R Vargas

J Neggers

R B Canto

J A Rodrigues

F Hild
email: hild@lmt.ens-cachan.fr

Comparison of two full-field identification methods for the wedge splitting test on a refractory

Keywords: Crack tip position, digital image correlation, finite element model updating, stress intensity factors, virtual test, crack propagation

Two full-field identification methods are applied to the Wedge Splitting Test (WST) to obtain crack tip positions, stress intensity factors (SIFs) and T -stress.

The first method is based on Finite Element Model Updating (FEMU), and the second is integrated digital image correlation (IDIC). Both are applied to a simplified virtual experiment and then to a cyclic WST. The gray level residuals are used to assess which results are more trustworthy. Fracture energy analyses are performed to validate the estimated R-curves.

Introduction

Refractory castables are ceramics that enable functional properties to be maintained in aggressive applications, generally at high temperatures [START_REF] Lee | Castable refractory concretes[END_REF]. Not only the chemical composition but also raw material characteristics should be considered when designing new formulations since they affect the resulting phases, microstructures and thermomechanical properties [START_REF] Luz | Refractory Castable Engineering[END_REF]. Different compositions may be needed for the same application because of several thermomechanical loading steps during industrial processes [START_REF] Lee | Evolution of in situ refractories in the 20th century[END_REF], which spur innovations to improve performance such as increasing reactiveness during processing by using nanomaterials [START_REF] Luz | Refractory Castable Engineering[END_REF] or mimicking microstructures observed in nature [START_REF] Studart | Bioinspired ceramics: Turning brittleness into toughness[END_REF]. In high risk applications, the material and mechanical properties should be well understood to better predict failures and thereby prevent accidents, while maximizing efficiency and lifetime [START_REF] Wachtman | Materials and Equipment -Whitewares -Refractory Ceramics -Basic Science: Ceramic Engineering and Science Proceedings[END_REF].

The Wedge Splitting Test (WST) is a mechanical experiment allowing crack propagation to be assessed in (quasi)brittle materials [START_REF] Tschegg | Prüfeinrichtung zur Ermittlung von bruchmechanishen Kennwerten sowie hiefür geeignete, Prüfkörper, Austrian Pat[END_REF][START_REF] Brühwiler | The wedge splitting test, a new method of performing stable fracture mechanics tests[END_REF]. It leads to stable crack propagation by decreasing the elastic energy stored in the machine using a wedge and cylinders to apply the load [START_REF] Harmuth | Stability of crack propagation associated with fracture energy determined by wedge splitting specimen[END_REF]. A high fractured-area to volume ratio leads to achieving representative results for coarser microstructures at laboratory scales [START_REF] Brühwiler | The wedge splitting test, a new method of performing stable fracture mechanics tests[END_REF]. The classical goal of WSTs is to obtain the fracture energy, by integrating the load vs. displacement curve and dividing by the projected fractured area [START_REF] Brühwiler | The wedge splitting test, a new method of performing stable fracture mechanics tests[END_REF]. Although important, the fracture energy itself is not the only information that can be extracted from WSTs. Recently, it is becoming common practice to seek more data from each of these tests to better understand the fracture process. Relationships between WST measurements with microstructures can be obtained [START_REF] Ribeiro | The influence of microstructure on the maximum load and fracture energy of refractory castables[END_REF], and various works have shown how different compositions affect crack propagation (with the WST) for magnesiaspinel [START_REF] Grasset-Bourdel | Tensile behaviour of magnesia-spinel refractories: Comparison of tensile and wedge splitting tests[END_REF], Al 2 O 3 -C [START_REF] Liao | Synergic effects of nano carbon sources on thermal shock resistance of Al 2 O 3 -C refractories[END_REF], Al 2 O 3 -MgO-CaO [START_REF] Long | Fracture behaviour and microstructure of refractory materials for steel ladle purging plugs in the system Al 2 O 3 -MgO-CaO[END_REF], and MgO-C [START_REF] Zhu | Fracture behavior of low carbon MgO-C refractories using the wedge splitting test[END_REF][START_REF] Zhu | Mechanical behavior and thermal shock resistance of MgO-C refractories: Influence of graphite content[END_REF] systems.

It is also possible to identify fracture properties using measured load data and compare them with numerical simulations of the WST [START_REF] Jin | Determination of Youngs modulus, fracture energy and tensile strength of refractories by inverse estimation of a wedge splitting procedure[END_REF].

More experimental data can be acquired from WSTs via full-field measure-ments. Among them, Digital Image Correlation (DIC) enables displacement fields to be measured [START_REF] Sutton | Computer vision-based, noncontacting deformation measurements in mechanics: A generational transformation[END_REF], and has been successfully used in crack propagation analyses [START_REF] Mcneill | Estimation of stress intensity factor by digital image correlation[END_REF][START_REF] Abanto-Bueno | Investigation of crack growth in functionally graded materials using digital image correlation[END_REF][START_REF] Forquin | A method to determine the toughness scatter of brittle materials[END_REF][START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF][START_REF] Mathieu | Identification of a crack propagation law by digital image correlation[END_REF][START_REF] Saracura | Surface crack network detection on MgO-based refractory castable by digital image correlation[END_REF]. DIC was already applied to WSTs to analyze the strain fields for microcrack formation in magnesia refractories, when a spinel phase was added [START_REF] Belrhiti | Investigation of the impact of micro-cracks on fracture behavior of magnesia products using wedge splitting test and digital image correlation[END_REF]. Similar ideas were used to quantify the fracture process zone of magnesia refractories, and highlight how microcracks tend to decrease the strength but increase the fracture energy [START_REF] Dai | Observation and quantification of the fracture process zone for two magnesia refractories with different brittleness[END_REF], and to estimate the crack growth resistance [START_REF] Dai | Determination of the fracture behaviour of MgO-refractories using multi-cycle wedge splitting test and digital image correlation[END_REF]. Another interesting approach is to measure crack propagation while checking the discontinuities in the displacement field [START_REF] Dupré | Detection of cracks in refractory materials by an enhanced digital image correlation technique[END_REF]. An integrated-DIC (IDIC) scheme to measure the R-curve behavior was also proposed [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF] by considering closed-form solutions of one propagating crack in an elastic medium [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF].

The aforementioned test [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF] will be considered herein in order to compare two different approaches for estimating fracture mechanics parameters using full-field measurements. It is important to note that the hypothesis of one straight crack is reasonable in experiments with the presence of a groove on the propagating faces, as commonly used for the WST to ensure more straight crack propagation [START_REF] Rieder | Notch sensitivity of ordinary ceramic refractory materials[END_REF]. However, crack branches may occur in the WST [START_REF] Dai | Determination of the fracture behaviour of MgO-refractories using multi-cycle wedge splitting test and digital image correlation[END_REF] and it should be checked for each test. With both approaches investigated herein, this check is part of the methodology.

In this paper, a procedure based on the methodology used in Ref. [START_REF] Mathieu | Identification of interlaminar fracture properties of a composite laminate using local full-field kinematic measurements and finite element simulations[END_REF] is applied to the WST. By using the outer measurements from DIC analyses as Boundary Conditions (BC) for a Finite Element (FE) analysis and using internal nodes for error estimators, it is possible to determine the crack tip position and calculate Stress Intensity Factors (SIFs), i.e., K 1 and K 2 , and the T-stress. It will then be compared with IDIC. First, the experiment, the DIC principles and both methodologies that will be compared are introduced. It is followed by an analysis of a virtual experiment. Then an experimental study is performed to compare both methods.

Methods

The two methods studied herein are summarized hereafter. Both of them were used independently to analyze various experiments with cracks [START_REF] Mathieu | Identification of interlaminar fracture properties of a composite laminate using local full-field kinematic measurements and finite element simulations[END_REF][START_REF] Affagard | Identification method for the mixed mode interlaminar behavior of a thermoset composite using displacement field measurements and load data[END_REF][START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF][START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF]. However, they were never compared with the same data set, be they synthetic or from an actual experiment. The first method couples FE analyses and DIC measurements in order to determine the crack tip position, stress intensity factors and T -stresses [START_REF] Mathieu | Identification of interlaminar fracture properties of a composite laminate using local full-field kinematic measurements and finite element simulations[END_REF]. It belongs to the class of finite element model updating techniques [START_REF] Grédiac | Full-field measurements and identification in solid mechanics[END_REF]. The second approach corresponds to integrated DIC [START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF], which is a standalone technique in comparison with the previous framework when applied to the analysis of cracked samples. Augmented Williams' series [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF] are used, in particular, for the determination of the crack tip position [START_REF] Roux | Digital image correlation and fracture: An advanced technique for estimating stress intensity factors of 2D and 3D cracks[END_REF][START_REF] Henninger | Enriched kinematic fields of cracked structures[END_REF].

Experiment

The WST analyzed herein was performed on a class C, anti-erosive commercial refractory, with ultra low cement content, whose typical mineralogical composition consists of quartz, mullite, kyanite, β-cristoballite and alumina [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF].

Water was added to the mixture up to 8.5 wt% of concrete. Drying was performed for 48h in humid environment at room temperature followed by 24h at 110 • C. A heat treatment was performed with a 1 • C/min rate and kept at 500 • C for 24h. The detailed chemical composition and the treatment of the material are reported in Ref. [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF]. Its processing and microstructure may lead to an increasing R-curve behavior, with weakly bonded grains and initiated microcracks due to anisotropic phases and differential thermal expansions.

The sample size is 100 mm in length, 100 mm in height and 72.5 mm in thickness. The geometry is shown in Figure 1, in which it is possible to see the contour of the sample and the loading devices (wedge, cylinders and blocks).

Two grooves (i.e., lateral notches, see dashed line in Figure 1) are machined on two opposite faces of the sample to reduce the local thickness and guide the crack propagation vertically. The two zones where the splitting displacement is evaluated via DIC are also shown in Figure 1 as yellow boxes. Considering δ as the initial distance between both zones, the horizontal displacements measured on these regions are averaged and their difference accounts for the splitting displacement ∆δ, which will be reported in Section 4. The Young's modulus (E) and Poisson's ratio used for the investigated methods are equal to 17 GPa (measured by the bar resonance method [START_REF] Diógenes | Determination of modulus of elasticity of concrete from the acoustic response[END_REF]) and 0.2, respectively. The test was driven by setting the velocity of the machine actuator to 1.3 µm/s, and 313 pictures were taken for both faces of the specimen at a rate of one picture each 8 s. The images were simultaneously acquired with two Canon T5 cameras with 28-135 mm lenses, with the illumination provided by LEDs. The 16-bit picture definitions are 2601 × 1733 pixels, with a dynamic range of approximately 60,000 gray levels. The imaged physical size of one pixel was 62 µm. A random speckle pattern was sprayed onto the specimen surfaces to increase the image contrast and improve the DIC resolutions.

The 5-cycle loading curve of the experiment, which corresponds to the vertical force F v vs. vertical actuator displacement ∆h, is shown in Figure 2. Further information on this test can be found in Ref. [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF], and further characterization of the studied material, processing and microstructure in Refs. [START_REF] Pereira | A study about the contibution of the α-β phase transition of quartz to thermal cycle damage of a refractory used in fluidized catalytic cracking units[END_REF][START_REF] Miyaji | The coke effect on the fracture energy of a refractory castable for the petrochemical industry[END_REF].

Digital Image Correlation

In global DIC, the displacement fields u(x) are measured by minimizing the L2-norm of the gray level residuals, φ 2 , between the image of the reference state f , and at the deformed state g

φ 2 = ROI [f (x) -g(x + u(x))] 2 . (1)
Searching for displacements at the pixel level is an ill-posed problem [START_REF] Hild | Digital image correlation[END_REF]. Parameterizations of the sought displacement u are used to regularize this problem and render the solution less affected by the acquisition noise

u(x) = N i=1 υ i Ψ i (x), (2)
in which υ i are the degrees of freedom, and Ψ shape functions (i.e., vector fields) that combine the displacement of a group of pixels in order to make the problem well-posed. The solution becomes

{υ DIC } = arg min {υ} φ 2 ({υ}), (3)
where {υ DIC } is the column vector gathering all amplitudes υ i . If the actual kinematics of the problem is not well known a priori, Ψ i can be chosen as finite element shape functions [START_REF] Besnard | Finite-Element" displacement fields analysis from digital images: Application to Portevin-Le Chatelier bands[END_REF]. In the present case, the DIC procedure is performed with 3-noded linear elements based on finite element discretization [START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF] and will be referred as T3DIC. In global approaches, the quality of the registration can be evaluated pixel-wise by computing the gray level residuals

ρ T3 = f (x) -g (x + Ψ T3 (x, {υ T3 })) , (4)
where Ψ T3 is the vector containing the shape functions converting nodal to pixel displacements, which depends linearly on the amplitude {υ T3 }. In the present case, the T3DIC mesh is composed of 3-noded elements whose average edge length is equal to 58 pixels (or 3.6 mm).

In the following, the global residual of DIC approaches will be compared. It is defined as the root-mean-square (RMS) gray level residual over the considered ROI

ρ T3 = RM S(ρ T3 ) ∆f , ( 5
)
where ∆f is the dynamic range of the picture of the reference configuration

∆f = max ROI f -min ROI f ≈ 60, 000 gray levels. (6)
The first step of any of the methods presented hereafter is to run T3DIC.

It provides displacement fields that can be compared with FE analyses, but also allows the crack path to be chosen for integrated DIC [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF]. The maximum eigen strain field is selected in order to check the validity of the straight crack propagation assumption and the presence of a single macro-crack. The two faces of the sample are analyzed with a very fine mesh of average element length of 8.5 pixels (or 530 µm). Figure 3 shows the results for both faces for the last image before the final unloading (Figure 2). The standard uncertainty of the maximum eigen strain is of the order of 3 × 10 -4 and the minimum strain level in the color bars of Figure 3 is set to 3 times this value. It was determined by correlating the two pictures shot for the reference configuration on each face.

Only one unique macro-crack is observed (guided by the groove).

Method 1: Finite Element Model Updating

The methodology described in this section is an adaptation of that proposed in Ref. [START_REF] Mathieu | Identification of interlaminar fracture properties of a composite laminate using local full-field kinematic measurements and finite element simulations[END_REF] to the WST. DIC results are taken as Dirichlet boundary conditions and FE formulation computes the displacement field over the considered surface. The crack tip position is identified as the one that provides the best fit between the simulated and measured displacement fields. The main idea of the method is schematically shown in Figure 4, where the outer contour of T3DIC measurements, represented in green, are prescribed to the FE model, and internal nodes (blue region) are used for comparison with FE analyses. Several crack tip positions are tested along the groove region (red dashed line), and the one that gives the least root mean squared displacement gap is considered the best estimate and thus, chosen as the crack tip position for the considered image.

The method is then repeated for every image taken during the test. It will be referred to as FEMU henceforth. It is important to highlight that such a procedure is run for each image taken during the experiment, so that one crack tip position is obtained for each considered step time. Within an FE code, it is generally possible to calculate quantities such as the mode I and II SIFs K I , K II , and the T -stress [START_REF] Simulia | Contour integral evaluation[END_REF]. Thus the change of these fracture mechanics parameters is evaluated for each analyzed crack length. Given the thickness of the sample, a plane strain hypothesis is considered in this paper for all the reported analyses.

The FEMU displacements and mesh are exported and interpolated onto the T3DIC mesh. The nodal displacement difference between T3DIC and FE analyses is computed, and the displacement gap consists in the RMS difference

φ 2 u = N i=1 υ FEMU i -υ T3 i 2 , (7)
in which υ i are nodal displacements. The superscript indicates if the displacements are obtained from FEMU or T3DIC analyses. It is worth noting that the gap estimate proposed in Equation ( 7) considers the same influence from every considered node. This definition may be improved using the T3DIC Hessian to augment the weight on nodes with respect to lower measurement uncertainties [START_REF] Mathieu | Estimation of elastoplastic parameters via weighted FEMU and integrated-DIC[END_REF]. Once different crack positions are tested, the one that provides the minimum φ u is taken as the estimated crack tip position, and the fracture mechanics properties are assessed with this configuration.

Although φ 2 u is considered for crack tip identification, the gray level residual ρ FEMU can be computed using the nodal displacements of the FE model {υ FE }

ρ FEMU = f (x) -g (x + Ψ FE (x, {υ FE })) , (8)
where Ψ FE is the vector containing the shape functions converting nodal to pixel displacements, which is linearly dependent on the measured degrees of freedom {υ FE }. The corresponding global residual reads

ρ FEMU = RM S(ρ FEMU ) ∆f . (9)
It will be compared with T3DIC results and the following integrated method.

Method 2: Integrated DIC

This section summarizes the methodology used to analyze a wedge splitting test with Integrated-DIC [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF] using a closed-form solution. Williams' series [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF] describe the kinematics in this case, and the gray level residual is minimized (Equation 1), instead of the displacement gap for FEMU (Equation ( 7)). The sought displacement field reads

u(z) = II j=I p f n=pi ω j n ψ ψ ψ j n (z), (10)
where the vector fields are defined in the complex plane

z = (x -x c ) + (y -y c )i = r exp (iθ), (11)
where (x c ,y c ) are the coordinates of the crack tip position, j = I is related to the mode I (opening) regime and j = II to mode II (shearing). The amplitudes ω j n become the unknown kinematic degrees of freedom of IDIC. The corresponding displacement fields are described by

ψ ψ ψ I n = A(n) 2µ √ 2π r n/2 κ exp inθ 2 - n 2 exp i(4 -n)θ 2 + (-1) n + n 2 exp - inθ 2 , (12)
and

ψ ψ ψ II n = iA(n) 2µ √ 2π r n/2 κ exp inθ 2 + n 2 exp i(4 -n)θ 2 + (-1) n - n 2 exp - inθ 2 , ( 13
)
where κ is equal to (3 -ν)/(1 + ν) for plane stress states or 3 -4ν for plane strain states, ν the Poisson's ratio and A(n) is defined by

A(n) = cos nπ 2 2 + sin nπ 2 . ( 14
)
The amplitude ω j 1 gives access to Stress Intensity Factors (SIFs), the amplitude ω I 2 provides the T -stress (positive in compressive) and ω II 2 the rigid body rotation. Higher order fields account for deviations from the theoretical assumption of an infinite medium [START_REF] Henninger | Enriched kinematic fields of cracked structures[END_REF]. With the use of two additional terms in the series, namely using p i = 0, ω I 0 and ω II 0 are related to rigid body translations.

Although not usual, negative values of p i can help to account for nonlinearities [START_REF] Henninger | Enriched kinematic fields of cracked structures[END_REF][START_REF] Mathieu | Identification of interlaminar fracture properties of a composite laminate using local full-field kinematic measurements and finite element simulations[END_REF]. It can be seen from Equations ( 12) and ( 13) that for negative values of n, super-singular solutions arise near the crack tip (r = 0). They are also important to locate the crack tip position, especially ω I -1 . With the assumption that the crack tip is perturbed by a small distance d, along with some recursive properties of the Williams' series [START_REF] Roux | Digital image correlation and fracture: An advanced technique for estimating stress intensity factors of 2D and 3D cracks[END_REF], it is possible to derive the offset

d = 2ω I -1 ω I 1 , (15)
which provides an estimation of crack tip shift to find the correct solution. In the sequel, p i = -3 is taken to account for nonlinearities [START_REF] Henninger | Enriched kinematic fields of cracked structures[END_REF]. The maximum value p f = 8 is chosen after a convergence analysis [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF]. A normalization of Ψ is also performed in order to decrease floating point rounding errors.

The pacman-like ROI used for IDIC1 is shown in Figure 5. An opening of 40 pixels (pacman mouth) is taken in order not to consider the cracked mouth.

The Williams' series are projected onto an FE mesh to allow the use of the same FE-DIC code as used in the previous section. This mesh is chosen sufficiently fine not to influence the results. (A convergence study was performed to check this statement.) The procedure to find the crack tip positions and the mechanical parameters of the fracture process can be summarized in the present steps:

1. A crack path is first defined as a straight line, in the groove region of the photographed sample face (Figure 4), as proved by the analysis of the maximum eigen strain fields (Figure 3);

2. The parameters defining the truncation of Williams' series are chosen, i.e., p i = -3, p f = 8, the radius for normalization and mesh parameters;

3. The calculation is initialized for the crack position assumed to be located at the notch root;

4. An optimization algorithm evaluates the amplitudes {ω} that decrease the global gray level residual [START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF];

5. The crack tip position is updated using Equation (15); 6. Steps 4 and 5 are repeated until convergence (i.e., d < 0.1 pixel).

After convergence, the corresponding gray level residual field is stored

ρ IDIC = f (x) -g (x + Ψ IDIC (x, {ω})) , (16)
where {ω} gathers all converged ω j n amplitudes (see Equation ( 10)), and Ψ IDIC are the shape functions described in Equations ( 12) and ( 13) evaluated for each pixel position x. From this information, the global IDIC residual becomes

ρ IDIC = RM S(ρ IDIC ) ∆f . ( 17
)
This methodology will be referred to as IDIC in Section 4.

Analysis of a virtual experiment

First, a virtual experiment is considered to test both approaches on a configuration for which the exact solution is known. It is discussed how to deform a reference image with displacements obtained from numerical simulations (e.g., Abaqus T M ). In the sequel, this virtually deformed image is then analyzed using the previous two methods. This virtual experiment will be referred to as VE in Section 3.2

Virtual experiment

A sketch following the instructions from Ref. [START_REF] Ribeiro | The influence of microstructure on the maximum load and fracture energy of refractory castables[END_REF] (width = 100 mm and height = 100 mm) is performed in Abaqus T M , with some adjustments such as the depth of the extrusion set to 72.5 mm related to the sample geometry [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF].

The numerical model is presented in Figure 6. Normal pressure was applied to the vertical faces onto which the rollers would apply the load. The line at the bottom of the sample does not move in the x and y-directions. In the zdirection, one single point has no motion. A straight crack is added and its tip is located in the middle of the crack propagation path of the sample (i.e., only one loading step is considered). The mesh is refined around the crack tip. A 2D mesh is extracted from the image plane presented in Figure 6 to consider a plane strain state. K I , K II , and the T -stress measurements at this plane are considered. For the present study, a reference image is also required. An actual image that shows the whole sample surface [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF] is used in the sequel. The 2D mesh is extracted and interpolated onto pixel coordinates such that it fits the sample in the image, as shown in Figure 7. Once the displacement field for each pixel is known, the deformed image is created. In DIC the gray level conservation reads

f (x) = g(x + u(x)), (18)
in which x contains integer pixel coordinates in the reference image. However the position x+u(x) = θ(x) is not necessarily an integer. The evaluation of g(θ(x)), which corresponds to the picture of the deformed configuration corrected by the displacement field u, requires an interpolation scheme of the gray levels.

In the present case, an inverse mapping is required, namely, integer valued positions x g are sought to construct the picture in the deformed configuration g. Consequently, the position θ -1 (x g ) has to be determined for computing the gray level f (θ -1 (x g )), which also requires an interpolation scheme [START_REF] Réthoré | Robust identification of elasto-plastic constitutive law parameters from digital images using 3D kinematics[END_REF].

To be more representative of a real experiment, it is chosen to add real noise to the deformed image. The gray level residual from T3DIC results for the first deformed image of the front face is added to the artificially deformed image.

The global residual is equal to 0.57%. This residual at the beginning of the test is considered to be a good approximation of the noise during the test since the load was small and no crack propagation had yet occurred.

Results

The results obtained from the VE analysis with different methods are gathered in Table 1. The global residuals, ρ method , in which method is replaced by VE, FEMU or IDIC, are assessed by correcting the image in the deformed configuration g with the corresponding displacement fields (see Section 3.1, Equations ( 8) and ( 16), respectively). For consistency along the discussion, all the reported residuals are calculated in a pacman-shaped area centered about the crack tip position for the discussed method (see Figure 5). The fact that the residuals for the exact solution are not vanishing is due to the added noise to g and gray level interpolation inaccuracies. Consequently, ρ VE defines the minimum level that can be achieved. Both methodologies (i.e., FEMU and IDIC) are consistent with the exact (i.e., VE) solution since the gray level residuals (ρ FEMU and ρ IDIC ) are only 1.5 times higher than the minimum level (ρ VE ), and that their difference is very small with a value that is slightly lower for IDIC. The later better predicts the crack tip position (∆a in Table 1)

and mode I SIF for an elastic body with a single crack, while the usage of the FEMU procedures provides T -stress levels more accurately. Mode II SIF is believed to be close to the resolution for both methods.

Since the exact solution is known in the present case, the discussion could stop here. However, in an actual experiment, the exact solution is unknown.

Since the ROIs of each method are not located at the same position in the reference image, the crack tip locations predicted by both methods are expected to be different. T3DIC will thus be used to independently assess global residuals computed over the same ROI as those in the considered methods. Since in T3DIC no mechanics-based assumptions are made on the displacement fields apart from their continuity,2 the global residuals ρ T3 method for the same ROI of each method are also evaluated. The difference in global residuals ∆ρ T3 method = ρ method -ρ T3 method [START_REF] Forquin | A method to determine the toughness scatter of brittle materials[END_REF] then assesses the overall identification quality (i.e., ρ T3 method is thus taken as the reference) and the smaller ρ T3 method , the better the identification result.

Table 1 shows that ρ T3 method is virtually identical for the three methods. This is expected since this virtual case only involves noise and gray level interpolation inaccuracies. The fact that ρ T3 VE is greater than ρ VE is due to the T3DIC mesh that is rather coarse. This choice was made since very small displacements are sought (see below) and a finer mesh would have induced higher measurement uncertainties [START_REF] Hild | Digital Image Correlation[END_REF]. This choice also explains why ∆ρ T3 VE < 0. Had a finer mesh been used, T3DIC would be expected to be closer to the VE solution.

Furthermore, ρ T3 FEMU is slightly higher than ρ T3 IDIC , and more importantly, IDIC is closer to T3DIC than FEMU (i.e., ∆ρ T3 IDIC = 0.03% in comparison with ∆ρ T3 FEMU = 0.06%). The difference between both methods remains very small, which validates both procedures. However, IDIC slightly outperformed FEMU in the present analysis.

Experimental study

The two methodologies described in Sections 2.3 and 2.4 are now applied to one wedge splitting test. Figure 8 shows the crack tip position for the first part of the experiment during which the crack has propagated. ∆a = 0 considers the crack tip to be located at the pre-notch root. Both methodologies have high uncertainties for the first two cycles, which are related to very small displacement ranges (i.e., 0.15 pixel, or 9 µm at the most). It is observed that FEMU identifies smaller crack lengths than IDIC. It is worth noting that both methodologies predict different crack propagation histories on the two analyzed faces and that, in the end, the crack propagated farther on the back face (Figure 3). Two features in Figure 8 need to be further commented. First, the negative values of ∆a at the beginning of the experiment, namely, a crack tip position out of the sample, and second, the crack closure in the unloading phase of each cycle.

These observations do not mean that crack healing occurs in the present case. It is believed that the negative values are due to the uncertainties associated with crack tip positioning when the displacement levels are very small (i.e., at the beginning of the experiment). The crack tip itself is a feature associated with the considered fracture mechanics model (here defined with Williams' series or finite element simulations). Although the main conclusions of this work

will not be affected, physically, crack propagation is believed to be stopped during unloading phases, and restarts once it reached a critical SIF level in the subsequent loading cycle.

The SIFs are reported in Figure 9. For this case in which a single macrocrack propagates guided by the groove (Figure 3), pure mode I is expected and is confirmed by both approaches. As K II is close to zero, it can be used as an evaluation of the resolution for SIF evaluations. The RMS of K II values measured by both methodologies and at both faces is of the order of 3 × 10 -2 MPa √ m.

The general tendency observed for K I is opposite in comparison with estimates of crack tip positions, namely, lower K I levels and larger crack lengths are reported by IDIC in comparison to FEMU results. However, crack tip positions and SIFs values obtained by both techniques are of the same order of magnitude. The T -stress histories are shown in Figure 10. The levels obtained via IDIC are generally higher, predominantly after the second cycle and for the front face. For the back face, the values are closer. The fluctuations could be related to mechanical features since some fluctuation were also observed on the load vs. crack mouth opening displacement curve for this test [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF], but further studies are needed to confirm this hypothesis. In order to have an absolute evaluation of the quality of both approaches ρ FEMU and ρ IDIC are reported. This type of analysis is no longer an inter comparison, but probes the individual merit of each technique with respect to the pictures acquired during the experiment with the same number of pixels. The RMS gray level residuals are reported in Figure 11. The overall levels remain very small for the whole sequence, which validates the reported results. For both cases, increasing residuals occur with crack propagation. The longer the crack, the bigger the fracture process zone and the assumption of an elastic medium is presumably less true. Further, 3D effects may also be more pronounced [START_REF] Réthoré | Optimal and noise-robust extraction of fracture mechanics parameters from kinematic measurements[END_REF],

which make the 2D assumptions used in Williams' series and FE calculations less accurate. To compare in a more quantitative way the two procedures the residual difference, ∆ρ, is computed

∆ρ = ρ IDIC -ρ FEMU (20)
Plots of ∆ρ for both faces of the sample are shown in Figure 12. The differences are in 0.2% range, which is lower than the acquisition noise for these images (i.e., ≈ 0.6%). However, IDIC shows lower residuals at the beginning the test.

The crack tip position is better captured (Figure 8), even though the SIF levels are rather consistent with both approaches (Figure 9). The fact that IDIC becomes less accurate than FEMU at the end of the propagation step may be related to the Williams' series not describing the boundary effects as the crack tip approaches the sample edge. The residuals reported in Figure 11 and 12 were calculated for the same ROI size but centered about different crack tip positions (Figure 8). In a virtual experiment ρ T3 method does not change considerably in different regions as reported in Table 1. In an actual experiment, not only slightly different image features at each ROI but also images artifacts such as lighting changes and lens focus may affect ρ T3 method . The residual difference ∆ρ T 3 method (see Equation ( 19)) is shown in Figure 13 and indicates how close each method was close to T3DIC. IDIC residuals are closer to T3DIC than FEMU. Small negative values seen for IDIC and FEMU in some images indicate that they outperformed T3DIC, which can be explained by the non-optimal T3DIC mesh. Although the residuals for both methodologies are in the same range (Figure 12), the ROIs in which FEMU converged were presumably less affected by image artifacts and the residuals were farther from T3DIC residuals. The main conclusion when considering different regions separately (Figure11 and 12) is that IDIC was performing better at the very beginning and then FEMU would be preferred for the rest of the test. However, using ρ T3 method as a reference (Figure 13) in order to account for image artifacts and textures on different regions, the conclusion is that, for the present case, IDIC is more accurate for the whole analysis.

Last, even though very small, K II is considered to analyze the R-curve behavior defined as

R = K 2 I + K 2 II E (1 -ν 2 ) ( 21
)
in a plane strain state, as considered in FEMU and IDIC. It is worth noting that although some deviation from linear elasticity may occur, the effective crack tip is measured with full-field approaches accounting for elasticity. In such cases, with the effective instead of visible crack tips, Equation ( 21) can be applied [START_REF] Bornhauser | R-curve evaluation with ceramic materials at elevated temperatures by an energy approach using direct observation and compliance calculation of the crack length[END_REF].

The R-curve is then plotted as a function of crack tip position in Figure 14. It shows that energy consumption increases as the crack propagates (with a steeper slope for FEMU), which is related to extrinsic toughening mechanisms such as crack branching, microcracking or bridges between aggregates [START_REF] Launey | On the fracture toughness of advanced materials[END_REF]. In the present case, it was checked that crack branching was not occurring out of the groove in the photographed surface of sample, at the scale of DIC measurements (see Figure 3), namely, of the order of the element length (i.e., ≈ 0.5 mm).

With the chosen magnification, the main purpose of DIC was to analyze the macroscopic fracture mechanics parameters, i.e., R-curve behavior , and could resolve the effect of the toughening mechanisms occurring at lower scales. The observed R-curve behavior may indicate crack branching, and although not seen on the strain fields reported in Figure 3, it may take place in the bulk of the material. For checking such mechanisms, in situ tomographies analyzed with Digital Volume Correlation would be a suitable approach [START_REF] Hild | Damage measurements via DIC[END_REF]. The present study enables to assess uncertainties associated with the use of two different identification techniques. The latter ones provided results that were both considered as realistic and may be used when comparing crack propagation in different materials compositions, for instance. However, they led to significant differences in the crack propagation resistance curve of the investigated material (Figure 14). Thanks to the use of gray level residuals, the merit of both techniques could be assessed, but more importantly it could be decided which one was more trustworthy (i.e., IDIC in the present case).

In order to validate the R-curves reported in Figure 14, a last study is performed. First, the loading history shown in Figure 2 may be integrated to estimate the fracture energy, γ wof (v)

γ method wof (v) = 1 2A method ∆h f ∆h0 F v d (∆h) (22)
where the interval [∆h 0 , ∆h f ] corresponds to the loading envelope and the final unloading, ∆h is the vertical displacement of the actuator, F v the vertical force, and method is substituted by IDIC or FEMU for the evaluation of the cracked area A. The latter is considered as the maximum crack length (∆a max for last point before final unloading) obtained for the method (IDIC or FEMU) multiplied by the thickness t g of the specimen inside the groove (i.e., 65 mm).

It is worth noting that γ wof (v) corresponds to an upper bound since it includes dissipated energy through friction of the loading parts.

The splitting force F h vs. displacement ∆δ curves are shown in Figure 15, in which the splitting displacement ∆δ is measured from T3DIC results at the locations shown in Figure 1, namely, at the same height as the splitting forces are applied. By integrating the data shown in Figure 15, the fracture energy γ wof reads

γ method wof = 1 2A method ∆δ f ∆δ0 F h d (∆δ) (23)
where the interval [∆δ 0 , ∆δ f ] corresponds to the loading envelope and the final unloading. In the present case, the DIC extensometry only accounts for the work performed on the specimen itself, thus is more representative of the work of fracture of the studied material.

To calculate the average of R, which is denoted by R, for IDIC and FEMU the R-curves shown in Figure 14 are integrated

R method = 1 ∆a max ∆amax ∆a0 R method d ∆a method (24)
Since the dissipated energy is consumed to create two cracked surfaces, R = 2γ wof (h) [START_REF] Sakai | Fracture toughness testing of brittle materials[END_REF].

The calculated energies are listed in Table 2. All reported values are expected to underestimate the fracture energy measured for the full crack propagation (with same methodology), which is not accessible in the experiment reported herein. The fracture energies have the same order of magnitude, which is a further validation of the R-curves predicted with IDIC and FEMU. However, IDIC predicts values that are closer to those obtained with DIC extensometry than FEMU. FEMU even provides estimates above the upper bound (i.e., γ wof (v) ). Even though FEMU and IDIC results are in the same range for the various quantities reported in the present section, IDIC results are more consistent with experimental data in terms of gray level residuals, but even more importantly when compared with independent estimates of fracture energies. For the latter ones, it is shown that about 20% is dissipated by friction when the upper bound estimate is compared with that derived from the splitting force vs. displacement data, or equivalently by R-curves provided by IDIC.

Conclusions

The FEMU methodology [START_REF] Mathieu | Identification of interlaminar fracture properties of a composite laminate using local full-field kinematic measurements and finite element simulations[END_REF] was adapted to analyze WSTs. An automated procedure was implemented to create an FE model with Dirichlet boundary conditions measured via T3DIC analyses. The crack tip position was identified, among several tested positions, as the one that provided the best kinematic fit.

Interaction integrals of the FE code were used to estimate fracture-related properties (i.e., SIFs and T -stress). This approach was compared with IDIC when applied to a WST [START_REF] Vargas | Analysis of wedge splitting test on refractory castable via integrated DIC[END_REF]. First, a virtual experiment was analyzed, then followed by experimental analyses for both faces of a refractory castable specimen.

FEMU allows for better T -stress estimates, while IDIC is more trustworthy for estimating the crack tip position and mode I SIF in the virtual experiment.

Gray level residuals were used to check the merit of each technique directly with experimental pictures. FEMU residuals were farther from T3DIC than IDIC when considering T3DIC as the reference to check the quality of measurements, proving that IDIC is slightly more trustworthy than FEMU in the present case.

IDIC provides longer crack lengths coupled with lower SIFs in comparison with FEMU. Consequently, the R-curve behavior is less steep for the former.

Different levels of properties on each side and measured by each method highlights the importance of such analyses performed on both sample faces. These differences are related to experimental imperfections such as the fine alignment of the wedge and microstructural heterogeneities.

Accessing gray level residuals is a powerful tool to check analyses with the experimental data when the actual solution is not known. It is of utmost importance to consider the measurement regions in such analyzes. IDIC residuals were closer to T3DIC resulting in more reliable measurements.

The evaluations of fracture energies were consistent with both discussed methods, namely, the same order of magnitude of the R-curves was obtained via IDIC and FEMU. When compared with FEMU, IDIC estimates were closer to those based on DIC extensometry. While IDIC was comparable to conventional methods, FEMU resulted in fracture energies greater than the experimental upper bound. All these differences show that the estimation of the crack tip location, which was different for both investigated approaches, has to be very accurate. The use of the first supersingular field in Williams' series for estimating the crack tip location via IDIC is further validated thanks to the present study.

Figure 1 :

 1 Figure 1: Detailed geometry of the wedge splitting test. The contour of the sample, including the pre-notch, is shown in cyan. The wedge, cylinders and blocks used to apply load are highlighted in white, blue and purple, respectively. The vertical groove in order to guide the crack vertically is shown with the dashed red line. The splitting displacement ∆δ corresponds to the difference of the mean displacement of the two yellow boxes. The thickness of the specimen is 72.5 mm. All dimensions are expressed in mm.

Figure 2 :

 2 Figure 2: Loading history for the test analyzed in Section 4. Each point represents two image acquisitions (one per face) performed during the test for DIC analyses. The red circle depicts the picture acquisitions used to evaluate the strain fields shown in Figure 3.

Figure 3 :

 3 Figure 3: Maximum eigen strain fields for both analyzed faces for the image before the 5th unloading. No macro-crack branches are seen and consequently, the hypothesis of a straight crack propagation guided by the groove can be made.

Figure 4 :

 4 Figure 4: Adaptation of methodology introduced in Ref. [30] to the WST. An area around the propagation path is chosen and the displacements measured by T3DIC at the outer boundaries (green) are used as boundary conditions. The internal measured displacements (blue) are used for comparison with FE outputs, and the crack tip position is tested in the groove region (red).

Figure 5 :

 5 Figure 5: Example of the pacman-like mesh used in IDIC analyses.

Figure 6 :

 6 Figure 6: 3D Abaqus T M model showing the applied pressure and the prescribed displacements. It is also possible to see the partition lines in the middle of the crack propagation path. The used image plane is also shown.

Figure 7 :

 7 Figure 7: Superposition of 2D mesh extracted from the 3D Abaqus T M mesh for the reference image (on the left), with a zoom about the crack tip (on the middle). Horizontal displacements ux (expressed in µm) (on the right) in the virtual experiment.

Figure 8 :

 8 Figure 8: Crack tip position for front and back faces with both methodologies for the first 312 images acquired during the analyzed wedge splitting test.

Figure 9 :

 9 Figure 9: Mode I and II SIF histories for both faces with both approaches.

Figure 10 :

 10 Figure 10: T -stresses measured with both methodologies for both faces.

Figure 11 :

 11 Figure 11: Dimensionless residuals for the two methods (FEMU and IDIC) and both faces.

Figure 12 :

 12 Figure 12: Residual difference for both faces. A negative (resp. positive) value corresponds to a more faithful evaluation via IDIC (resp. FEMU).

Figure 13 :

 13 Figure 13: Differences from IDIC and FEMU residuals in comparison with T3DIC. Positive values are related to smaller residuals measured with T3DIC.

Figure 14 :

 14 Figure 14: R vs. crack length for both faces with both approaches. Images taken during unloading steps are not accounted for the computation of R.

Figure 15 :

 15 Figure 15: Splitting force (i.e., 5.715 times the vertical force) versus splitting displacement ∆δ measured at the locations shown in Figure 1.

Table 1 :

 1 Virtual experiment results. Calibrated fracture mechanics parameters and corresponding global gray level residuals.

		∆a [mm] [MPa K I √	K II m] [MPa √	m]	T -stress ρ method ρ T3 method [MPa] [%∆f ] [%∆f ]	∆ρ T3 method [%∆f ]
	VE (Sect. 3.1)	36.0	3.8	3 × 10 -4	6.4	1.68	2.45	-0.77
	FEMU (Sect. 2.3) 38.9	3.1	3 × 10 -3	7.8	2.52	2.46	0.06
	IDIC (Sect. 2.4)	36.4	3.9	-6 × 10 -2	8.8	2.48	2.45	0.03

Table 2 :

 2 Fracture energies (expressed in J/m 2 ) calculated with different approaches.

		Fract. energy front back mean
		2γ wof (v)	114	83	99
	FEMU	2γ wof	92	71	82
		R	162	97	130
		2γ wof (v)	85	69	77
	IDIC	2γ wof	69	60	64
		R	84	52	68

This ROI is always centered about the evaluated crack tip position

Note that the cracked area is masked by the pacman mouth (Figure5)

Acknowledgments

JAR thanks CNPq for the productivity scholarship, grant #307127/2013-3.

RV's stay at LMT was supported through an RIA scholarship, grant #2017/20911-9, São Paulo Research Foundation (FAPESP).


David Glassmeyer

Kimberly Cortes

Jie Hao

Herman Ray

Gregory Rushton
email: gregory.rushton@stonybrook.edu

A Longitudinal Analysis of First-time Praxis® II Mathematics Content Knowledge Examinees

HAL is

The U.S. uses the Praxis® tests to measure academic skills and the subject-specific content knowledge needed to be a teacher. This study examined Praxis® Mathematics Content Knowledge scores from 2006 to 2016 to describe patterns in who are becoming certified to teach mathematics in the country (n=89,693). Longitudinal analyses were used to discern patterns in the demographics of examinees and trends in exam performance across several demographic characteristics. The results reveal substantial differences in performance and pass rates between examinees of different genders, races, undergraduate majors, undergraduate GPAs, and census regions. From our analyses, we suggest several measures for the improvement of mathematics teacher preparation. Keywords: assessment practices, longitudinal analysis, mathematical content knowledge, teacher preparation.

DESCRIPTION OF POSTER CONTENT

This poster will provide information on the U.S. Praxis® (ETS, 2017) tests to an international audience. A small section will describe the theoretical perspective [START_REF] Hill | Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study[END_REF][START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] and how it was used in this study. Our research questions will be included on the poster, as well as how data were collected and analyzed to answer these questions. The middle portion of the poster will include several graphics depicting the model resulting from the analysis, specifically the statistically significant factors in whether test takers pass the exam and the demographic characteristics of those who pass. These figures will illustrate the substantial differences in performance and pass rates between examinees of different genders, races, undergraduate majors, undergraduate GPAs, and census regions.

We will offer several measures for the improvement of the mathematics teaching workforce and establish potential leaks in the teacher pipeline that may impact the quality and diversity of U.S. mathematics teachers [START_REF] Gitomer | Useful signal or unnecessary obstacle? The role of basic skills tests in teacher preparation[END_REF]. Specifically mathematics education faculty would do well to recognize that students who have sat in their classrooms and lecture halls are represented in this sample and that these scores are at least partially reflective of what they have retained from their courses [START_REF] Kleickmann | Teachers' content knowledge and pedagogical content knowledge: The role of structural differences in teacher education[END_REF]. The adoption of instructional strategies that promote students' long-term conceptual understanding over those that validate rote memorization may be a critical choice on the part of mathematics faculty in improving the preparedness of future mathematics educators.

NOTES

1. In full disclosure, the U.S. Educational Testing Service reviewed this paper, though the analyses were conducted by our research team.


Olov Viirman
email: olov.viirman@hig.se

Irina Pettersson

Johan Björklund

Jonas Boustedt

Programming in mathematics teacher educationa collaborative teaching approach

Keywords: novel approaches to teaching, teaching and learning of mathematics in other fields, team teaching, algorithmic thinking, programming

The digitalization of modern society is having a major impact on schools and schooling, and in recent years, programming and algorithmic thinking is increasingly seen as topics to be included in curricula at all levels. In Sweden, for instance, the National Agency for Education have instigated a revision of the national curricula to emphasize what they denote as "digital competence" (Skolverket, 2017a, b). In particular, the mathematics curriculum now includes the use of algorithmic thinking and programming as tools for problem solving. Since these topics have not been a mandatory part of the education of mathematics teachers in Sweden, this creates a need for such courses aimed both at practicing and prospective teachers. Given the direction taken in the revised curriculum, these courses will need to take an integrated approach, focusing on how programming and algorithmic thinking can be used to learn mathematics and to solve mathematical problems. This places particular demands on the instructors teaching these courses. Not only do they need to have knowledge of programming, but also of mathematics and mathematics education. Most probably, courses in programming aimed at practicing mathematics teachers will attract teachers from all levels of the educational system, placing additional demands on the mathematical competence of the instructors. In the light of these challenges, we are considering the idea of collaborative, or team, teaching. Having mathematicians and/or mathematics educators collaborating with computer scientists can provide different viewpoints on the topics considered and problems posed in the course, thus serving to more closely connect the programming to the mathematics. Moreover, having two teachers in the classroom naturally leads to discussions, showing by example the type of questions experts pose when engaging with a topic, and hopefully fostering a classroom climate where students also engage in discussion. Before outlining the planned project, we sketch what team teaching is about, and exemplify how it has been used in university mathematics education.

Various forms of collaborative teaching are already used in schools, where teachers collaborate on course preparation, implementation, and assessment. There are several models for collaborative teaching (Friend, Cook, Hurley-Chamberlain & Shamberger, 2010, p. 12), ranging from "one teacher, one assistant" models, via parallel and alternative teaching through to team teaching proper, with two teachers in the classroom together, taking shared responsibility for content. The research literature on team teaching of mathematics at the university level mostly consists of case studies in the context of teacher education. For instance, [START_REF] Clarke | A collaborative teaching approach: Views of a cohort of preservice teachers in mathematics and technology courses[END_REF] describe a project where two lecturers collaborated on the planning of courses on mathematical methods and instructional technology, emphasising the crossdisciplinary character of the courses. However, one teacher for each course did the actual teaching. What we aim to do is slightly different. A team of instructors with backgrounds in mathematics/mathematics education and computer science will coteach a course on programming in a mathematical context, sharing responsibility for content, planning and assessment. For part of the course, two teachers will lecture jointly, presenting and discussing content together, providing different perspectives on programming in school mathematics. For instance, in the context of probability theory, the computer scientist can show how to program simulations of probabilistic experiments, with the mathematician pointing out how the underlying mathematical ideas influence the design of the simulation. The mathematician can then go deeper into the mathematical theory, with the computer scientist highlighting the algorithmic aspects. In this way, we aim to provide students with opportunities to develop a deeper understanding of how to integrate programming and computational methods in their mathematics classes. The programming course is still in its planning stages, but we are currently piloting the team teaching approach in the context of a first-year calculus course within the engineering programs.

From a research perspective, taking a discursive view on learning [START_REF] Sfard | Thinking as communicating. Human development, the growth of discourse, and mathematizing[END_REF] we wish to investigate to what extent the team teaching might support students' active participation in mathematical and computational discourse. We are also interested in how the discourse of the students develops through participation in such a course, particularly concerning the interplay of mathematical and programming discourse. Furthermore, we view team teaching as having great potential for teacher development, with participating teachers being able to learn from one another as well as jointly developing innovative teaching practices.


Olov Viirman
email: olov.viirman@hig.se

Characteristics of participation -A mathematician and a mathematics educator collaborating on a developmental research project

Keywords: teaching and learning of mathematics in other fields, teachers' and students practices at university level, cross-disciplinary collaboration, mathematical modelling, communities of practice

In this paper, a developmental research project involving offering mathematical modelling (MM) activities to university biology students is presented, and a particular aspect is studied, namely the project as a collaboration between mathematicians and mathematics educators. The aim of the paper is to investigate what characterizes their participation in the project, and how the characteristics of the project and its development might influence this participation. Interview data as well as observation data from the MM sessions are analysed, and findings show that the mathematics educator served as a broker influencing the practice of the mathematician. It is hoped that the findings of the study can be of use when planning future collaboration between mathematicians and mathematics educators.

MATHEMATICIANS AND MATHEMATICS EDUCATORS

There is a close relationship between mathematics and mathematics educationas remarked by Kilpatrick (1998, p. 36), "neither can exist without the other". However, collaboration between mathematicians and mathematics educators is still relatively rare, although it has received attention in the literature (e.g. Fried & Dreyfus, 2014), and its importance and relevance to both communities have been emphasized (e.g. [START_REF] Nardi | Amongst mathematicians: Teaching and learning mathematics at university level[END_REF]. [START_REF] Dörfler | Mathematics and mathematics education: Content and people, relation and difference[END_REF] has pointed out several obstacles to collaboration. For instance, the communities of mathematics and mathematics education are often quite separated: mathematicians and mathematics educators work in different departments, teach different subjects and have different educational backgrounds. Furthermore, there are few arenas where they can meet professionally: they mostly publish in different journals and attend different conferences. Dörfler also highlights prejudices about the other field as a possible obstacle to collaboration, something also discussed by other researchers (e.g. [START_REF] Ralston | Research mathematicians and mathematics education: A critique[END_REF]. On a related note, the mathematician in [START_REF] Nardi | Amongst mathematicians: Teaching and learning mathematics at university level[END_REF] book states the need for mathematics educators to "be able to talk to mathematicians about mathematics", for there to be a basis for collaboration (p. 270). At the same time, "there is no such thing as the mathematician, the mathematics educator, or the mathematics teacher" (Thompson, 2014, p. 319). What is described above is only the general picturethere are numerous examples of functioning collaborative relationships. Most of these, however, take place on the individual rather than the organizational level. Indeed, the importance of individual relationships and trust is pointed out repeatedly in the literature. As [START_REF] Thompson | Reflections on collaboration between mathematics and mathematics education[END_REF] puts it: "successful collaboration requires mutual trust and respect among collaborators in the context of a shared commitment to solving a problem" (p. 331).

UNIVERSITY MATHEMATICS TEACHING AS SOCIAL PRACTICE

The MM activities forming the basis of the project were developed and conducted by a MatRIC team consisting of a mathematician and three mathematics education researchers, and in order to study what characterized this collaboration I will adopt a Communities of Practice perspective [START_REF] Wenger | Communities of practice: Learning, meaning and identity[END_REF]. A community is "a group of individuals identifiable by who they are in terms of how they relate to each other, their common activities and ways of thinking, and their beliefs and values" (Biza, Jaworski & Hemmi, 2014, p. 162), and a community of practice (CoP) is a community characterized by mutual engagement, joint enterprise and shared repertoire (Wenger, 1998, p. 73). Mutual engagement concerns, for instance, norms and social relationships within the community; joint enterprise refers to common understandings of the aims and ideals of the practice; and shared repertoire concerns what and how various resources are used in the practice (Biza et al., 2014, p. 163). Participation in a CoP can vary from the central participation of an experienced "old timer" to the more peripheral participation of a newcomer (ibid, p. 162). An individual's sense of belonging to a CoP involves engagementactive involvement in mutual negotiation of meaning; imaginationextrapolating from your own experiences to form an image of your own place within the CoP; and alignmentcoordinating your activities to fit within the structures of the CoP and contribute to the enterprise (Wenger, 1998, p. 173-174).

The CoP perspective has been used successfully in research on university mathematics teaching [START_REF] Biza | Communities in university mathematics education[END_REF]. For example, [START_REF] Jaworski | How we teach mathematics: Discourses on/in university teaching[END_REF], in a study of university teachers' lecturing, found little indication of a joint enterprise of teaching. Also originating in research using elements of social practice theory to investigate university mathematics teaching is the Spectrum of Pedagogical Awareness (SPA) [START_REF] Nardi | A spectrum of pedagogical awareness for undergraduate mathematics: From "tricks" to "techniques[END_REF]. The SPA provides a means of describing and reflecting upon university mathematics teachers' pedagogical thinking and practices, and consists of four levels ranging from Naïve and Dismissive, through Intuitive and Questioning and Reflective and Analytic to Confident and Articulate (ibid, p. 293). There are also several case studies using socio-cultural perspectives to investigate university mathematics teachers' teaching practices (e.g. [START_REF] Treffert-Thomas | Conceptualising a university teaching practice in an activity theory perspective[END_REF]. However, there is little research on the type of situation investigated in the present paper. Here, a team of mathematicians and mathematics educators collaborate on developing and conducting teaching activities at the university level, not primarily aimed at introducing new mathematics but rather at getting students to apply the mathematics they already know in new contexts. Of relevance for its focus on the collaborative aspect is the study by [START_REF] Cooper | A Mathematics Educator and a Mathematician Co-Teaching Mathematics: Affordances for Teacher Education[END_REF], investigating a mathematician/mathematics educator co-teacher partnership in a course on mathematical proof.

The present study seeks to answer the following questions: What characterizes the participation of the mathematician and mathematics educator in the project? How might the characteristics of the project and its development influence this participation?

THE PROJECT -METHODS, DATA AND PARTICIPANTS

As mentioned above, the project involves MM activities aimed at improving biology students' motivation for, interest in, and perceived relevance of mathematics in biological studies. The planning and teaching was carried out by a MatRIC team consisting of one mathematician and three mathematics educators. The mathematician, Yuriy Rogovchenko, is a professor of mathematics with extensive experience of MM both in teaching and in research. He is also coordinator of the MatRIC network on MM. Yuriy had the main responsibility for planning the mathematical content of the sessions and selecting tasks. The mathematics educators were: Simon Goodchild, professor of mathematics education and leader of MatRIC, with extensive experience of teaching mathematics in secondary school, but whose teaching experience at the university level mostly consists of courses in mathematics education; the author, a post-doctoral researcher in mathematics education with a background in mathematics, whose role in the project was mostly data collection and research support to Yuriy; and a doctoral student in mathematics education who was mostly acting as an observer and research assistant in this iteration, in preparation for the second iteration which would form a major part of his doctoral project. Since writing about their collaboration at the level of detail necessary for this paper would make anonymization practically impossible, professors Goodchild and Rogovchenko kindly agreed not only to participate in the research, but also to make their identities known. For simplicity and brevity, however, in what follows they will be referred to as ME and M, respectively. The project was conducted at a well-regarded Norwegian university where biology students take one compulsory mathematics course, taught in the first semester, designed not specifically for the biology undergraduate program but for students from about twenty different natural science programs. Hence, the course provides little opportunity for focusing on issues specific to biology.

So far, the project has been through two cycles of development; however, in this paper only the first iteration is considered. It consisted of a pilot, where the team met with 10 volunteer students for one three-hour session, a regular sequence of four three-hour sessions with a different group of 11 volunteer students concurrently with their compulsory mathematics course, and a follow-up meeting with the second group of students the following semester. All sessions were taught in English, but student group-work was conducted in Norwegian. The basic structure of the sessions was similar throughout, with an introductory lecture conducted by M introducing some ideas and tools of MM, followed by small-group work on various MM tasks set in a biological context, with whole-group follow-up, led by M but with some input from ME. Examples of tasks included estimating the density of a rabbit population based on the number of roadkill rabbits, and investigating the growth of a yeast culture in a petri dish. All sessions were video recorded. Data analysis is still ongoing, but initial results have been presented elsewhere [START_REF] Viirman | From ritual to exploration: The evolution of Biology students' mathematical discourse through Mathematical Modelling activities[END_REF].

The analysis presented in the present paper draws on the video-recorded data from the four regular sessions in the first cycle of the project. To provide further insight into the collaborative process involved in developing the project, after the conclusion of the second iteration of the project the author of the present paper conducted an audio-recorded, semi-structured group interview with the two principal members of the team, M and ME. It was decided not to include the fourth member of the team (the doctoral student) in the interview, since his involvement in the first iteration of the project was peripheral. A thematic content analysis was then conducted, of the interview data and the video data from the whole-class sessions, using the CoP framework as a tool for structuring themes, focusing on signs of mutual engagement, shared enterprise and joint repertoire, central and peripheral participation, and possible obstacles to a CoP developing. For instance, concerning shared enterprise, I looked at what M and ME said about their aims when developing the project, and then examined the video-recordings of the sessions looking at how these aims translated into what was emphasised in their teaching practice, comparing and contrasting the practices of M and ME. Some consideration of my own role in the project is appropriate. Although I was involved in the project from an early stage, and am well acquainted with its aims, my role has been that of the researcher. I did not participate in the planning of the content of the sessions, and my involvement in the sessions was for the most part restricted to data collection.

RESULTS

Looking at the video recordings from the sessions, certain patterns in the behavior of M and ME could be discerned, where M often had a very clear focus on the mathematics and the tasks, while ME assumed responsibility for the students' wellbeing and the nature of their learning. For instance, in the second session work on the first task had taken longer than expected. Still, M, enthusiastic about the mathematics and eager to get through all that he had planned for the session, begins introducing the next task:

M:

You will be solving an important medical problem. Similar situations occurred on several occasions during the sessions, with M getting carried away by the mathematics, and ME intervening on the students' behalf. At the same time, this pattern was not entirely consistent. There were occasions where M showed a clear concern about the students' wellbeing and enjoyment, telling jokes and striving to make the students feel at ease. Similarly, there were occasions where ME got to present his solutions to some of the tasks, displaying an obvious enthusiasm for and a deep engagement with the mathematics. Still, the overall tendency was relatively clear.

Another more distinctive difference between M and ME concerned their engagement with student contributions to the solution of the MM tasks. Again, M had a strong focus on the mathematics, to the extent that he displayed signs of what [START_REF] Ralston | Research mathematicians and mathematics education: A critique[END_REF] In line with this, on the few occasions during the sessions where ME conducted discussions, he instead let the students lead. He interrupted students' presentations less frequently, and was not so quick to evaluate their contributions. When presenting solutions to tasks he emphasized the process of doing mathematics over the result, presenting his mistakes and dead-end attempts, seeing these as opportunities for students' learning.

Still, there were signs of M gaining an increased understanding of students' needs and potential for contribution as the project progressed. Preparing the last session, he decided to involve the other team members more in the session, providing support to the groups in their work, stating: "We should have done this in the first session". He also expressed this willingness to reflect on his own practice in the interview:

M: Feedback was essential, so it was like my teaching was shaped during those four sessions towards some goals -I was experimenting, I was changing - At this point, ME commented that this was "very much the didactician's view", and that it showed "a readiness to reflect" upon his own teaching. This readiness to reflect and change was further emphasized by how M later in the interview talked about his plans for a new course he would be teaching, where he intended to incorporate more student-centred teaching methods.

In the interview, when discussing the collaboration M emphasized the particular qualities of himself and ME: "We are both weird people". When asked what this "weirdness" consisted in, he alluded to the stereotypical views mentioned by [START_REF] Dörfler | Mathematics and mathematics education: Content and people, relation and difference[END_REF], saying that he has a (for mathematicians) uncommon interest in teaching, partly due to the influence of ME, while ME is (for a mathematics educator) uncommonly knowledgeable in mathematics. Furthermore, both M and ME strongly emphasized the value they place on their professional relationshipthat two people from adjacent, but still different fields can work so productively together. M added that in no department where he had worked (and he has worked in numerous places, in many different countries) had educational matters been on the agenda. The collaboration was always on the individual level, never on the departmental, and as ME added, this holds true even at their current university, where mathematicians and mathematics educators work in the same department. At the same time, M expressed the view that the differences between mathematicians and mathematics educators had helped improve teamwork within the current project, but emphasized how this was dependent upon all participants being able to engage with the mathematics. Concerning other aspects of successful collaboration, ME highlighted the need for mutual motivation: "Maybe this interdepartmental work has worked so well because bioCEED wants something, and we [MatRIC] want something, and those two wants coincided." When asked what the mutual motivation was that enabled M and ME to work together, ME highlighted the will to see MatRIC succeed, and how this success was in part dependent on the collaboration between ME and M. M struggled with formulating an answer, but emphasized interest in and curiosity about mathematics.

The main obstacle to collaboration mentioned in the interview was lack of time for preparation:

ME:

In terms of preparation, one of the things that I (…) found a bit frustrating, was that the preparation was very often done between three and four o'clock in the morning on the day of the session, and therefore there was very little opportunity for discussion between the three of us [M, ME and the author] about what was going to happen in that session.

ME attributed this to differences in individual styles of working, but still pointed out that because of this and other time constraints, discussions about the mathematical and didactical aspects of the sessions rarely managed to go below the surface level.

In response to this, M stated that his initial plan was to have the whole sequence of activities pre-planned, but that the feedback he got while conducting the sessions led to continuous refinement and change. Still, he agreed that providing the rest of the team with the tasks further in advance would have been preferable. Generally, both M and ME stated that the main difficulty with the project was the lack of timeeveryone involved were so busy with other things that they found it difficult to engage with the project as deeply as should have been needed.

CONCLUSIONS AND DISCUSSION

In the findings from the actual teaching sessions, M was seen to a large extent emphasising the mathematics and ME showing more concern with the students, their wellbeing and learning experiences. These characteristics of the participation of M and ME in the sessions align well with observations made by [START_REF] Cooper | A Mathematics Educator and a Mathematician Co-Teaching Mathematics: Affordances for Teacher Education[END_REF], where a similar pattern could be seen. At the same time, these characteristics were not uniform throughout the sessions, and the overall picture of their engagement with the content and the students is not quite as simplistic as that. Still, the differences are discernible and should not be downplayed.

Furthermore, one might ask whether a CoP developed around the project. From what M and ME expressed in the interview there are definite signs of mutual engagement they both emphasize the close professional relationship, and there is a dedicated support between the members of the project in making it work. There is however, less evidence when it comes to the joint enterprise and shared repertoire. In particular, practical circumstances regarding the projectthe late preparation, the lack of time for discussion and reflectionmade the development of a shared repertoire difficult. The responsibility for developing activities and tasks fell mainly on M, and the ways of interacting with students differed quite substantially, as the findings above show. It is also difficult to find evidence of joint enterprise. On a surface level, this could of course be said to be the project itself, but looking at what M and ME say about the aims and motivations behind the project, the existence of a joint enterprise is less clear. For ME, as leader of MatRIC, it is about the success of the centre, whereas for M it has more to do with engagement in mathematics at a personal level. Hence, it is difficult to claim that a CoP has developed.

Rather, what happened was that ME, occupying a more central position within the mathematics education community, acted as a broker (Wenger, 1998, p. 105), introducing elements of mathematics education practice and thereby contributing to changing M's practice. In this way, M displays the beginnings of a trajectory from a peripheral position towards a more central. In the interview, M clearly states how participation in the project has changed the way he views teachinghe talks about "seeing things through students' eyes", described by ME as "a didactician's view". These changes are perhaps less clearly visible in the video recordings from the sessions, but as mentioned above there are signs of an increased awareness of students' needs as the project progressed. This could also be understood in terms of the Spectrum of Pedagogical Awareness. When M talked about reflecting about his own practice, and "seeing through students' eyes", this fits well with the Reflective and Analytic level of the spectrum, which is characterized in part by "awareness in starting to articulate pedagogical approaches and of reflection enabling making strategies explicit" (Nardi, Jaworski & Hegedus, 2005, p. 293). At the same time, much of what M did in the sessions would rather fit with the Intuitive and Questioning level of the spectrum, with a less explicitly articulated pedagogical thinking and a more intuitive recognition of students' difficulties and needs (ibid, p. 293). Hence, a movement along the spectrum could be discerned as the project progressed, and there is some support for claiming that M's pedagogical awareness has increased as a consequence of the project.

Regarding what factors contribute to functioning between mathematicians and mathematics educators, what has been reported here is only a single case, and one should be wary of drawing strong conclusions. Still, it is worthy of note how well findings from literature, about mathematicians demanding that mathematics educators know the mathematics [START_REF] Nardi | Amongst mathematicians: Teaching and learning mathematics at university level[END_REF] and the need for individual relationships and trust [START_REF] Thompson | Reflections on collaboration between mathematics and mathematics education[END_REF] resonate with what M and ME say in the interview. Both point out these factors as crucial to their collaboration, while lack of time for joint preparation and reflection is pointed out as the main obstacle. An awareness of the importance of these factors will be highly useful when planning and conducting further collaborative efforts involving mathematicians and mathematics educators. Indeed, providing institutional means and resources for preparing and developing projects of the kind discussed in this paper might be one way of moving parts of the responsibility for establishing collaboration from the individual to the institutional level, something which is necessary if such collaboration is to be sustainable.

  partly I was listening to what you [ME and the author] were saying, partly I was trying to do what I felt was appropriate (…) I had very useful advice which might have affected my way of teaching. Interviewer: In what way? M: I probably more often look at what I'm doing from the other side. From the students' perspective.

  calls the One Right Answer Syndrome, clearly having one particular solution in mind. This affected the way he lead discussions. For instance, when students came up with solutions or suggestions that fit with this expected solution, they were received as being the "correct" ones, as in these examples from sessions 1, 2 and 3, respectively:

	M:	I will show you how the reasoning goes
	---
	M:	When we look at the answers, the second group gave the correct one
	---
	M:	This is the perfect one (…) the only thing we need to check is, how far is
		this from my computation
	Particularly the last quote displays how M used his own solution as the yardstick
	against which student solutions were measured. On the other hand, when student
	contributions did not resemble what he had in mind, M often did not comment on
	them at all, or used a different terminology: "Well, that was a very constructive way
	to solve the problem." This contrasts with how ME talked about the students' work
	on the tasks, as in this example from session 3:
	M:	Any other assumptions you were using in your work?
	S:	Yeah, that 97 dead rabbits were per 24 hours
	M:	OK, that's a good one
	S:	Because it said that they were easily recognizable
	M:	Correct, you read exactly what was meant there
	(…)

ME:

We really, really, REALLY would like you to share with us solutions that you have worked on within groups on those tasks. Now, it doesn't matter at all whether they are good or wonderful solutions, perfectly correct solutions, whatever perfectly correct means, whether they look like his [points at M] solutions or look like my solutions, which will be quite different -it really doesn't matter. What matters is the engagement, and the thinking processes, and sharing with each other, and with us, the thinking processes.

ACKNOWLEDGMENT

My most sincere thanks are due to my collaborators in the project, both within MatRIC and bioCEED, and to the students who kindly agreed to participate. Thanks also to the participants of the NORMA 17 symposium "Researching university mathematics education", in particular Professor Elena Nardi, for insightful critique of an earlier version of this paper.