA) of Grade 4 & 8 Students, Data Analytic Report[END_REF]. According to the FNLA reports in 2013, the education of mathematics should receive special emphasis in all deliberations, for example, the use of teaching and learning resources, and professional development of teachers about innovative pedagogies.

The overall objective of this study is therefore to make a contribution to developing the quality of mathematics teaching and learning in Ethiopia. This pilot study is to test the usefulness of the variation theory as a theoretical framework in guiding the researcher to design and implement problem-based tasks to improve students learning in mathematics. In this paper, we mainly focus on the following two research questions: (1) to what extent does the theoretical framework help to design the instruction of problem-based tasks? (2) to what extent does the theoretical framework help to analyze the implementation of such design in the classroom?

Literature Review and Theoretical Framework Marton's Variation Theory of Learning (VToL)

Ference Marton and his colleagues in Sweden and Hong Kong have developed Variation Theory of Learning (VToL) to describe how a learner might come to see, understand, or experience a given phenomenon in a certain way [START_REF] Marton | Necessary conditions of learning[END_REF][START_REF] Marton | Learning and awareness[END_REF][START_REF] Marton | On some necessary conditions of learning[END_REF][START_REF] Marton | Classroom discourse and the space of learning[END_REF]. For instance, [START_REF] Marton | Learning and awareness[END_REF] emphasized that one cannot become aware of new aspects without becoming aware of differences. If certain aspects of a phenomenon are varied while its other aspects are kept constant, then those that vary will be noticed. Every concept, situation or phenomenon has particular or critical aspects, and it is important to identify the critical aspects and the focus of students' attention should always be directed to the critical aspects [START_REF] Lo | Variation theory and the improvement of teaching and learning[END_REF]. In order for students to discern the critical aspects of a concept(s), they need to experience different forms of variation in which the critical aspects of the concept(s) will be varied and not varied. Moreover, VToL addresses that for learning to occur, some critical aspects of the object of learning must vary while other aspects remain constant. According to VToL, the specific knowledge/content or capability to be gained through learning is called the object of learning [START_REF] Lo | Variation theory and the improvement of teaching and learning[END_REF][START_REF] Marton | On some necessary conditions of learning[END_REF][START_REF] Marton | Classroom discourse and the space of learning[END_REF]. VToL takes the object of learning as the point of departure and claims that what is being learned is highly influenced by what actually comes to the forefront of students' attention. Kullberg, Runesson and Marton (2017, p. 561) explained that "when analyzing how the object of learning is handled during teaching, the 'intended', the 'enacted', and the 'lived' objects of learning are used to differentiate between the teacher's particular goal regarding what the students should learn, what is made possible to learn in the lesson, and what the learners actually learn". The theory further suggests that how students perceive a specific object of learning depends on what pattern of variation and invariance is provided by the teacher. Four patterns of variation and invariance are identified: contrast (i.e., recognizing differences between two values of an aspect), generalization (i.e., keeping the focused feature invariant and varying other out of focus aspects), separation (i.e., separating aspects with varying values from invariant aspects), and fusion (i.e., experiencing several critical aspects simultaneously) [START_REF] Leung | Variation and mathematics pedagogy[END_REF][START_REF] Lo | Variation theory and the improvement of teaching and learning[END_REF][START_REF] Marton | Classroom discourse and the space of learning[END_REF]. [START_REF] Marton | Necessary conditions of learning[END_REF] further proposed a new pattern of variation in which only three of the four patterns, i.e., contrast, generalization, and fusion exist as the basic ones, which are observed by learners.

Gu's Teaching with Variation (TwV)

In parallel with Marton's theory of variation, a theory of mathematics teaching and learning, called Teaching with Variation (TwV) (bianshi jiaoxue in Chinese), has been developed by Gu Ling-yuan and his colleagues through a series of longitudinal mathematics teaching experiments in China [START_REF] Gu | Theory of teaching experiment: The methodology and teaching principles of Qingpu[END_REF]. According to Gu's theory, which was highly inspired by the theory of cognitive constructivism, meaningful learning empowers learners to establish a significant and non-arbitrary connection between their new knowledge and their previous knowledge [START_REF] Gu | Teaching with variation: An effective way of mathematics teaching in China[END_REF]. Classroom activities can be designed to help students to create this kind of connection by experiencing certain dimensions of variation. Gu's theory of TwV suggests that two types of variation are helpful for meaningful learning. One is called "conceptual variation", and the other is "procedural variation" [START_REF] Gu | Teaching with variation: An effective way of mathematics teaching in China[END_REF]. Conceptual variation refers to understanding concepts from multiple perspectives. Procedural variation is progressively unfolding mathematics activities. That is, teaching process-oriented knowledge by enhancing the formation of concepts step-by-step, and experiencing problem-solving from simple problems to complicated problems [START_REF] Gu | Teaching with variation: An effective way of mathematics teaching in China[END_REF].

Based on the above literature, we formed an initial analytical framework (see Table 1) to guide us in designing an instruction and implementing such design in the pilot study.

Theories Theoretical terms

The use of the theoretical terms in both design and implementation of an instruction of problem-based tasks in the pilot study Gu's TwV [START_REF] Gu | Teaching with variation: An effective way of mathematics teaching in China[END_REF]  conceptual variation  procedural variation  non-arbitrary connection b/n new and prior knowledge  systematic and step-by step way of problemsolving 1. Design of the instruction of the problem-based tasks:  Conceptual variation (concept and non-concept variation).  Procedural variation (varying methods, problem transformations, connection within and between the problems) (for details see finding one below).

Marton's VToL [START_REF] Marton | Necessary conditions of learning[END_REF][START_REF] Marton | Classroom discourse and the space of learning[END_REF]  object of learning (intended, enacted and lived object of learning)  critical aspects of the object of learning  variation and invariance  Three patterns of variation (contrast, generalization, and fusion)

2. Implementation of such design  The analysis focused on how the designed tasks are planned (intended), taught (enacted), and learned(lived) (explained in the findings section).  Two critical aspects of an object of learning were designed (see Table 2).  Each critical aspect was brought into focal awareness through the systematic use of variance and invariance of values of each aspect, and all were experienced and discerned simultaneously (for details see finding two below).

Methodology Research Design

This study was based on the principles of design research [START_REF] Cobb | Design research: An analysis and critique[END_REF]. According to [START_REF] Cobb | Design research: An analysis and critique[END_REF], design research is a research approach that has been increasingly appreciated in mathematics education research. It can help to construct the instructional ways of promoting learning of a particular concept, while regularly studying the development of that learning, considering all components of the instructional ways, including the context in which it is carried on. According to [START_REF] Cobb | Design research: An analysis and critique[END_REF], some of the main characteristics of design research approach is its highly interventionist nature, iterative design and cyclic nature. Each cycle progresses in three phases: (i) preparation and design of teaching experiments; (ii) implementation of teaching experiments; and (iii) retrospective analysis that can lead to revisions and a new cycle [START_REF] Cobb | Design research: An analysis and critique[END_REF]. In this pilot study, two teaching experiments, in two fourth grade classes of 106 students, were conducted in their regular classes, during a 45 minutes period for each class. The two classes were taught by the same teachers and the study design is illustrated in Figure 1.

Data Analysis Procedure

We followed the three phases of each cycle progresses [START_REF] Cobb | Design research: An analysis and critique[END_REF] in our data analysis. The data analysis mainly focused on how the object of learning was designed and implemented in terms of patterns of variation and invariance of critical aspects of the object of learning within and between units of teaching, such as the problem-based tasks (for details see the analytical framework in Table 1). The analyses were made by watching the video recorded lessons and meetings, repeatedly focusing on what was planned (the intended), what was possible for students to discern (the enacted) and what the students actually learned (the lived object of learning). In addition, pre and posttests, students' worksheets and students' interview were also analyzed to find out students' discernment of the object of learning. The analysis of the first class (see the intervention cycle 1 in Figure 1) is based on the data of the study group's initial lesson plan, namely the problem-based task design and implementation in the class. The analysis of the second class (see the intervention cycle 2 in Figure 1) concerns more on the usefulness of the redesign of tasks, the effective design and use of tools, and the impact on students learning. Given the space of this paper, we will chiefly focus on our research questions in the analysis of the first class.

Findings

Finding one and two below focus on reporting our retrospective analysis of the problem-based task design (namely the stage of the preparation and design of teaching experiments of the first lesson); and classroom implementation of such design in the first lesson.

Finding one: The usefulness of the variation theory in designing the problem-based tasks

In this pilot study, we focused on designing problem-based tasks for improving students' mathematical understanding of the concept of area and perimeter of rectangles. The problem-based tasks were designed by referring to both Gu's terms of conceptual and procedural variations and Marton's terms of variation/invariance (see Table 1). The sequence of tasks used in the classroom instruction were designed with reference to the two problems and critical aspects shown in Table 2.

Object of learning

Critical aspects Problems used to design sequences of tasks in the lesson

The relationship between area and perimeter of rectangles

(1) If the perimeter of rectangles is the same, then the area does not have to be the same.

(1) The figures below are two bars of biscuits. If the price of these biscuit bars is the same, which biscuit do you want to buy? Why do you choose that biscuit? What is your strategy for choosing the biscuit you want to buy? Explain your answer. Use the given grid paper to help you compare the shapes.

(2) A rectangle with dimensions closest together or the same (square) is the largest.

(2) A farmer has a 16 meters long fence and he plans to enclose a rectangular grazing field with it. Can you help the farmer to think of a way to enclose the largest grazing field? At what condition is the field smallest?

Table 2: The link between the problems and critical aspects of the object of learning

Conceptual variation was carried out to firstly emphasize certain concepts of rectangle such as its area and perimeter and hence the concepts could be more readily picked up for use in finding the relationship between area and perimeter of rectangles. In terms of procedural variation, the problems were transformed into simpler problems, so that students could solve them step by step in a progressive way, using different methods and that helped them to answer the original problem. In addition, the second more advanced problem was designed to scale up the methods/solution techniques used by students in the first problem (connecting new and prior knowledge). Students were also used their prior measurement techniques as a procedure to find area and perimeter of rectangles. Both procedural and conceptual variation was also carried out in connecting the abstract concept, for example, mathematical formula of area (A = length x width), directly to its concrete correspondence (counting unit squares).

Marton's theoretical notion 'the object of learning' [START_REF] Marton | Classroom discourse and the space of learning[END_REF] was also used in the pilot study in that the study group aimed to develop students' learning of the relationship between the concepts of area and perimeter of rectangles. Moreover, the study group also applied the critical aspects of the object of learning to further understand students' learning (see Table 2). The two problems shown in Table 2 were designed to bring the two critical aspects into the focal awareness of learners so that they could discern them step by step. The two problems were also used to guide the design of the sequence of tasks in the classroom instruction.

Finding two: The usefulness of the theories of variation in analyzing the implementation of the designed tasks.

The implementation of the designed instruction was guided by both Marton's VToL and Gu's TwV. For example, the teacher used procedurally varied approaches in implementing each of the designed tasks, in a systematic and progressive way. At the beginning of each task, the teacher asked students to conjecture the answers to each problem individually and then to compare it with their colleague's work. The teacher then conducted a discussion with students on their answers without giving any hint on the correctness of students' suggestions and did not give them the correct answers. The teacher then asked students to start a group investigation of the posed problems in a small group. Students were also encouraged to use the different tailor-made resources made available by the teacher to discover patterns in their answering. Each task was finally concluded by a whole class discussion followed by summary given by the teacher. [START_REF] Marton | Necessary conditions of learning[END_REF] term of 'contrast' (i.e., recognizing differences between two values of an aspect) was used to guide the task implementation. For example, students were asked to draw rectangles by fixing one dimension of the rectangle (say, length) and varying the other (width) and vice versa to find the area and perimeter of each using resources such as strings and unit square tiles. At this stage the concepts of perimeter and area of rectangles were discerned as area is the amount of surface of a region, and perimeter is the distance around the region. In addition, students were able to realize the change in area/perimeter due to the change in length/width of rectangles.

Secondly, [START_REF] Marton | Necessary conditions of learning[END_REF] term of 'generalization' (i.e., keeping the focused aspects invariant and varying other out of focus aspects) was focused in the classroom. The analysis of students recording sheet showed that most students discovered the formula for finding the area and perimeter of rectangles easily, and they were also able to understand how area formula is related to counting unit squares covering the rectangular region. Some of students' correct findings and generalizations presented in class and written in the record sheet were: the perimeter of a rectangle equals the sum of all of its sides and this can be calculated as: perimeter of a rectangle = (2 x length + 2 x width); the area of a rectangle is the total number of unit grid-squares inside the rectangles and this can be calculated as: area of a rectangle = length x width. Students were also able to compare rectangular shapes with fixed perimeter. At this stage students discerned the first critical aspect, that is, if the perimeter of rectangles is the same, then the area does not have to be the same. Most students were able to change their conjecture for problem one that the biscuit with largest side (the first one) was chosen to buy. In the recording sheet, most students wrote a generalization that rectangles with the same perimeter can have different area and vice versa.

Finally, [START_REF] Marton | Necessary conditions of learning[END_REF] term of 'fusion' (i.e., experiencing several critical aspects simultaneously) was highlighted. For instance, the relationship between the two concepts was shown by varying certain aspects (length and width) while keeping an aspect fixed (perimeter of rectangles). At this stage students were able to identify the largest/smallest rectangle among all rectangles with the given fixed perimeter. Most students were also changed their conjecture on identifying the largest grazing field with the given fixed length fencing material in problem two. In the recording sheet, most students wrote a generalization that a rectangle with a fixed perimeter would have the largest area when its length and width are equal (a square).

Discussions and Conclusions

In this paper, we reported our retrospective analysis [START_REF] Cobb | Design research: An analysis and critique[END_REF] of using the theory of variation in both the designed instruction and the implementation of such design in the pilot study (see Table 1 & Figure 1). Our data analysis shows that the joint consideration of the two different but compatible frameworks of variation (see Table 1) helped the study group to focus on the systematic use of patterns of variation and invariance in the design and analysis of the implementation of the problem-based tasks in the classroom learning. In particular, it enables the researcher to analyze the implementation of such design to see how the students may experience and discern the critical aspects of the object of learning and to improve their learning in mathematics. The retrospective analysis of the pilot study would lead the researcher to revisions and new cycles for the main study. In particular, we aim to develop a more sufficient understanding of interrelationship of Marton's terms of 'the critical aspects of the object of learning', and the patterns of variation and invariance [START_REF] Marton | Necessary conditions of learning[END_REF]) and Gu's terms of conceptual and procedural variation [START_REF] Gu | Teaching with variation: An effective way of mathematics teaching in China[END_REF] with the design-based approach [START_REF] Cobb | Design research: An analysis and critique[END_REF], to enable us to design and implement the problem-based tasks in the main study for engaging students in mathematical learning in a systematic process and at the same time to develop their own learning in mathematics.

The analysis of the pilot study also showed that the implementation of problem-based tasks with patterns of variation and invariance helped students to improve their understanding of the critical aspects of the object of learning (the relationships between area and perimeter of rectangles) significantly. After the implementation of the first lesson, as can be evident from the researcher's observation and video recording of lessons, most students were actively engaged in individual and group discussions and some students were able to grasp the concept quickly and began to make interesting conjectures from experiencing variations in the tasks. For example, most students showed progress in answering each question of the post-test with correct explanations. It was found that there were percentage increases of students understanding in discerning all the critical aspects (49% and 26.4% of students provided correct explanations, 47.2% and 56.6% of students provided partially correct answers, in posttest questions related to the first and second critical aspects respectively), and paired-samples t-test result informed that it was statistically significant (with the sig. level, 0.01) when it is compared with their pre-test results. In addition to their significant progress in the post-test, during the post-lesson interview students said that they had achieved a better understanding of area and perimeter of rectangles by actively involving in the tasks. However, there are also few students who do not discern the critical aspects (3.8% and 17% of students were not able to answer the first and second questions of the posttest respectively). This is supported by the researcher's qualitative analysis result that the intended, enacted and lived object of learning sometimes did not coincide. According to [START_REF] Kullberg | What is made possible to learn when using the variation theory of learning in teaching mathematics?[END_REF], well-designed tasks and examples, and patterns of variation (differences) and invariance(sameness) in teaching, however, are insufficient and does not guarantee learning, it can, at best, make it possible for learning to happen. From our analysis, the two theories can, however, offered tools that the study group can use to focus on the mathematical content taught, students' understanding of it and how to enable possibilities for learning.

Figure 1 :

 1 Figure 1: The research process in the pilot study

Table 1 : The analytical framework of Gu's and Marton's work of 'variation'

 1


Dániel Katona
email: danikatona@gmail.com

Web of problem threads (WPT) -a theoretical frame and task design tool for inquiry-based learning mathematics

Keywords: web of problem threads, connected task-design, anthropological theory of the didactic, reverse didactic engineering

Web of problem threads (WPT) -a theoretical frame and task design tool for inquiry-based learning mathematics

Goal, Context and Methodology

The poster presented the basic concepts and a sample of the web of problem threads (WPT), the core element of a theoretical framework under development, and a tool for task-design in mathematics education, based on the qualitative analysis of the Pósa-method for inquiry-based learning mathematics. The WPT is the result of the research aiming at, on the one hand, theorizing the 'intuitively' developed Pósa method, a 'good practice' as it is widely used in Hungarian talent care education, and on the other hand, providing theoretical background for the development and reconstruction of the method to be applied in public education.

The term Pósa method refers to the nationally well-reputed three-decade-long teaching practice of Lajos Pósa, in Hungarian out-of-school weekend mathematics camps for highly talented 12-18 years old students, who form a study group for 6 years. They solve connected mathematical problems (tasks) of various kinds (regarding content area), but the focus is on discovering and discussing the mathematical ideas connecting the problems and the corresponding 'ways of thinking', or kernels, according to our WPT framework. The Pósa-method was developed "based in teaching as craft knowledge" (Watson & Ohtani, 2015, p. 5), lacking the construction of, or building on any theoretical framework. There is also a demand (by the Hungarian Academy of Sciences) for the re-design of the method to be applied in public education, based on a theoretical background to be built. The main goal of the research is to subsequently (re)construct the theoretical frame and the tools of the task-design of the Pósa method. The term 'reverse didactic engineering' is suggested for this research methodology (also based on the discussion in TWG 17 of CERME11).

Theoretical Background, Steps of Theorizing and some Results

Based on mathematical content analysis, the web of problem threads has been theorized as the first step of theorizing, focusing on specific kinds of connections, common features (called the kernels) between the mathematical tasks (problems). According to this theorization, a set of connected tasks, in a partially fixed order, creates a thread of the problems. Kernel is the manifestation of a kind of connection (common feature) that creates the thread (that can actually have multiple kernels). As some problems belong to several threads, threads cross each other, forming a web, the WPT. The problems are selected and created (partially) for giving birth to the kernels, and not (usually) for their own sake.

In the 2 nd chapter of the 22 nd ICMI study (Watson & Ohtani, 2015), C. Kieran, M. Doorman and M. Ohtani categorize theoretical frames into grand, intermediate-level, and domain-specific frames.

Our study aims at (re)constructing an intermediate-level framework, which we call the Theory of WPTs (TWPT), and links this to already established frameworks, such as the anthropological theory of the didactic (ATD) and its application, known as study and research paths (SRP) (Chevallard, 2007;[START_REF] Bosch | Introduction to the Anthropological Theory of the Didactic (ATD)[END_REF]Watson & Ohtani, 2015, pp. 260-272). The first step of theorizing will in a second step be analyzed through the lenses of ATD, where WPT is considered as part of the technology and theory elements of the studied praxeologies [START_REF] Bosch | Introduction to the Anthropological Theory of the Didactic (ATD)[END_REF], which, similarly to SRPs, focuses on the connections between tasks (or questions). Therefore, considering 'distribution as a dilemma' in the categorization of design elements of tasks along 5 dilemmas by P. Sullivan, L. Knott and Y. Yang in the 3 rd chapter of the 22 nd ICMI study, both taskdesign approaches favour creating "doing mathematics" tasks (Watson & Ohtani, 2015, pp. 91-94).

The poster is to present a sample of a WPT (see Figure 1 below) linked preliminarily to the ATD, with tasks, solutions, and the analysis of the highlighted 'kernels' yieldingness and invariant (quantities). In Task B (Can you tell a power of 3 that ends with 127?), we do not need to consider divisions by 10 3 , it is enough (yieldingness) by it's divisor, 8, as any power of 3 divided by 8 gives 1 or 3 as the remainder.

Figure 1 :

 1 Figure 1: A sample of the Pósa WPT Some other kernels in the TWPT are experimentation, bounds (upper and lower), recursion (recursive thinking), induction, and proof of impossibility. The collection and analysis of a (more) complete set of kernels is one of the main future goals of the present research.


Zeger-Jan Kock

Birgit Pepin

Secondary school mathematics teachers' selection and use of resources

Keywords: Teacher resources, Secondary school mathematics, Survey

Introduction

Secondary school mathematics teachers rely on multiple resources to select materials for classroom use, prepare lessons, and design their curriculum. These include traditional curriculum resources (e.g. textbooks), social resources (e.g. discussions with colleagues) and various digital resources, such as mathematics apps, e-textbooks [START_REF] Pepin | Digital curriculum resources in mathematics education: Foundations for change[END_REF], and online platforms for sharing digital materials, and constructing lesson series and curricula [START_REF] Gueudet | Online platforms: New connections and new possibilities for teacher documentation work? Poster presented at the 5th ERME topic conference: Mathematics education in the digital age[END_REF]. A plethora of digital resources have become available for teachers (Pepin & Gueudet, 2018). Understanding the role of resources in mathematics teachers' work may inform the design of (digital) resources and platforms. To foster this understanding knowledge about the actual use of the various resources is important. In the Netherland, a survey on the use of learning materials has taken place periodically [START_REF] Blockhuis | Leermiddelenmonitor 15/16: Leermiddelen in het po en vo[END_REF]. However, this general survey does not focus on mathematics teachers and the specific combinations of resources they select.

In this exploratory survey study, we focused on the resources selected and used by secondary school mathematics teachers to prepare their lessons, including resources they use to find materials for classroom use (e.g. student tasks). The research question was: which kinds of resources are selected by secondary school mathematics teachers for lesson preparation, and what do they use them for?

Theoretical framework: the lens of resources

In line with the documentational approach to didactics [START_REF] Trouche | Documentational Approach to Didactics[END_REF] we assume that teachers' use of resources has an impact on their teaching. Teachers look for resources, adapt them to their needs and arrange selected resources into learning sequences for their students. This documentation work is an essential characteristic of a teacher's work and takes place at the different levels: curriculum, lesson series, and (parts of) individual lessons. The types of resources used by teachers fall into different categories. [START_REF] Pepin | Curriculum resources and textbooks in mathematics education[END_REF] distinguish (1) curriculum resources, (2) social resources, and (3) cognitive resources. Curriculum resources are "developed and used by teachers and students in their interaction with mathematics in/for teaching and learning, inside and outside the classroom". They are further categorized as (a) text resources (e.g. textbooks, syllabi, websites), (b) digital curriculum resources (e.g. e-textbooks, educational platforms), and other material resources (e.g. calculators, digital instructional technology). What distinguished digital curriculum resources from digital instructional technology is the possibility of arranging content and activities in line with the curriculum requirements. Social resources refer to human interactions, for example conversations with colleagues, or feedback from students. Cognitive resources refer, for example, to the mathematical frames and routines used by teachers, and are not considered in this study. By use of resources, we denote the selection of resources, their appropriation, and eventually modification by teachers to fulfill particular educational goals (e.g. determining mathematics content, designing classroom activities or homework tasks). The results of a recent Dutch learning material survey show that textbooks and related materials have still been among the most important resources used by secondary school teachers independent of the subject, although teachers and school managers expect their importance to go down with time [START_REF] Blockhuis | Leermiddelenmonitor 15/16: Leermiddelen in het po en vo[END_REF]. The most important digital resources for teachers are interactive student exercises. Teachers have indicated that they need guidance, particularly in selecting digital resources.

Method

A survey was administered in June/July 2018 to secondary school mathematics teachers, who were invited to participate by means of a brief text and a link in an electronic newsletter 1 . We received 78 complete responses from mathematics teachers. The survey consisted of 14 questions and was partly based on a survey on digital resources for teachers in France [START_REF] Gueudet | Online platforms: New connections and new possibilities for teacher documentation work? Poster presented at the 5th ERME topic conference: Mathematics education in the digital age[END_REF]. Items included, among others, teacher background information; (digital) resources selected to prepare lessons; the purpose of the selected resources; and classroom material retrieved or adopted from digital resources.

Results and conclusion

The survey results indicate that the most widely used resources for lesson preparation were national documents (e.g. syllabi), teachers' own notes, interactions with colleagues, mathematics apps, and the website belonging to the textbook. General educational platforms and social media were mentioned least often. Resources were mostly consulted to find materials for classroom activities, although interactions with colleagues covered all aspects of lesson preparation,. In terms of digital resources, Geogebra was mentioned most often, followed by a digital version of the textbook with its associated website, and online video clips. Least often, general educational platforms were used, in which teacher could create and share materials, in spite of efforts to promote these platforms and their transformative potential [START_REF] Pepin | Digital curriculum resources in mathematics education: Foundations for change[END_REF]. It appears that the mathematics teachers were mostly led by the school's mathematics textbook and its associated (digital) materials.


Dr Helen Martin
email: h.martin@abdn.ac.uk

Multiplicative reasoning task design with student teachers in Scottish schools: valuing diversity, developing flexibility and making connections

Keywords: multiplicative reasoning, teacher education, task design

Multiplicative reasoning task design with student

teachers in Scottish schools: valuing diversity, developing flexibility and making connections

Introduction

The aim of this poster is to present the development of a synthetic landscape of multiplicative reasoning constructed by student teachers. It explores the implications of working in a way that integrates literature from three different perspectives: maths recovery (MR), cognitively guided instruction (CGI) and realistic mathematics education (RME). The aim is to better understand how working in this way affects the way student teachers interact with learners, the questions they ask, the tasks they design: their capacity to value diversity in literature and develop their flexibility in practice.

There is a pattern of superficial adoption of findings from research in teaching mathematics in Scottish schools that often relies on dissemination of a product where teachers have not been a part of the process of constructing understanding. This often leads to a series of short lived interventions that rarely produce the positive effects expected. In which case, what can we use within teacher education to inform ourselves, our student teachers, our professional colleagues that accepts the natural dilemmas of teaching and might improve professional and ethical decision-making in an ever-increasing political world of curriculum design and change?

To address this question the author alongside a teaching colleague developed and have redeveloped a year 3 undergraduate mathematics education course over the past four years to consider how to better support student teachers entering this environment. They need to be able to adapt quickly to changing priorities using informed professional decision-making. To create the space and time to think more slowly, the course focuses on the key shift in thinking from additive to multiplicative reasoning. The first semester explores additive reasoning working with children aged 4-7 years old and the second semester focuses on multiplicative reasoning working with children aged 8-12 years old. Each semester consists of six 3-hour sessions followed by five weeks in a school. This poster uses the second semester course where the assignment requires student teachers to report on their analysis of the children's multiplicative reasoning.

Theoretical Background

Rather than taking one piece of major research and focusing on this to the exclusion of others we looked at using several significant studies. In this study, alongside the formal curricular guidelines in Scotland, Curriculum for Excellence (Scottish Government, 2009), three different theoretical perspectives are used: Realistic Mathematics Education [START_REF] Fosnot | Young mathematicians at work: Constructing multiplication and division[END_REF], Cognitively Guided Instruction [START_REF] Carpenter | Children's mathematics: Cognitively guided instruction[END_REF] and Maths Recovery [START_REF] Wright | Developing number knowledge: Assessment, teaching and intervention with 7-11 year olds[END_REF]. The course is designed to consider the benefits and limitations of each in a complementary rather than a competitive manner. It is by synthesizing ideas from different perspectives that have different foci that student teachers can become much more aware of the similarities and differences: complementary strengths and non-overlapping weaknesses. Similarly, the sessions include tasks focused on student teachers as learners of mathematics, researchers and teachers of mathematics to build a sense of themselves as mathematicians, researchers and teachers.

Methods

This design research [START_REF] Gravemeijer | Design research from a learning design perspective[END_REF] takes an iterative approach to studying and refining task design within a mathematics education classroom: to develop our understanding of the embodied process of developing theory through practice and the means that are designed to support that learning. The design principles are based on mathematics as a constructive activity [START_REF] Watson | Mathematics as a constructive activity: Learners generating examples[END_REF] with an emphasis on mathematical practices and a more participatory view of learning and teaching mathematics. This poster will illustrate the latest cycle of this design research using multimodal data provided by the most recent cohort of 38 students during academic session 2017/18: student teacher's assignments, planning documents, videos and transcripts of task-based interviews with children and images of child jottings to better understand how the design of the course influences the tasks designed by student teachers.

Findings

Preliminary findings indicate that most student teachers have shifted their focus to a more participatory view of learning mathematics, discussing a range of strategies and models used by the children. Some of the student teachers take informed risks, experiment with different pedagogical approaches and are beginning to design their own tasks however very few student teachers understood the underlying structure of number strings [START_REF] Fosnot | Young mathematicians at work: Constructing multiplication and division[END_REF].