Edris Pouresmaeil

Majid Mehrasa

Radu Godina

Ionel Vechiu

Rubén L Rodríguez

João P S Catalão

Double Synchronous Controller for Integration of Large-Scale Renewable Energy Sources into a Low-Inertia Power Grid

Keywords: Large-scale renewable energy sources, double synchronous controller (DSC), active and reactive power. I

Renewable energy sources are normally connected to the power grid via power electronic converters. High penetration of these energy sources into the power grid leads to high instability in voltage and frequency. This issue is caused by neglecting the inherent characteristics of synchronous generators i.e., inertia, damping and proper active and reactive power sharing in the structure of the used control technique in the control loop of the interfaced converter between power grid and renewable energy sources. This paper presents a powerbased control technique based on a double synchronous controller (DSC) for interfaced converter between the renewable energy sources and the power grid, including an active-reactive power based dynamic equation. Through the proposed DSC, a decoupled control method is performed in which both active and reactive power can be injected from renewable energy sources into the power grid by the interfaced power converter with the inherent features of synchronous power generators. By using the proposed control technique, a stable operation of the power grid can be guaranteed during the integration of large-scale renewable energy sources. Stringent simulation results performed in MATLAB/SIMULINK environment verify the proficiency of the proposed control technique.

INTRODUCTION

Because of environmental reasons and also the lack of fossil fuel sources in the near future, using power converter based distributed generation (DG) sources [START_REF] Pouresmaeil | Control scheme of three-level H-bridge converter for interfacing between renewable energy resources and AC grid[END_REF] in the frame of microgrid systems has attracted significant attention in recent years [START_REF] Mehrasa | Passivitybased control technique for integration of DG resources into the power grid[END_REF]- [START_REF] Mehrasa | A control plan for the stable operation of microgrids duringgridconnected and islanded modes[END_REF].

However, increasing the penetration of renewable energy sources in power grids by use power electronic converters leads to major problems, e.g. power flow fluctuations, unstable operation, oscillations of system frequency and out-of-rage of converter capability curve, which have been presented recently in several papers [START_REF] Mehrasa | A control plan for the stable operation of microgrids duringgridconnected and islanded modes[END_REF]- [START_REF] Zeraati | Distributed Control of Battery Energy Storage Systems for Voltage Regulation in Distribution Networks with High PV Penetration[END_REF].

On the other hand, focusing on the design of an autonomous unified reactive power control method with the aims of the system voltage regulation and unintentional islanding problems linked in presence of high penetration of distributed photovoltaic (PV) generation systems are discussed in [START_REF] Zhou | Integrated Autonomous Voltage Regulation and Islanding Detection for High Penetration PV Applications[END_REF].

Generally, in control loop of most of the power generators with interfaced power electronic converters, the intrinsic kinetic energy buffer that is rotor inertia is not considered, which lack of this inertia leads to instability issues in power grid during connection of large scale renewable energy sources.

In [START_REF] Wesenbeeck | Grid Tied Converter with Virtual Kinetic Storage[END_REF], a short-term energy buffer is added to the system by emulation of rotor inertia, called as Virtual Synchronous Generator (VSG) that can reduce instability problems of system operation level.

A synchronous power controller has been proposed in [START_REF] Zhang | Synchronous Power Controller With Flexible Droop Characteristics for Renewable Power Generation Systems[END_REF] for control of grid-connected converters for integration of renewable energy sources to the power grid.

In this paper, inertia, damping, and flexible droop characteristics are considered to meet the requirements for connection of power converters into the power grid and both dynamics and frequency regulations.

In [START_REF] Díaz | A Robot-Soccer-Coordination Inspired Control Architecture Applied to Islanded Microgrids[END_REF] a novel robot-soccer-coordination inspired control architecture has been proposed for a coordinated operation of an islanded microgrid which this controller is able to perform the voltage and frequency restoration for reaching a complete regulation of the common bus without losing the coordination between distributed generation units.

For a large-scale photovoltaic (PV) farm/plant, a distributed voltage control (DVC) scheme has been presented in [START_REF] Chamana | Distributed Control of Voltage Regulating Devices in the Presence of High PV Penetration to Mitigate Ramp-Rate Issues[END_REF] to decrease high ramp-rate issues.

Reference [START_REF] Wang | Probabilistic Analysis of Small Signal Stability for Power Systems With High Penetration of Wind Generation[END_REF] employs probabilistic analysis to evaluate the small signal stability of power systems in presence of wind generation high penetration.

Also distributed energy storage systems (DESS) are highly utilized for making power grid stable in large scale integration of renewable energy units into power grid [START_REF] Liu | A Stochastic Stability Enhancement Method of Grid-Connected Distributed Energy Storage Systems[END_REF]- [START_REF] Torres | Selftuning virtual synchronous machine: A control strategy for energy storage systems to support dynamic frequency control[END_REF].

To deal with the high penetration of renewable energy systems into power grid, a double synchronous controller (DSC) is performed by analyzing and considering the features of synchronous power generator in this paper, in order to provide coordinated behavior for operation of the interfaced converters. This controller is designed using a new dynamic equation based on active and reactive power. As can be seen, a DC-DC converter with a photovoltaic (PV) source is responsible to provide DC-link voltage for a three-phase dc-ac voltage source converter.

To draw a control technique for control of the interfaced converter for integration of renewable energy sources into the grid, dynamic model of the whole model should be extracted.

Therefore, the dynamic equations of the proposed model in d-q reference frame can be expressed as, 0

d d q d d c d di L Ri Li u v v dt ω + - - + = (1) 0 q q d q d c q di L Ri Li u v v dt ω + + - + = (2) 0 dc d d q q dc dv C u i u i i dt + + + = (3)
where, vdq and idq are the output voltages and currents of dc-ac converter in d-q reference frame.

Also, vdc, idc, and udq are dc-link voltage, dc-link current, and switching state functions of the dc-ac converter in d-q reference frame, respectively.

By multiplying d v in equations (1)-(3) and assuming

P=idvd , Q=-iqvd and neglecting partial variations, a set of new dynamic equations in terms of active and reactive power of the dc-ac converter is achieved as,

1 0 d c pd L dP L P Q u P P R dt R ω + + - + = (4) 1 0 q c pdq L dQ L Q P u P P R dt R ω + - + - = (5) 1 2 0 c d q c dP RC u P u Q P dt + - + = (6)
where, Pc1 and Pc2 are power associated with the dc-link voltage and current, and Ppd and Ppdq are the active power associated with the d and q voltage components of interfaced converter, respectively, which can be calculated as,

1 dc d c v v P R = (7) 2 c d c d P i v = (8) 2 d pd v P R = (9) d q pdq v v P R = (10)

III. THE PROPOSED CONTROL TECHNIQUE

In this section, the proposed control technique is presented based on the new dynamic equations and also the proposed double synchronous controller.

In the following subsection, DSC is designed in the first step and then the overall structure of the final control method is discussed in the next step with all details.

A. The Proposed Double Synchronous Controller (DSC) Model Analyze

The main concern about the large-scale integration of renewable energy sources to the power grid is the unstable behavior of the utilized control techniques for control of dc-ac converters. To solve this problem, all properties of synchronous generators (SG) are considered in the final proposed control technique as following.

As it is known, the so-called swing equation can be expressed as,

m P P d J dt ω ω - = (11)
where, J ,ω , m P and P are the values of inertia, angular speed, mechanical power and electrical power of SG, respectively. By applying small-signal linearization to [START_REF] Díaz | A Robot-Soccer-Coordination Inspired Control Architecture Applied to Islanded Microgrids[END_REF] and doing some simplifications, ( 12) can be driven as, ( )

* * * * 2 1 . / m m P P J s P P J ω ω ω Δ -Δ Δ = + - (12)
In other words, the relationship between the power and the angular speed variations can be demonstrated as, ( )

* * * * m m P P P P J s s s ω ω ω   - Δ Δ   = - + Δ     (13)
By noticing ( 12) and ( 13) and adding some control considerations, the proposed DSC can be achieved as shown in Fig. 2. Based on Fig. 2 According to Fig. 2, DSC2 consists of a mechanical power error and a low pass filter (LPF). In fact, DSC2 is aimed to generate the moment angular speed employed for both active and reactive-axis of the proposed control method. Also, the angular speed driven by DSC2 is used for DSC1 as well.

By noticing Fig. 3, the switching functions of dc-ac converter can be achieved as ( 15) and ( 16), (

)

1 / 1 / m p p i p d c pm i m p d P P k k s u L P k k s Q P R ω ω   Δ -Δ + +   =   Δ + + +     (15)
( )

1 1 / q p q i q p d q c L u Q k k s P P P R ω -  = Δ + + +     (16)
The proposed switching functions of ( 15) and ( 16) are made to provide the synchronous behavior for PWM dc-ac converter.

B. The Proposed Control Technique

By use of the power based dynamic equations of ( 4)-( 6) and the DSC illustrated in Fig. 2, the proposed control method can be performed as depicted in Fig. 3. As can be seen from this figure, a modified feed-forward (decoupled) power-based control technique included both DSC1 and DSC2 is performed to control the proposed converter-based model shown in Fig. 1, with the aim of synchronous regulation performance analogous SG operation. To further evaluate the DSC responsibilities, it can be understood from Fig. 3 that DSC1 can exclusively impact on the accuracy of synchronous active power sharing of DG unit. Moreover, DSC2 has high effect on the decoupling properties of the proposed control technique which should be considered in more regulations of DSC.

IV. RESULTS AND DISCUSSION

The complete system model proposed in Fig. 1 is simulated using the 'Power System Blockset' simulator operating under the Matlab/Simulink environment, in order to demonstrate the high performance of the proposed control technique for control of the interfaced converters and integration of renewable energy sources into the power grid and/or loads, in grid connected and islanded operating modes.

The network parameters, including the ratings and impedances of the proposed model components for simulation analysis, are given in Table I.

As the first step, the renewable energy source is regulated to supply a load of 6kW+j3.5kVAR in grid-connected mode. At the same mode, another load with the value of 4kW+j2kVAR is added at t=0.1s. This operation is continued until t=0.2s, while the operation of interfaced converter is changed to the islanded mode with the same load and should continue its performance for supplying the load in this operating mode. Fig. 4 shows the output ac voltages of the interfacedconverter during both operating modes. As can be seen, in grid-connected mode, three phase ac voltages are kept in its steady state waveforms without any transient time.

Immediately after the operation mode changes to the islanded mode, a fast-transient response can be observed and then the desired waveforms are achieved in an acceptable response range. The output ac currents of the interfaced-converter is illustrated in Fig. 5. Based on this figure, increasing the ac currents is happened proportional to the load changes and after making islanded mode, this increment is remained with a proper transient response.

The frequency of three phase ac voltages is illustrated in Fig. 6. When the load increment occurs at t=0.2s, a drop happens for PCC voltage frequency as depicted in Fig. 6. As it can be seen from this figure, the ac voltages frequency is placed closed to the 50 Hz after a very short transient time, which very proper from the view point of controller operation.

In the time of load increment, the voltage magnitude experiences some fluctuations which will be damped in very short transient time as shown in Fig. 7. Also, when islanding mode occurs, some slight undershoot and overshoot happen and then the magnitude completely tracks its reference value. The ability of the proposed control technique for tracking the active and reactive power desired values is depicted in Fig. 8 and Fig. 9.

As it can be seen from the Fig. 8, the tracking of load active power variations is accurately done in grid connected mode which follows all the active power of grid connected loads before and after load increment. After the operation mode changes to the islanded mode, the proper active power tracking is continued with very low fluctuations. The reactive power injection from the interfaced-converter to the loads, before and after load increment, and for both of grid-connected and islanded operating modes is shown in Fig. 9.

As shown in this figure, the same consequence happens for the reactive power compensation, in which the tracking of desired values is properly executed for reactive power of the system. In both active and reactive power, the proposed controller is able to quickly damp the transient responses.

V. CONCLUSION

An innovative double synchronous control technique has been proposed in this paper for the integration of large-scale renewable energy sources into the power grid. The features of synchronous generators have been considered in the structure of the proposed control technique for the interfaced converter to integrate renewable energy sources into the power grid. The control technique has been developed based on a new dynamic model made by active and reactive powers state variables. This control technique is influenced by the proposed DSC in both its active and reactive control axis. The application of the proposed control technique can guarantee the stability of the power grid during the integration of large-scale renewable energy sources into the power grid. The high performance of the proposed control technique was validated through stringent simulation results in MATLAB/SIMULINK environment under different operation conditions.

  By the use of the proposed dynamic equation and DSC, the DSC based control technique is able not only to execute accurate active and reactive power sharing, also can perform fast reaction to voltage magnitude and frequency instability to reach desired values for them. The ability of the proposed control technique is validated through stringent simulation results via MATLAB/SIMULINK. II. THE PROPOSED MODEL General schematic diagram of the proposed converterbased model is presented in Fig. 1.

Fig. 1 .

 1 Fig. 1. the proposed model (a) single phase diagram of the grid-connected PV based model, (b) three-phase model included with the interfaced dc-ac converter between renewable energy sources and power grid.

  , DSC1 is consisted of all properties of SG which is used for active power-axis of the proposed controller. In this part of DSC, the proportional-integral coefficients of angular speed error PI controller are equal to,

FirsFig. 2 .Fig. 3 .

 23 Fig. 2. General structure of the proposed Double Synchronous Controller.

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig.[START_REF] Yan | Pilot-Bus-Centered Automatic Voltage Control With High Penetration Level of Wind Generation[END_REF]. Three phase ac voltages of interfaced-converter in both grid-connected and islanded operating modes.

Fig. 7 .Fig. 8 .Fig. 9 .

 789 Fig. 7. Voltage Magnitude of three phase ac voltages.

TABLE I SIMULATION PARAMETERS

 IPARAMETERS

	Parameter	Value
	dc-link voltage (vdc)	850 V
	ac voltage	310 V
	Fundamental frequency	50 Hz
	Switching/Sampling frequency	10 kHz
	SPVSI resistance	0.25 Ω
	SPVSI inductance	30 mH
	J	7.5 s
	m P Δ	10 W
	Load 1	6+j3.5
	Load 2	4+j2

ACKNOWLEDGMENTS

This work was supported by INSUL'GRID project, France, by FEDER funds through COMPETE 2020 and by Portuguese funds through FCT, under Projects SAICT-PAC/0004/2015 -POCI-01-0145-FEDER-016434, POCI-01-0145-FEDER-006961, UID/EEA/50014/2013, UID/CEC/50021/2013, UID/EMS/00151/2013, and SFRH/BPD/102744/2014. Also, the research leading to these results has received funding from the EU Seventh Framework Programme FP7/2007-2013 under grant agreement no. 309048.


B Corcoran

Christelle Monat
email: monat@physics.usyd.edu.au

C Grillet

D J Moss

B J Eggleton

T P White

L O'faolain

T F Krauss

Green light emission in silicon via slow light enhanced thirdharmonic generation in photonic crystal waveguides

Slow light has attracted significant interest recently as a potential solution for optical delay lines and time-domain optical signal processing 1,2 . Perhaps even more significant is the possibility of dramatically enhancing nonlinear optical effects 3,4 due to the spatial compression of optical energy 5,6,7 . Two-dimensional (2D) silicon photonic crystal (PhC) waveguides have proven to be a powerful platform for realizing slow light, being compatible with on-chip integration and offering wide-bandwidth and dispersion-free propagation 2 . Here, we report the slow light enhancement of a nonlinear optical process in a 2D silicon PhC waveguide. We observe visible third-harmonic generation (THG) at a wavelength of 520nm with only a few watts of peak power, and demonstrate strong THG enhancement due to the reduced group velocity of the near-infrared pump signal. This demonstrates yet another unexpected nonlinear function realized in a CMOS-compatible silicon waveguide.

Main text

Although silicon has been the material of choice for the CMOS industry and more recently for integrated photonics, its optical properties -e.g light emission -still provide major challenges. In addition to an indirect band-gap and inversion symmetry, its strong absorption in the visible restricts the potential emission window to wavelengths above ~800nm. Nonlinear optical effects such as stimulated Raman scattering and THG offer new "tricks" for light emission 8 , thereby extending the functionality of silicon photonics.

2D PhCs have recently attracted considerable attention by controlling the propagation of light in unprecedented ways 9,2 . In particular, they can produce "slow"

3 light [START_REF] Krauss | Why do we need slow light?[END_REF][START_REF] Baba | Slow light in photonic crystals[END_REF] with the ability to dramatically enhance nonlinear optical phenomena [START_REF] Soljacic | Photonic-crystal slow-light enhancement of nonlinear phase sensitivity[END_REF][START_REF] Bhat | Optical pulse propagation in nonlinear photonic crystals[END_REF][START_REF] Mcmillan | Enhanced stimulated Raman scattering in slow-light photonic crystal waveguides[END_REF] .

However, although nonlinear optical processes have been widely demonstrated in silicon nanowires, [START_REF] Bogaerts | Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology[END_REF][START_REF] Lin | Nonlinear optical phenomena in silicon waveguides: Modeling and applications[END_REF][START_REF] Jalali | Teaching silicon new tricks[END_REF][START_REF] Baba | Slow light in photonic crystals[END_REF] corresponding demonstrations in silicon PhCs -especially with slow light -have been elusive, only having been reported in the context of the electro-optic coefficient [START_REF] Jacobsen | Strained silicon as a new electro-optic material[END_REF] .

Here, we report visible (green light) THG in slow light silicon PhC waveguides via end-fire coupling of ~ 10 W peak pump power near-infrared pulses. This power is 5 to 6 orders of magnitude lower than that of earlier THG demonstrations in bulk silicon [START_REF] Wynne | Optical third-order mixing in GaAs Ge Si and InAs[END_REF][START_REF] Wang | Optical Third Harmonic Generation in Reflection from Crystalline and Amorphous Samples of Silicon[END_REF][START_REF] Moss | Third Harmonic Generation as a Structural Diagnosis of Ion-Implanted Amorphous and Crystalline Silicon[END_REF][START_REF] Moss | Dispersion in the anisotropy for optical third harmonic generation in Si and Ge[END_REF] , and arises from a combination of extreme mode confinement by the PhC waveguide, slow light enhancement, and extraction of the visible light via the PhC lattice.

The generation of third-harmonic light (electric field intensity I 3ω , frequency 3ω) from a fundamental pump beam (I ω , ω) propagating in a lossless homogeneous waveguide of length L is given by 17 :

) , ( ] 2 )
( [ ) ( ) 2 ( ) 3 ( 3 2 2 ) 3 ( 2 3 4 2 3 ω ω ω ω ω χ π ω A A f L k sinc L I nc I ⋅ ⋅ Δ ⋅ ⋅ ⋅ ⋅ ⋅ = (1)
where χ [START_REF] Soljacic | Photonic-crystal slow-light enhancement of nonlinear phase sensitivity[END_REF] and n are the 3 rd order nonlinear susceptibility and refractive index of silicon, Δk=k 3ω -3k ω is the phase mismatch between the fundamental mode and the third-harmonic wavevectors and f(A ω ,A 3ω ) accounts for the spatial overlap between the two modes. For perfect phase matching (Δk=0) and maximum mode overlap (f=1),

Equation (1) primarily reflects the THG cubic dependence on I ω , which arises from the basic nature of the THG process -i.e., converting three ω photons into a single 3ω photon (Fig. 1a). Clearly, then, increasing I ω within the nonlinear material is crucial to enhancing the THG efficiency. This has been achieved in the past by employing ultrahigh-Q, small modal volume silica microtoroids, [START_REF] Carmon | Visible continuous emission from a silica microphotonic device by third-harmonic generation[END_REF] although these tend to yield narrow bandwidth, and dispersive features. Another approach has been to exploit the high density of states at the band-edge of periodic structures in various materials and geometries such as Bragg gratings in porous silicon [START_REF] Martemyanov | Third-harmonic generation in silicon photonic crystals and microcavities[END_REF][START_REF] Dolgova | Giant third-harmonic in porous silicon photonic crystals and microcavities[END_REF] , 2D GaN PhCs 21 and 3D polystyrene PhCs [START_REF] Markowicz | Dramatic Enhancement of Third-Harmonic Generation in Three-Dimensional Photonic Crystals[END_REF] . These schemes generally involve a free-space configuration (reflection or diffraction surface probe experiments) where the lack of optical confinement limits the potential enhancement of I ω , thus requiring megawatt peak pump powers.

Combining optical confinement and dispersion engineering through the use of optimized 2D PhC waveguides [START_REF] Frandsen | Photonic crystal waveguides with semi-slow light and tailored dispersion properties[END_REF][START_REF] Li | Systematic design of flat band slow light in photonic crystal waveguides[END_REF][START_REF] Kubo | Low-group-velocity and low-dispersion slow light in photonic crystal waveguides[END_REF] is highly promising because I ω is related to the peak power (P ω ) through

n n A P I g ⋅ ∝ ω ω ω (2)
where A ω and n g are the effective area and group index of the fundamental mode, repectively. Hence, by exploiting the extreme concentration of optical energy afforded by: (i) the tight confinement of light within the high index, sub-μm scale (A ω ~ 0.4 µm 2 ) silicon PhC waveguides and; (ii) spatial pulse compression in the slow light (v g =c/40) regime, we significantly reduce the peak pump power required to observe THG to 10W. In addition, the PhC structure provides a mechanism for light extraction at a wavelength that would otherwise be strongly absorbed, thereby opening the spectral emission window of silicon to the visible.

Our device consists of an 80 μm long W1 PhC waveguide in a 220nm-thick air suspended silicon slab, coupled to two tapered ridge waveguides (Fig. 1a,b). Unlike the highly dispersive slow light mode associated with the band edge of typical PhC waveguides [START_REF] Notomi | Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs[END_REF][START_REF] Engelen | The effect of higher-order dispersion on slow light propagation in photonic crystal waveguides[END_REF] , here the fundamental mode is engineered to display both low group velocity and low dispersion [START_REF] Li | Systematic design of flat band slow light in photonic crystal waveguides[END_REF] (Fig. 2a,b). Large dispersion typically broadens and distorts short pulses, which tends to compromise the benefits of slow light for nonlinear applications. In our case, however, we focus on the spectral window (1550nm to 1559nm) where the measured group velocity of the fundamental mode [START_REF] Gomez-Iglesias | Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry[END_REF] varies almost linearly by a factor of 4 from c/10 to c/40 (Fig. 2b), enabling us to investigate the effect of group velocity.

When launching a near-infrared 1.5ps pulse train (4MHz) into the PhC waveguide, we observe green light emitted from the surface of the chip by eye (Fig. 3a). The emission is directional, being at an angle ~ 10º from the vertical, in the backward direction, as indicated by the schematic in Fig. 1a. Imaging the emission onto a calibrated linear CCD camera with a 0.25 N.A. microscope objective reveals that it is localized above the PhC waveguide, and decays exponentially along its length (Fig. 3b). The total emitted green power (integrated spatially over the CCD image, Fig. 4) shows a cubic dependence on the coupled pump power up to ~ 65 μW and is verified to have a wavelength of 520nm ± 5nm using bandpass filters, both of which are expected for a THG process driven by a 1560nm pump. At higher pump powers, a slight saturation occurs in the fundamental power transmission (see Fig. 4) due to two-photon and subsequent free-carrier absorption. We observe a maximum THG output of ~10pW for 80μW (10W) average (peak) pump power.

In general, there are a number of effects that can contribute to a variation in the THG efficiency along the waveguide such as group velocity dispersion (GVD) and nonlinear absorption, both of which degrade the fundamental pulse intensity along the PhC waveguide. It potentially also includes phase matching between the fundamental and third-harmonic modes, although here the extremely short (~ 1μm) absorption length of the third-harmonic light in silicon dampens this effect (see Methods). In order to minimize these effects and determine the dependence of the THG efficiency solely on group velocity, we therefore restricted our measurements to a region within 6 5μm of the PhC waveguide entrance, which is much smaller than the dispersion length associated with the GVD, even in the "fast light" regime (see Methods).

Figure 5a,b shows that the observed THG power displays a clear enhancement for pump wavelengths near 1557nm where the group velocity is lowest. Equation ( 1) predicts a cubic dependence on n g of the THG power obtained at a fixed pump power.

In order minimize the nonlinear loss saturation effect discussed above at all wavelengths though, we chose instead to plot on Fig. 5c,d the input power density (P ω /A ω ) required to produce a constant (and sufficiently low) THG output power (~0.2 pW) as the wavelength (hence group index n g ) is varied. By plotting the power density (P ω /A ω ) rather than power (P ω ) we factor the spectral dependence of A ω out in order to focus on the variation due solely to group index (see Methods). The results show very good agreement with a 1/n g variation, as expected from Equation (2) -P ω /A ω ∝ I ω /n g when considering a constant intensity I ω responsible for a given THG signal. Note both the trend and the variation in enhancement is well accounted for using only the experimentally measured group velocity dispersion of Fig. 2b. It is clear, then, that any contribution from other effects would cause a discrepancy with experiment, even if (by coincidence) its wavelength dependence happened to be identical to that of n g . Hence we believe these results conclusively demonstrate direct slow-light enhancement of this nonlinear process.

Our experiments were performed with ~10W peak pump powers, corresponding to a reduction of 5 to 6 orders of magnitude compared to previous reports [START_REF] Wynne | Optical third-order mixing in GaAs Ge Si and InAs[END_REF][START_REF] Wang | Optical Third Harmonic Generation in Reflection from Crystalline and Amorphous Samples of Silicon[END_REF][START_REF] Moss | Third Harmonic Generation as a Structural Diagnosis of Ion-Implanted Amorphous and Crystalline Silicon[END_REF][START_REF] Moss | Dispersion in the anisotropy for optical third harmonic generation in Si and Ge[END_REF] of THG in silicon. Even more significantly, this work represents a nearly 100-fold reduction in pump power relative to fully phase-matched THG in PPLN/KTP waveguides [START_REF] Rusu | Efficient generation of green and UV light in a single PP-KTP waveguide pumped by a compact all-fiber system[END_REF] . Although a comparable power density (~GW/cm 2 ) has been achieved in ultra-high Q (>10 7 ) cavities [START_REF] Carmon | Visible continuous emission from a silica microphotonic device by third-harmonic generation[END_REF] , the advantage of the PhC waveguide approach is that the full bandwidth of short optical pulses can be accommodated. We estimate our conversion efficiency η to be ~10 -7 (or 5 x 10 -10 for 1W of peak pump power), which represents an increase of 5 orders of magnitude over that reported in 3D polystyrene PhCs (η ~10 -15 for 1W peak pump power as inferred from the quoted value of η ~10 -5 at P ω =10MW [START_REF] Markowicz | Dramatic Enhancement of Third-Harmonic Generation in Three-Dimensional Photonic Crystals[END_REF] ). This efficiency could be further improved, e.g. by decreasing the effective area (η ∝ 1/A eff 3 ) or the group velocity (η ∝ n g 3 ). A group velocity of c/80 can be reasonably well achieved with this PhC waveguide design [START_REF] Li | Systematic design of flat band slow light in photonic crystal waveguides[END_REF] -this would provide an order of magnitude improvement in η. Efficiency could also be improved at high pump powers by reducing nonlinear losses (in particular due to free carriers) through techniques such as ion implantation.

Besides tight optical confinement and slow light, the 2D PhC geometry offers additional versatility to improve the third-harmonic generation and extraction efficiency. In periodic structures, the phase matching condition, Δk=0, is relaxed to Δk=±mG, where mG can be any reciprocal lattice wave-vector, increasing the possibilities for phase matching (see Methods). Perhaps more importantly in our case, since the absorption length at 3ω is extremely short (3dB/µm absorption loss at 520nm in silicon [START_REF] Green | Optical Properties of Intrinsic Silicon at 300K[END_REF] ), the PhC also provides a suitable platform for extracting light by coupling to surface radiating modes above the light-line. The directive nature of the emission (~10 o ) as well as the absence of green emission from the access waveguides, suggests that a component of the third-harmonic Bloch mode in the PhC lies above the light line, as illustrated by the band structure in Fig. 2a. This provides a mechanism for the THG to be extracted out-of-plane. However, because the 3ω Bloch mode also contains harmonic components well confined in the PhC slab below the light line, the measured pico-watt level of green emission is expected to be significantly lower than the total THG power generated in the PhC waveguide. The 8 conversion efficiency reported above is therefore a conservative estimate. In addition, the PhC waveguide geometry was not optimized for third-harmonic extraction in this study, so modification may well increase overall device efficiency. Finally, we note that the optimization of all of these processes can be done across a wide range of frequencies, allowing one to address the entire visible spectrum, or to extend the THG to other spectral windows.

In conclusion, we report optical third-harmonic generation of visible (green) light, enhanced by a reduction in the group velocity, or "slow light" in 2D silicon photonic crystal waveguides. We observe visible green emission for only ~10W peak pump powers due to both the tight light confinement within the PhC waveguide and the energy density enhancement provided by the slow light mode. Following recent observations of other nonlinear effects such as efficient Raman amplification and four wave mixing, the THG observed here further highlights the rich photonic functionality available with the silicon photonics platform.

Methods

Device structure and fabrication

The 2D PhC structure consists of a triangular lattice of air holes with lattice constant a=414nm and hole radius 118nm (0.286a) etched into a 220nm thick silicon suspended membrane. A W1 waveguide is introduced by omitting a single row of holes along the ΓK direction to form a linear defect. The total PhC length is 80µm, and the lattice period of the first and last 10 periods is increased to 444nm parallel to the waveguide to enhance coupling to the slow light mode [START_REF] Hugonin | Coupling into slow-mode photonic crystal waveguides[END_REF] . The dispersion of the PhC waveguide is engineered by shifting the first two rows of holes adjacent to the guide perpendicular to the direction of propagation [START_REF] Li | Systematic design of flat band slow light in photonic crystal waveguides[END_REF] . For the waveguide used in this experiment, the first and second rows are shifted 52nm away from and 12nm toward the axis of the waveguide, respectively. Light is coupled in and out of the PhC waveguide via 2mm long ridge access waveguides whose width is tapered from 3µm to 0.7µm over 200μm close to the PhC waveguide, in order to improve coupling to the PhC. The group velocity dispersion β 2 for the engineered PhC guided mode ranges between 3x10 -21 to 2.5x10 -20 s 2 /m for the spectral window of interest, providing an associated dispersion length for 1.5ps pulses from 90µm to 750µm. This is longer than the entire PhC waveguide length and much longer than the small 5µm region considered in Fig. 5, and so we expect the effects of dispersion on our measuments to be negligible.

We note that this device was not optimized in terms of losses, which are dominated by coupling losses. There are several obvious routes to further improve this issue, for example through the use of inverse tapers at the end-facets. The loss of 17dB extracted from Fig. 4 The device was fabricated from a SOITEC silicon-on-insulator wafer by electron-beam lithography (hybrid ZEISS GEMINI 1530/RAITH ELPHY) and reactive ion etching using a CHF 3 /SF 6 gas mixture. The silica layer under the PhC slab was selectively under-etched using a HF solution to leave the PhC section in a suspended silicon membrane. More details of the procedure are given in Ref [32].

Transmission experiment

The device was probed using a polarization controlled, near transform-limited, figureof-8 fibre laser, tunable over the C-band. The pulses were sech 2 shaped, ~1.5ps long, with a spectral full-width-at-half maximum of ~2nm and were amplified through an Er-doped fibre amplifier. The pulses were launched into the TE-like mode of the waveguide, using lensed fibres with a 2.5μm focal length. To calculate the "coupled power" on Fig. 4 and 5 from the power launched into the input lensed fibre, we estimated an in-coupling insertion loss of 8dB. This is a typical value obtained from independent Self Phase Modulation measurements on the same waveguide. Because this coupling loss is nonetheless the lowest typically measured, our estimation of the THG efficiency represents a low, or conservative, limit of the third harmonic efficiency.

Green light measurement

The linearised, fixed gain, CCD camera (Cohu 4910) was calibrated using a low RIN doubled Nd:YAG laser diode, with the power externally controlled by an attenuator assembly and monitored with a thermopile detector (Coherent PS10 and PowerMax meter).

Corcoran et al.

NPHOT-2008-10-00588A 11

Calculations

The bandstructures in Fig. 2a were calculated with a 2D Finite Difference Time Domain (FDTD) method using effective material indices of 2.81 and 4.1 at the fundamental (~1550nm) and third-harmonic (~520nm) wavelengths respectively to account for the effects of confinement by the slab and the material dispersion of Si [START_REF] Green | Optical Properties of Intrinsic Silicon at 300K[END_REF] .

Calculations were performed for a lattice period of a= 414nm and hole radius r= 0.286a, given by the experimental parameters. The calculated dispersion curves for the fundamental mode at 1550nm are red-shifted by less than 4% relative to the experimental measurements in Fig. 2b. The bandstructure shows that several symmetric modes (H y symmetric across the waveguide, like the fundamental mode) exist at wavelengths close to 520nm that can sustain the THG emission. We believe that the third-harmonic light is most likely coupled to the refractive-like mode highlighted by the dotted red line of Fig. 2a, which has the same symmetry as, and significant overlap with, the fundamental mode at 1550nm. Observation also yields an exit angle for the third-harmonic light at ~10º from the normal which is reasonably consistent with coupling to this mode. We observe no noticeable change in exit angle with pump wavelength over the 10nm wavelength range near 1560 nm studied here, leading us to believe that there is no significant change in the third-harmonic mode over this range.

The effective mode field area ( A ω ) of the fundamental mode was calculated using 3D Plane Wave Method (PWM) and the following equation: 12 yields an increase of A ω from 0.4µm 2 to 0.6µm 2 for increasing wavelength over the range studied.

dV E dV E a A Vol Vol 4 2

Phase matching considerations

One might expect that, since the fundamental wavevector varies by ~10% as the wavelength is tuned over a 10nm range near 1560 nm, it may influence the wavelength variation of the THG efficiency via the phase matching term. In order to shed some light on the conclusion of our work that the observed enhancement in THG efficiency over the wavelength range studied is solely due to the group index 1/n g , we find it useful to compare characteristic length scales in our device. In periodic structures, the phase mismatch Δk between the fundamental and the third harmonic modes is defined at ±mG, where mG can be any reciprocal lattice wave-vector. Therefore, Δk < π/a (where a is the PhC lattice constant) and the associated coherence (or beating) length between the two modes, L coh , is thus always > 2a (~0.83μm). From the specific band structure of Fig. 2a we further expect the coherence length to be above 2μm. In silicon, since the absorption length of the third-harmonic light at 520nm is ~ 1μm < L coh , the influence of phasematching variation on the THG efficiency in our device is expected to be negligible. Further, notwithstanding this argument, or the experimental evidence, we investigated the wavelength dependence of phasematching in our device by calculating it from the bandstructure, based on coupling to the third-harmonic mode discussed above, and assuming no losses at the third-harmonic wavelength, and we found that it was quite different to what was observed. Hence, for all of these reasons we believe these results conclusively demonstrate direct slow-light enhancement of this nonlinear process.

10 (

 10 is comprised of out-coupling losses (typically ~10 dB and measured to be at best of ~ 8dB), propagation loss in the ridge and nanowire segments NPHOT-2008-10-00588A ~2-3dB), scattering loss at the nanowire/ PhC interface (<1 dB) and the propagation losses in the PhC waveguide in the slow light regime (2-3dB).

  the electric field amplitude of the mode and Vol is the volume of a unit cell of length a associated to the PhC waveguide. This Corcoran et al. NPHOT-2008-10-00588A

Figure captions : Figure 1 -

 :1 Figure captions: Figure 1 -Green light emission through third-harmonic generation in a slow

Figure 2 -

 2 Figure 2 -Photonic crystal waveguide dispersion. a, 2D FDTD band structure

Figure 3 -

 3 Figure 3 -Observation of green light. a, Visible green emission as seen by eye

Figure 4 -

 4 Figure 4 -Power dependence of the green light emission. The left axis ( )

Figure 5 -

 5 Figure 5 -Slow light enhancement of green light emission. a,b, Green light power

Acknowledgements

This work was supported by the EU-FP6 Marie Curie Fellowship project "SLIPPRY", the EU-FP6 Network of Excellence "ePIXnet" and the EU-FP6 "SPLASH" project.

The fabrication was carried out in the framework of the ePIXnet Nanostructuring Platform for Photonic Integration. We would also like to acknowledge the Australian Research Council (ARC) through its Federation Fellow, Centre of Excellence and Discovery Grant programs as well as the International Science Linkages program of the Australian Department of Education, Science and Technology.

Competing financial interests

There is no competing financial interest.


domain: math shs

text:
Stephanie Treffert-Thomas
email: s.thomas@lboro.ac.uk

Getting into university: From foundation to first year engineering

Keywords: Engineering mathematics, foundation degree, transition, motivation, activity theory

I present a longitudinal study with engineering students taking a (science) foundation course (FC). The aim of the research is to explore the reasons why students decided to take the FC going beyond the rather obvious reason, "because they did not have the necessary qualification for direct entry". Secondly, I explore students' mathematical progression into first year engineering. Many UK universities have addressed students' lack of mathematical preparedness (e.g. [START_REF] Hawkes | Measuring the mathematics problem[END_REF] by establishing drop-in centres or one-to-one support with specialist tutors; some offer a oneyear FC as an alternative route into higher education, primarily aimed at students wanting to study a STEM subject. While the FC is often seen as aimed at students who 'missed' (perhaps narrowly) their target grades at A-level, it became clear to me that students came from varied backgrounds and qualifications, work experience, and included students with health problems that impacted on their progression. Thus I became interested in the motivational factors that led students to take a FC and the mathematical progression that students make. The FC at my institution consists of a number of modules, of which mathematics and physics are compulsory for engineering students, the largest group in the cohort. Topics studied include indices, logarithms, differentiation, integration, matrices and complex numbers. I pose two research questions: (1) Why did students take the FC? (2) What is students' mathematical experience when moving into first year engineering study?

Methodological considerations

So far two cohorts of students have been interviewed at the end of their FC; one cohort has been reinvited for interview at the end of the first year of their engineering course. Ten to eleven students were interviewed each year. I, therefore, pursue a case study methodology within an interpretive paradigm. The study is longitudinal in design with further interviews planned over the next years. Data analysis is qualitative with a focus on the reasons that students gave for taking the FC. I take an activity theory perspective since identifying reasons means identifying the motive of activity, hence closely linked to characterising activity [START_REF] Leontiev | The problem of activity in psychology[END_REF]. Action-goals can be used to discuss students' mathematical goals, and actions taken in pursuit of these, a focus in future analyses. Research was with students I had taught. Interview questions were communicated in advance and followed a fixed order. All interviews were recorded and subsequently transcribed and analysed.

Research findings and conclusions

I report on two separate findings. First, in interviews in 2015, students were asked why they had chosen a FC. Students replied in a variety of ways, giving one or more reasons. As part of analyses, these were categorised using an open coding procedure and summarised in Figure 1. Most students cited career advancement and gaining entry to their engineering course when A-level grades had not been good enough. Some did so strategically, i.e. there was no need to re-apply through UCAS, the UK body overseeing university applications, since passing the FC guaranteed entry. Also represented were six students who took the view that it gave them an advantage over other students when entering the first year of engineering. Thus students' motives were nuanced, going beyond "because they did not have the necessary qualification for direct entry". Most (not all) students reported having difficulties with the mathematics content of the FC, and some students said they were overwhelmed by them. Second, I report on three of seven students who were re-invited for interview and asked to compare current experience with how they recalled their experience during the FC. The second and third column (Figure 2) relate to students' experience and achievement during the FC (as recalled by the student one year later); the last column to students' current experience. Clearly there are some differences in how students perceived their transition. The FC is well established at my university and students are thought successful in progressing into their engineering courses. Students cited different motivational factors for taking a FC. While some were not surprising (e.g. change of career, A-level grades not good enough) others were, and related to students employing a more strategic decision (e.g. to gain an advantage later on in their studies). Considering students' progression it is clear that this is far from linear. Both mathematically strong and not so strong students cited struggling with the mathematical demands placed on them, while a mathematically weaker student reported feeling settled and coping well. This raises some questions to explore further, e.g. to what extent can the FC provide a good mathematical basis for all students.

Figure 2 :

 2 Figure 2: Results from three interviews

Proceedings of CERME10


text:
Peter Stender
email: peter.stender@uni-hamburg.de

Heuristic strategies in mathematics teacher education

Keywords: Heuristic strategies, teacher education, tutorial in mathematics

Here a tutorial is described that focuses on mathematical methods that implicitly appear in the mathematics shown in a Linear Algebra lecture, and makes these methods explicit for the students.

Interviews with students at the end of their first semester show some effects.

Theoretical framework

Mathematics, as all sciences, comprises content (e.g. definitions, theorems, finished proofs, …), but also mathematical methods. Pólya (1945) underlined the importance of the mathematical methods in the subtitle of "How to solve it -A new aspect of mathematical methods". We use a broad approach of heuristic strategies as a central part of the mathematical methods shown in the following list:

 Organise your material / understand the problem: change the representation of the situation if useful, try out systematically, (Pólya, 1945) use simulations with or without computers, discretize situations,  Use the working memory effectively: combine complex items to supersigns, which represent the concept of 'chunks' (Miller, 1956), use symmetry, break down your problem into subproblems,  Think big: do not think inside dispensable borders, generalise the situation (Pólya, 1945),  Use what you know: use analogies from other problems, trace back new problems to familiar ones, combine partial solutions to get a global solution, use algorithms where possible (Pólya, 1945),  Functional aspects: analyse special cases or borderline cases (Pólya, 1945), in order to optimise you have to vary the input quantity,  Organise the work: work backwards and forwards, keep your approachchange your approachboth at the right moment (Pólya, 1945).

Other mathematical methods are e.g. "proof strategies" which make extensive use of formal language such as mathematical induction, proof by contradiction, proof by exhaustion, the invariance principle or others. Another kind of methods are the use of mathematical language e.g. in depth reading mathematical texts or writing down mathematical proofs correctly.

Tutorial example

In the tutorial we made mathematical methods implicitly used in the lecture explicit. We also used these methods giving additional explanations and reflected this use afterwards in the tutorial. The concept of quotient spaces in linear Algebra was established on rings. One example given in the lecture was ℤ/𝑚ℤ . In the tutorial we reduced this example to the case ℤ/4ℤ and gave additional representations of this concept, some of them shown in figure 1.

Thematic Working Group 14

Proceedings of CERME10 Here the four residue classes are shown in several representations, together with the way of calculating on that structure. The similarity of the calculation on a common clock was stressed as a possible link to school mathematics and further explanations were given. Calculating on the equivalence class that are supersigns from the heuristic point of view was difficult for the students and the similarity to calculating with fractions that are equivalence classes and supersigns as well was discussed. Furthermore, moving between representations was stressed as a method to deepen the understanding of the content or to find a way to solve a problem.

Students' feedback

At the end of the first semester we interviewed five students who participated in the tutorial over the whole semester. The interviews were transcribed and analyzed using quantum content analysis.

All students remembered well the connection to school mathematics based on mathematical methods that appear in university and school as well. The students also emphasized that visualized explanations with less formal language were very helpful. These explanations using "a change of representation" were remembered well.

The input related to the heuristic strategies was remembered by the students. The concept of "heuristic strategy" was remembered and the students could describe examples given in the tutorial, connecting the examples to the appropriate heuristic strategy.

Figure 1 :

 1 Figure 1: ℤ/𝟒ℤ in several representations.

Conclusions and looking ahead

Supporting students by stressing mathematical methods, and particularly by stressing heuristic strategies, seems to have a positive impact, but is not realized easily. For this we will continue with this tutorial in the third and fourth semester bringing in heuristic strategies wherever possible. Miller, G. A. (1956). The Magical Number Seven, Plus or Minus Two: Some Limits on our Capacity for Processing Information. Psychological Review, 63, 81-97. Pólya, G. (1945). How to solve it. A new aspect of mathematical methods. Princeton University Press.

Thematic Working Group 14

Proceedings of CERME10


text:
Mária Slavíčková
email: slavickova@fmph.uniba.sk

Developing conceptual knowledge by using ICT on mathematics lessons

Keywords: course evaluation, diagnostic teaching, active learning, Bloom taxonomy

Theoretical background and research question

According to our previous study [START_REF] Slavíčková | Changes in teaching of calculus at the university level[END_REF] we state the following research questions: "How to design a course of calculus to help students obtain higher level of understanding of the topic? How to implement an ICT into this process? Will be using of and ICT helpful?" To find the answers we used mathematical and didactical software in the lectures for presentation of mathematical terms, definitions, properties etc. We also created an e-learning course in the Learning Management System Moodle (LMS). We used this LMS as a primary communication channel between the teacher and the students, we published there all the materials from the lectures, as well as some extra tasks for the students (solving of these tasks was not obligatory).

Activities implemented into the educational process were based mostly on constructivism. Our goal was prepare such activities which help students obtain cognitive dimension "Apply" according to the RBT (Revised Bloom Taxonomy) according to [START_REF] Anderson | A Taxonomy of Learning, Teaching and Assessing: a Revision of Bloom's Taxonomy[END_REF] and Web-link Bloom taxonomy (2015). Most of the activities were supported by ICT and we used mostly the works of [START_REF] Jonassen | Computers as Mindtools for Schools[END_REF] and [START_REF] Kadijevic | Coordinating the process and object features of mathematical knowledge by CAS[END_REF].

The important notes concerning the learning/teaching process according to RBT:  Before you can understand a concept, you must remember it.  To apply a concept you must first understand it.  In order to evaluate a process, you must have analyzed it.  To create an accurate conclusion, you must have completed a thorough evaluation.

Intervention

We used Derive, Graphic Calculus, MS Excel and GeoGebra software in the lessons to demonstrate mathematical properties of functions, sequences, limits of a sequence, a function etc. We prepared small environments for observing, modeling and exploring.

Organization of the teaching -learning process

Lectures: lectures were lead by us; we used the computer and data projector to project mathematics formulas, theorems, some parts of the proofs and the most importantto demonstrate the relationships between the theory and the praxis.

Seminars:

The main work was on the students. We use data projector and computer to project the tasks, questions, some interesting schemas etc. Students can use computers to solve them or to help themselves to get deeper view into the problem. The materials from the activities were uploaded to the LMS Moodle, so students can use them at home preparation.

Thematic Working Group 14

Proceedings of CERME10

The Table 1 shows the focus on the lessons and teston the lessons we focused every field marked with "x". On the test were only task highlighted by red.

Results

The implementation ICT and using different type of software we obtained the good results. The most important results from the observations in our group are:

 students take an active part of the teaching/learning process,  students wanted to discuss the interesting mathematical topics on the lessons,  students started modelling of the situation without a teacher's command,  students interest in the topics was higher than in the other group (taught by our colleaguesit could be also by the personality of the teacher, but previous years showed us, that in our and colleagues group were similar result),  some students started studying more outside the classroom so they can follow the topics without the troubles (we avoided the troubles which still resist in colleague group, like if we change the task a little, students do not know what to do)

Discussion

The preliminary results shows that there is potential in using ICT on calculus lessons to obtain deeper knowledge and better understanding of the topic. Using different kind of software shows students different approaches to the different issues. The next step in our research will be to enlarge the sample and to prepare more technology oriented materials for the research.

Table 1 : Cognitive and the knowledge dimension according to RBT

 1

		Remember Understand Apply	Analyze Evaluate Create
	Factual	x	x	x	x
	Conceptual	x	x	x	x	x
	Procedural	x	x	x	x	x
	Metacognitive


text:
Tore Alexander Forbregd

Magdalini Lada
email: magdalini.lada@ntnu.no

The flipped learning model in teaching abstract algebra

Keywords: College mathematics, algebra, flipped learning

Introduction

Flipped learning is a relatively new model of instruction currently growing both in popularity and success. In a flipped classroom the elements of typical lecture and homework are reversed. Students are introduced to new material at home, mostly through videos prepared by the teacher, while the classroom time is reserved for solving problems, group work, discussions and other activities that help students control, deepen and extend their understanding and knowledge.

The board of the Flipped Learning Network has defined Flipped Learning as … a pedagogical approach in which direct instruction moves from the group learning space to the individual learning space, and the resulting group space is transformed into a dynamic, interactive learning environment where the educator guides students as they apply concepts and engage creatively in the subject matter (FLN, 2014) The number of college and university instructors who practice Flipped Learning has increased over the past two years and has expanded in all subject areas. Research in Flipped Learning has shown positive impact both on students' achievement and engagement [START_REF] Overmeyer | Research on Flipping College Algebra: Lessons Learned and Practical Advice for Flipping Multiple Sections[END_REF].

Methodology

During our study we collected data from the students, the lecturer and the class. We conducted a semistructured interview with the lecturer in the middle of the semester at the university campus and had several informal discussions through Skype and emails at the end and after the end of the semester. The students were given evaluation forms to fill out both in the middle and in the end of the semester, the latter being enriched with open questions where the students were asked to describe their experience with the flipped course. In addition, we visited the class and observed how in-class time was spent and how the students interacted with each other and with the teacher.

The researchers in this study have direct experience with the course material and the way it was previously taught, as they attended it as students. The second author has also taught the course.

The structure of the course

The lecturer-informant recorded videos, using a mobile phone, while writing down on a blanc A4 paper what he would otherwise write on the blackboard in a traditional lecture. The videos were then made available to the students through the course's webpage. The problem sheets, which are, except from small changes, the same sheets that were used in previous years, were also put out on the web. The students were instructed to watch the videos at home and work with the problems when they met in class.

Ten to twelve students on average were normally attending the class, which is an expected number for this course. In a relaxed and informal environment, students formed groups freely and worked with the problems, while the lecturer was walking around answering questions and guiding them through the problems. The groups were let to progress through the problem sheets at their own tempo and already in the middle of the semester different groups were working with different parts of the curriculum. Thirteen students took the exam at the end of the semester.

Findings

Even on this primitive form the Flipped Learning Model turned out to be beneficial to the learning outcome of the students. The students spent more time working on computational problems, concretizing difficult abstract concepts such as for example the radical of rings and modules, projective and injective modules, exact sequences, resolutions and dimensions, clarifying in this way the connection between theory and concrete problems on special classes of algebras. Some more advantages pointed out by the lecturer include, among others:  Differentiated guidance according to the level of each student  Students forced to work more actively with the course, during the whole semester  Better balance between learning the theory and working on problem solving What the students found more beneficial was, among others,  Watching the videos at their pace. Pause it, think, watch it again.  Thorough explanations from the lecturer  Working the problems in groups, learning from each other.

There was a slight improvement on the average of the grades, compare to the previous years the lecturer-informant had taught the course, though the class is too small to make any safe conclusions.

Proceedings of CERME10


text:
Mitsuru Kawazoe
email: kawazoe@las.osakafu-u.ac.jp

Masahiko Okamoto
email: okamoto@hs.osakafu-u.ac.jp

A practical study of a design of university mathematics courses for humanities and social sciences students

Keywords: Mathematical literacy, decontextualization, structuralization

Problems in university mathematics education: in the Japanese context

It is becoming more important for all students regardless of educational level to acquire the ability to use mathematics in a variety of contexts, especially in real world situations [START_REF] Niss | Mathematical literacy[END_REF]. From this viewpoint, mathematics education for non-mathematics students at university is becoming an important issue; however, it is still under-investigated [START_REF] Artigue | Mathematics education research at university level: Achievements and challenges[END_REF]. In Japan, it is an emergent issue that we have no successful mathematics curriculum designed especially for humanities and social sciences students. Many of those students have math anxiety and difficulty in doing and learning mathematics, and do not understand how mathematics is used in the real world. Hence mathematics education for those students is a challenging issue.

Design principles of mathematics courses for humanities and social sciences students

Continuing empirical studies from our previous research [START_REF] Kawazoe | A design research in the course of mathematics with meaningful learning for humanities students[END_REF], we have developed the following design principles of mathematics courses for humanities and social sciences students: (1) Design lessons according to mathematical modelling processes; (2) Choose topics and contexts by considering which mathematical knowledge students will encounter in real life and in which situations they will encounter it; (3) Present problems in different contexts associated to the same mathematical knowledge; (4) Connect different areas of mathematical knowledge by using different mathematizations of the same problem or mutually related contexts; (5) Explain mathematical concepts and operations in both mathematical language and everyday language; (6) Engage students in group activities rather than individual activities; (7) Design worksheets based on hypothetical cognitive processes of students' understanding and use them as tools for formative assessment.

The above principles originate in the four perspectives of learning environments developed in learning science (Bransford, 2000, Chapter 6). The first four principles (1)-( 4) also originate in discussions on mathematical literacy (cf. [START_REF] Sfard | Reflections on mathematical literacy[END_REF] and mathematical modelling (cf. [START_REF] Kaiser | Mathematical modelling and applications in education[END_REF]. Especially, (3)-( 4) are aimed at making students' knowledge decontextualized and structuralized respectively, because it is often claimed that teaching mathematics in real world contexts makes students' knowledge restricted to the learning contexts (cf. [START_REF] Sfard | Reflections on mathematical literacy[END_REF]. In a previous study [START_REF] Kawazoe | A design research in the course of mathematics with meaningful learning for humanities students[END_REF], we showed that mathematics courses designed according to the above principles are successful in reducing students' math anxiety and motivating them to learn math. However, their effectiveness for decontextualization and structuralization was not examined. To examine this effectiveness is the research objective of the present study.

Evaluation of decontextualization and structuralization

To evaluate the effectiveness of our design principles for decontextualization and structuralization, we analyzed the performance of examinations and the free descriptions in the self-report questionnaires conducted at the end of each semester in Basic Math I and Basic Math II, which are successive one-semester mathematics courses for humanities and social sciences students designed according to the above principles. The data were collected in the academic year 2012, when 300 and 244 students took the courses respectively, with students divided into four classes.

The examinations mainly consisted of problems in real world contexts, but the contexts were different from the learning contexts. The mean scores of the exams of Basic Math I & II were 85.9 and 75.4 out of 100, respectively. The results of the exams indicated that students were able to use mathematics in contexts different from those they had studied before, suggesting that they acquired abilities to use mathematics in real world situations.

In the qualitative analysis of the self-report questionnaire, we found evidence of structuralization of mathematical knowledge only in a small number of descriptions. Here we show an example:

Student:

In this course, it seemed that mathematical formulae, which were just isolated pieces of knowledge, had been changing into meaningful ones. I felt that the knowledge base of mathematics that I had became harder and stronger; in that respect, it was a meaningful time.

In sum, the design principles can be considered to be effective for decontextualization, however, the structuralization is still a challenge. More study is needed for structuralization.

Proceedings of CERME10


text:
Diarmaid Hyland
email: diarmaid.hyland3@mail.dcu.ie

Paul Van Kampen

Brien Nolan

Investigating students' difficulties with differential equations in physics

Keywords: Mathematics education, physics education, APOS theory. Research questions

We have investigated the difficulties encountered by undergraduate physics students when studying differential equations, and how these are best addressed. We developed a survey to identify these difficulties that was administered to a pilot cohort of students. The results were used to develop an instructional intervention, informed by APOS Theory, that seeks to address the difficulties uncovered by the survey. The intervention comprised fourteen one-hour tutorials. The tutorials were trialled for the next iteration of the module in question and the survey was given to the students who had completed the module. Applying a design based research approach, the results from these surveys were used to improve the intervention which is being evaluated using a combination of pre-and posttesting and interviews with students in the coming academic year. The interviews will be analyzed using the APOS framework, which acts as the overall conceptual framework for this research

1. What is the precise nature of the difficulties encountered by physics students in using (identifying the need for/setting up/solving) differential equations?

2. How may these be addressed?

These questions lead to the following specific aims for the project which are: (1) identify the areas students struggle with and excel in during their study of differential equations;

(2) develop a set of tutorials based on the data from the survey; and (3) evaluate and improve the tutorials that were developed.

Conceptual framework

In 1984, Dubinsky began developing a theory of how mathematical concepts may be learned. This eventually became known as APOS Theory and is a constantly evolving model developed and refined through application in research and instructional design. As described by Arnon et al.

APOS Theory focuses on models of what might be going on in the mind of an individual when he or she is trying to learn a mathematical concept and uses these models to design instructional materials and/or evaluate student successes and failures in dealing with mathematical problem situations. (2014, p.1)

As explained by Dubinksy (1991), APOS Theory describes the mental structures and mechanisms an individual constructs and applies when trying to understand a problem in mathematics. This project primarily uses APOS Theory as an analytical, evaluative tool. The language of APOS Theory is used to describe the level of understanding displayed by students during their interviews.

Methodology

Achieving the aims outlined in the opening sections will require the combination of both qualitative and quantitative data. The design diagram below shows how the project is structured. The first aim, identifying difficulties, gave rise to both quantitative and qualitative data, gathered using a survey. The Diagnostic Survey was divided into four separate sections assessing different aspects of the students' learning in differential equations: prior mathematical learning; conceptual issues in the study of differential equations; transfer issues; and modelling. To evaluate the tutorials a series of pre-and post-tests and interviews are being used to assess the effectiveness of the intervention. These results feed into the evaluation and improvement of the tutorials.

Results

The results obtained from the Diagnostic Survey show that of the four sections contained in the survey, students struggle most with conceptual understanding, indicating (in terms of APOS Theory) shortcomings in their ability to encapsulate Actions and Processes as Objects. Comparing to the results of the students who completed the tutorials, we saw a dramatic increase in their conceptual understanding, in addition to improvements in the three other sections of the survey. The pre-and post-test data were also used to amend the tutorials for the second research cycle.

Figure 1 :

 1 Figure 1: Design diagram

Proceedings of CERME10