Keywords: Tension wood, Tropical rainforest species, UV microspectrophotometry, Scanning Electron Microscopy, Cellulose microfibril angle

Tension wood of Laetia procera (Poepp.) Eichl. (Flacourtiaceae), a neo-tropical forest species, shows a peculiar secondary wall structure, with an alternance of thick and thin layers, while opposite wood of this species has a typical secondary wall structure (S1+S2+S3). Samples for the study of microstructural properties were collected upon estimation of growth stresses in the living tree, in order to analyse correlation of the former with the latter. Investigation using optical microscopy, scanning electron microscopy and UV microspectrophotometry allowed the description of the anatomy, ultra-structure and chemistry of this peculiar polylaminate secondary wall. In the thick layers, cellulose microfibril angle is very low (i.e., microfibril orientation is close to fibre axis) and cellulose microfibrils are well organised and parallel to each other. In the thin layers, microfibrils (only observable in the inner layer) are less organised and are oriented with a large angle relative to the axis of the cell. Thick layers are lightly lignified although thin layers show a higher content of lignin, close to that of opposite wood secondary wall. The more the wood was under tensile stress, the less the secondary wall was lignified, and the lower the syringyl on guaiacyl lignin units ratio was. The innermost layer of the secondary wall looks like a typical S3 layer with large microfibril angle and lignin occurrence. The interest of this kind of structure for the understanding of stress generation is discussed.

INTRODUCTION

In order to restore their verticality after accidental leaning, to maintain the branch at a given angle or to change axis orientation to reach the canopy for better access to light trees are able to bend progressively their trunk or branches by a very active mechanical action driven by cambial activity [START_REF] Sinnott | Reaction Wood and the Regulation of Tree Form[END_REF]. This reorientation is associated with the formation of a peculiar type of wood, called reaction wood. In gymnosperm species, reaction wood is formed on the lower side of the tilted axis (compression wood), while in angiosperm species it is formed on the upper side (tension wood). Whatever the species considered, the process of axis reorientation is always based on circumferential heterogeneity in cambial region (cambial zone, differentiating and maturing zone) activity occurring at 3 distinct structural levels:

-at the macroscopic level, the division and differentiation of the cambial initials is controlled differently between the upper and the lower side of the trunk. This can lead to an eccentric growth that causes the reaction wood side to be often wider than the opposite side [START_REF] Dadswell | What is reaction wood?[END_REF][START_REF] Almeras | Effect of circumferential heterogeneity of wood maturation strain, modulus of elasticity and radial growth on the regulation of stem orientation in trees[END_REF]);

-at the mesoscopic level, proportions of the various cells types (fibres, vessels, ray and axial parenchyma) constituting secondary xylem can vary substantially between normal and reaction wood [START_REF] Onaka | Studies on compression and tension wood[END_REF][START_REF] Jourez | Anatomical characteristics of tension wood and opposite wood in young inclined stems of poplar (Populus euramericana cv "Ghoy")[END_REF][START_REF] Ruelle | Tension wood and opposite wood in 21 tropical rainforest species. 2. Comparison of some anatomical criteria[END_REF]);

-at the microscopic level, fibres produced during the reaction process strongly differ structurally from normal fibres. This occurs through the modulation of various structural features: (i) secondary wall fibre thickness, that tends to increase in some species [START_REF] Ruelle | Tension wood and opposite wood in 21 tropical rainforest species. 2. Comparison of some anatomical criteria[END_REF]); (ii) size and orientation (MFA) of secondary wall cellulose microfibrils: [START_REF] Washusen | The association between cellulose crystallite width and tension wood occurence in Eucalyptus globulus[END_REF] reported an increase of microfibrils size in tension wood and MFA is known to be lower in tension wood and larger (up to 45°) in compression wood [START_REF] Timell | Compression wood in gymnosperms[END_REF][START_REF] Yoshida | Tension wood and growth stress induced by artificial inclination in Liriodendron tulipifera Linn. and Prunus spachiana Kitamura f. ascendens Kitamura[END_REF][START_REF] Yoshizawa | Anatomy and lignin distribution of reaction wood in two Magnolia species[END_REF][START_REF] Barnett | Cellulose microfibril angle in the cell wall of wood fibres[END_REF]Clair et al. 2006a); (iii) organization and chemical composition of hemicellulose/lignin matrix surrounding cellulose microfibrils [START_REF] Pilate | Lignification and tension wood[END_REF][START_REF] Gorshkova | Secondary cell-wall assembly in flax phloem fibres: role of galactans[END_REF].

The mechanism allowing reorientation of the axis originates in structural modifications at the cell-wall level. Indeed, these micro-structural modifications induce in wood a spontaneous tendency to strain during its maturation process [START_REF] Boyd | Basic cause of differentiation of tension wood and compression wood[END_REF][START_REF] Yamamoto | Generation mechanism of growth stresses in wood cell walls : roles of lignin deposition and cellulose microfibril during cell wall maturation[END_REF][START_REF] Bamber | A general theory for the origin of growth stresses in reaction wood : how trees stay upright[END_REF][START_REF] Yamamoto | Origin of the Biomechanical properties of wood related to the fine structure of the multi-layered cell wall[END_REF]. The maturation process of cell wall can be subdivided in steps from cell expansion, secondary wall formation and lignification to cell death [START_REF] Plomion | Wood formation in Trees[END_REF]. During this process compression wood tends to swell and tension wood tends to shrink. This tendency is impeded because reaction wood is stuck to the core of old wood, resulting in a state of mechanical stress (compression or tension), called maturation stress. The asymmetry of longitudinal maturation stress around the circumference results in a bending moment generating a change in curvature and thus a reorientation movement [START_REF] Archer | Growth stresses and strains in trees[END_REF]. The efficiency of this mechanism depends on the difference between the force acting on the reaction wood side and the force acting on the opposite side. The magnitude of the force acting on one side is the integral of the product of the area of maturing wood by the magnitude of the maturation stress in wood. The magnitude of maturation stress in wood, in turn, can be viewed as the product of the maturation strain by wood Young's modulus of wood.

Finally, Young's modulus can be expressed as the product of wood density by the specific modulus of wood material elasticity [START_REF] Almeras | Effect of circumferential heterogeneity of wood maturation strain, modulus of elasticity and radial growth on the regulation of stem orientation in trees[END_REF].

All four of these biomechanical factors, i.e. maturation stress asymmetry and magnitude, Young's modulus of wood and specific modulus of wood material can be controlled through modulations of cambial activity at the above-mentioned levels.

Eccentric growth controls the area of reaction wood and opposite wood. Proportions of each cell type in the xylem partly control the specific modulus of elasticity of wood.

Fibre wall thickness controls wood density. Cellulose microfibril geometry and matrix composition partly control specific modulus of elasticity and directly control sign and magnitude of maturation strain [START_REF] Okuyama | An estimation of the turgor pressure change as one of the factors of growth stress generation in cell walls[END_REF][START_REF] Yamamoto | Generation mechanism of growth stresses in wood cell walls : roles of lignin deposition and cellulose microfibril during cell wall maturation[END_REF]. The mechanical effect of these structural modifications can be predicted by mechanical models acting at different levels. At the macroscopic level, reaction efficiency can be computed using beam theory and the principles of its application to a growing structure (Fournier et al. 1994a). Fournier and coworkers model (1994a) takes into account the effects of eccentric growth, modulus of elasticity and maturation strain [START_REF] Almeras | Effect of circumferential heterogeneity of wood maturation strain, modulus of elasticity and radial growth on the regulation of stem orientation in trees[END_REF]. At the mesoscopic level, the effect of wood anatomy (i.e. the proportions and organisation of the different cell types) can be predicted by homogenization procedures [START_REF] Badel | Détermination des propriétés élastiques et du retrait d'un cerne annuel de chêne dans le plan transverse : description de la morphologie, mesures des propriétés microscopiques et calculs d'homogénisation Baillères H[END_REF]. At the microscopic level, cell wall micromechanical models allow to predict wood specific modulus of elasticity and magnitude and sign of maturation strain [START_REF] Yamamoto | Growth stress generation and microfibril angle in reaction wood. Microfibril Angle in Wood[END_REF][START_REF] Yamamoto | Origin of the Biomechanical properties of wood related to the fine structure of the multi-layered cell wall[END_REF]. These models take as input data quantitative information about cell wall organisation at the microscopic and ultra-structural levels, cell wall chemical composition, and timing of cell wall differentiation and lignification.

The structure of reaction wood fibres generally differs from that of normal wood fibres. In gymnosperms, compression wood fibres typically have a round shape, intercellular spaces and cracks in the cell wall [START_REF] Timell | Compression wood in gymnosperms[END_REF]). Their wall is thick and heavily lignified, and the microfibrils are oriented at a wide angle with respect to the fibre axis. Among angiosperms, diversity in the form of tension wood fibres has long been recognized [START_REF] Onaka | Studies on compression and tension wood[END_REF]Clair et al. 2006b). The most typical form of angiosperm tension wood is characterized by the development of a so-called gelatinous layer (Glayer). The G-layer is essentially made up of highly crystalline cellulose [START_REF] Norberg | Physical and chemical properties of the gelatinous layer in tension wood fibres of Aspen (Populus tremula L.)[END_REF][START_REF] Côté | A contribution to the ultrastructure of tension wood fibers[END_REF], with a very low microfibril angle [START_REF] Fujita | Electron Microscopy of Microtubules and Cellulose Microfibrils in Secondary Wall Formation of Poplar Tension Wood Fibers[END_REF].

However, several species do not develop G-fibres, while showing evidence of tension wood production [START_REF] Fisher | Occurrence of reaction wood in branches of Dicotyledons and its role in tree architecture[END_REF]Clair et al. 2006b).

During an exploration of biomechanical strategies of tropical rainforest species, trees from Laetia procera species proved to be very efficient in restoring verticality after accidental leaning [START_REF] Almeras | Effect of circumferential heterogeneity of wood maturation strain, modulus of elasticity and radial growth on the regulation of stem orientation in trees[END_REF]. Moreover, dissymmetry in the magnitude of maturation strain was identified as the leading factor determining efficiency. The maturation strain of tension wood was especially high in this species and tension wood fibres presented a peculiar polylamellated structure. In order to check whether this macroscopic behaviour was related to characteristic structural features, tension wood and opposite wood were investigated using complementary techniques. In this way, optical and electronic microscopy, histo-chemical reaction and UV microspectrophotometry allowed us to obtain quantitative measurement about the structure and qualitative and semi-quantitative information about the chemical composition of the secondary wall and of the fibres.

MATERIAL AND METHODS

Plant material and sampling

Laetia procera (Poepp.) Eichler (Flacourtiaceae) is rare to locally frequent in primary and secondary neo-tropical forests, on sandy soil. Five trees were selected in the same area in French Guyana, near Kourou. Their diameter at breast height (DBH) ranged between 19 and 28 cm (Table 1).

All trees were chosen because they were exhibiting a reorientation process after some accidental inclination. This was verified in situ by mechanical estimation of growth strain (GS). Experimentally, maturation strain can be estimated by releasing the longitudinal stress at the surface of wood and measuring the resulting strain, referred to as residual growth strain (GS). Growth strain is the sum of the maturation and support strains. At the periphery of the trunk, where measurements where done, support strain are due to the support of the newly formed layer and so are close to zero. Thus, growth strain (GS) is very close to maturation strain. In the following text we will use GS, since it is what has been experimentally measured. GS were evaluated using the "single hole" method (Fournier et al. 1994b;[START_REF] Clair | Relationship between growth stresses, mechanophysical properties and proportion of fibre with gelatinous layer in chestnut (Castanea Sativa Mill[END_REF][START_REF] Almeras | Effect of circumferential heterogeneity of wood maturation strain, modulus of elasticity and radial growth on the regulation of stem orientation in trees[END_REF].This method gives the value of the displacement between two pins, hammered onto the trunk (after local debarking) at a 45 mm distance from each other. A 20mm-deep, 20mm-wide hole is drilled at the mid-point between the two pins. A displacement is measured (in µm) and converted into a strain (in %) using a calibration factor: 9.6×10 -4 corresponding to a calibration made on Eperua Falcata (Fournier et al. 1994b), a tropical hardwood species with properties similar to those of Laetia procera.

Eight measurements (equally spaced around the circumference, i.e. every 45°) were performed at breast height on each tree. The first measurement position was located on the upper side of the leaning trunk. Two wood samples were taken, above and below the holes resulting from the measurement method, for anatomical and structural studies.

Observations were made on both samples to ensure homogeneity of the studied wood (Fig. 1).

Methods

First, classical optical microscopy on stained sections (including Wiesner reaction) was used for all the 5 trees, in order to check if there were main differences between trees or if the variation varied from one tree to another. Then, other techniques (Scanning Electron Microscopy, UV microspectrophotometry) were only applied to the tree with the highest contrast in GS between upper and lower part, Lp1 (Table 1).

Optical microscopy

Cross sections (thickness: 24 µm) were made with a sliding microtome using disposable razor blades (Feather N35). Sections were stained with safranin/fast green according to the protocol described by [START_REF] Yoshida | Tensile growth stress and lignin distribution in the cell walls of yellow poplar, Liriodendron tulipifera Linn[END_REF]. Safranin stains lignified tissues in red and fast green stains both lignified and un-lignified tissues in green.

Wiesner reaction

Cross sections (thickness: 12 µm) were made on opposite and tension wood specimens.

Wiesner reaction was performed on these sections by pouring a few drops of 2 % phloroglucinol ethanol solution on the section mounted on a glass-slide, adding one drop of 35 % HCl and covering the section with a cover slip.

The Wiesner reactive reacts with coniferyl (G) and synapyl (S) aldehyde units in lignin. The higher the Klason-lignin content the stronger the intensity of the red coloration [START_REF] Yoshizawa | Anatomy and lignin distribution of reaction wood in two Magnolia species[END_REF]. This result provide us to do a comparison between samples, i.e. a semi-quantitative analysis. As the coloration is not permanent, observation was performed during the 20 minutes after the beginning of the reaction.

UV microspectrophotometry

After dehydration through a graded ethanol series, the sections were embedded in epoxy resin. Thin cross sections (thickness: 1 μm) were cut with a diamond knife, mounted on quartz microscope slides, overlaid with a drop of non-UV-absorbing glycerin, and covered with a quartz cover slip [START_REF] Okuyama | Relation between growth stress and lignin concentration in the cell wall: Ultraviolet microscopic spectral analysis[END_REF]). The sections were observed under a microspectrophotometer (Zeiss MPM800). The scanning range of the wavelength was 250-350nm, the step of the wavelength scanning was l nm, and the bandwidth was adjusted to 5 nm. UV absorption spectra were obtained at various locations inside the secondary wall of opposite and tension wood fibres using a beam spot of 0.5 μm diameter. The absorption spectra directly provide information on cell wall lignin content [START_REF] Okuyama | Relation between growth stress and lignin concentration in the cell wall: Ultraviolet microscopic spectral analysis[END_REF][START_REF] Gindl | Comparing Mechanical properties of normal and compression wood in Norway Spruce: the role of lignin in compression parallel to the grain[END_REF][START_REF] Yoshida | Tensile growth stress and lignin distribution in the cell walls of yellow poplar, Liriodendron tulipifera Linn[END_REF]; the higher the absorption, the more lignified the wall. Results from this technique can be used to make a comparison between specimen; ie a semi-quantitative analysis among samples.

Each measurement for one position (one part of fibre wall) was repeted 8 times.

For each specimen the absorption spectrum of the secondary wall was taken at least on 5 different fibres and averaged to determine the UV absorption spectrum of the specimen.

The microspectrophotometer settings were: objective lens magnification: ×100; program: Lambdascan; (for more detailed information see [START_REF] Okuyama | Relation between growth stress and lignin concentration in the cell wall: Ultraviolet microscopic spectral analysis[END_REF].

Field-Emission Scanning Electron Microscopy (FE-SEM)

Observations were made in both transverse and longitudinal planes. Sample geometries were _7×5×1 mm 3 and 5×1×7 mm 3 , respectively (R×T×L). Samples were dehydrated through a graded ethanol series and then processed using the t-butanol freeze-drying method. In order to observe cellulose microfibrils of lignified layers, a lignin extraction treatment (NaCl 0.6%, CH 3 COOH 0.13% in distilled water during 40 hours) was performed on longitudinal sections. The dried samples were mounted on aluminium stubs and lightly sputter-coated with platinum. Samples were observed by FE-SEM (Hitachi, S-4500) at an accelerating voltage of 3 kV.

Microfibril angle (MFA) and diameter of cellulose aggregates were measured from direct observations by SEM on samples from tree Lp1. These measurements were made on 10 pictures per sample with magnifications from ×30k to ×70k and on about 20 microfibrils per picture for MFA and on 10 areas of about 5000 nm² for the diameter of cellulose aggregates. Examples of images used for these measurements are shown in Fig. 2.

Statistical analysis

Results from MFA and cellulose aggregates measurements were compared to highlight significant differences between tension and opposite wood samples. We used the bilateral Student test to account for the significance of these results.

RESULTS

Growth strain measurements

Measurements clearly show that wood located on the upper side exhibits much higher tensile growth strains than wood located in all other position (Table 1). Measurements performed on the lower (opposite) side do not present significant difference in mechanical stressing with lateral position. Upper wood layer positions with very high growth strains are called tension wood (TW) in the next paragraphs while other positions are named opposite wood (OW) or lateral wood.

Considering data reported by [START_REF] Archer | Growth stresses and strains in trees[END_REF] and more recent studies [START_REF] Yoshida | Tension wood and growth stress induced by artificial inclination in Liriodendron tulipifera Linn. and Prunus spachiana Kitamura f. ascendens Kitamura[END_REF][START_REF] Yoshizawa | Anatomy and lignin distribution of reaction wood in two Magnolia species[END_REF]Clair et al. 2006b), GS observed in tension wood are in the upper range of reported values.

Structure of the fibre wall

Polylaminate secondary wall of tension wood fibres

Optical microscopy observations after safranin/fast green staining show a homogeneous typical secondary wall in opposite wood fibres, while a peculiar polylaminate secondary wall structure is observed in tension wood fibres (Fig. 3). This structure consists of an alternance of thick and thin layers. This peculiar secondary wall stains as a typical gelatinous layer (G-layer), i.e. in green without any touch of red as in Populus euramericana [START_REF] Jourez | Anatomical characteristics of tension wood and opposite wood in young inclined stems of poplar (Populus euramericana cv "Ghoy")[END_REF] or Eperua falcata (Satiat-Jeunemaitre 1986).

Observation of transversal and longitudinal sections by Scanning Electron

Microscopy (SEM) confirms that the polylaminate structure of the secondary wall occurs in tension wood but not in lateral or opposite wood fibres (Fig. 4 and5). The number of thin layers has been counted on tension wood specimens; results are given in Table 2.

There was an average of 5 to 6 thin layers with thick layers between them. Thick layers are approximately ten times thicker than thin layers (Table 2).

Inner thin layer

Observations of longitudinal sections (Fig. 6a) also show a lignified layer inside the lumen of tension wood fibres; this inner layer allowed us to prospect the nature and the structure of thin layers observed in the polylaminate secondary wall. The aspect of this layer before (Fig. 6b) and after (Fig. 6c) lignin extraction treatment highlights its lignified feature and its large MFA. These features are typical of the S 3 layer commonly observed in the cell wall of opposite wood fibres (Fig. 7a).

Organisation of cellulose in the secondary wall

MFA was very low (close to fibre axis) in the thick layers of tension wood fibres and more than three fold larger (15 to 20°) in opposite wood (p<0.001) (Table 3).

Unfortunately we were unable to accurately measure MFA in intermediate thin layer.

Diameter of cellulose aggregates is in the range of values reported by [START_REF] Fahlen | Cross-sectional structure of the secondary wall of wood fibers as affected by processing[END_REF] on Picea abies, i.e. between 18 and 23 nm, and is lower in opposite wood (p<0.001) than in tension wood, respectively 18.4 and 21.9 nm.

Lignification features of tension wood fibres

Wiesner reaction

Intensity of the Wiesner reaction gives qualitative information on cell wall lignification features. Polylaminate tension wood fibres appear less lignified than opposite wood fibres (Fig. 8). In TW fibres the reaction is stronger in the outer part and becomes weaker towards the centre. Some TW fibres have stronger reaction intensity than others; some of them even show a lack of reaction on the polylaminate structure although the S 1 layer and the primary wall are stained in red by the reaction (Fig. 9). This could mean that various types of tension wood fibres can be observed.

Compared to typical G-layers, known to have very low lignin contents, the secondary wall of Laetia TW presents lignin within the thick layers and the coloration inside thin ones indicates a higher amount inside them.

UV microspectrophotometry

The average absorption spectra for the S 2 layer of a opposite wood sample (Lp1-5) and for thick and thin layers of the secondary wall of a tension wood specimen (Lp1-2) are given in Fig. 10.

In tension wood, the average absorption, and therefore polylaminate layer lignin content, is lower than in the S 2 layer of opposite wood. Actually absorbance values in tension wood specimen ranges from 0.15 to 0.24 at 280 nm and from 0.14 to 0.22 at 270 nm. In opposite wood absorbance values are higher than in tension wood and ranges from 0.34 to 0.53 at 280 nm and from 0.31 to 0.47 at 270 nm. Moreover the average absorption of the whole fibre at 270 and 280 nm decreases with increasing growth strain values (Table 4 and fig. 11).

UV microspectrophotometry also shows that lignification of TW is stronger in secondary wall thin layers than in the thick layer. Thus, intermediate thin layers are chemically different.

The absorption ratio A 280 /A 260 markedly depends on the ratio of syringylpropane (S) to guaiacylpropane (G) units. Actually the decrease of this ratio corresponds to an increasing S/G ratio [START_REF] Okuyama | Relation between growth stress and lignin concentration in the cell wall: Ultraviolet microscopic spectral analysis[END_REF][START_REF] Yoshida | Tensile growth stress and lignin distribution in the cell walls of yellow poplar, Liriodendron tulipifera Linn[END_REF]. This ratio for tension wood specimens is largely dominated by the properties of thick layers, because of the prominent proportion of these layers compared to thin layers. According to our results, S/G ratio increases with growth strains, with a good relationship (Table 4 and Fig. 12). This evolution has been also described in some other species differentiating or not a Glayer in their tension wood, such as eucalyptus (Baillères et al. 1995) or Liriodendron tulipifera [START_REF] Yoshida | Tensile growth stress and lignin distribution in the cell walls of yellow poplar, Liriodendron tulipifera Linn[END_REF].

DISCUSSION

Observations made on Laetia procera tension wood show a polylaminate structure of the secondary wall with the alternance of lightly lignified thick layers, with microfibrils almost aligned to the cell main axis, and more lignified thin layers in which it was not possible to measure microfibrils orientation. This kind of polylaminate structure has previously been observed in bamboo cells [START_REF] Parameswaran | On the Fine Structure of Bamboo Fibres[END_REF]. In a recent study screening the anatomical diversity of tension wood among tropical dicotyledonous species (Clair et al. 2006b), similar features were observed in tension wood of an other Flacourtiaceae, Casearia javitensis. Moreover [START_REF] Daniel | Polylaminate concentric cell wall layering in fibres of Homalium foetidum and its effect on degradation by microfungi[END_REF] observed this structure in a species from the Flacourtiaceae family, Homalium foetidum, but in their study its occurrence was not identified as a tension wood feature. Observation of this peculiar structure in tension wood fibres emphasises the idea exposed in Clair et al. (2006b) on the difficulty to classify tension wood structures.

The very low MFA observed in thick layers is similar to what is usually observed in tension wood fibres, with or without G-layer [START_REF] Norberg | Physical and chemical properties of the gelatinous layer in tension wood fibres of Aspen (Populus tremula L.)[END_REF][START_REF] Chaffey | Microfibril orientation in wood cells : new angles on an old topic[END_REF][START_REF] Ruelle | Tension wood and opposite wood in 21 tropical rainforest species. 2. Comparison of some anatomical criteria[END_REF]. The microfibrils of the inner thin layer of these cells appear less organised and lie at a larger angle than those of thick layers. In their study on Homalium foetidum, [START_REF] Daniel | Polylaminate concentric cell wall layering in fibres of Homalium foetidum and its effect on degradation by microfungi[END_REF] observed differences in microfibril angle between thick and thin layers. These various observations lead us to hypothesise that cellulose organisation in the thin layers is (i) different from the one of thick layers and (ii) similar to the one observed in the inner thin layer, so that all these thin layers look like successive S 3 layers between thick layers. [START_REF] Prodhan | Ultrastructure investigation of tension wood fibre in Fraxinus mandshurica Rupr. var. japonica Maxim[END_REF] shows that G-layer in Fraxinus mandshurica is lignified, in the same way as in Laetia procera, with a higher content of lignin in the outer part of the G-layer. Similar observations were made by [START_REF] Gierlinger | Chemical Imaging of Poplar Wood Cell Walls by Confocal Raman Microscopy[END_REF], i.e. a higher lignification in the outer part of G layer, but in their work they found a very weak content of lignin in the rest of the G layer. These observations support the statement [START_REF] Terashima | A new mechanism for formation of a structurally ordered protolignin macromolecule in the cell wall of tree xylem[END_REF]) that there is a lignification gradient from outer to inner part of the cell wall during fibre development. However the alternance of more lignified layers raises several questions about the development of the secondary wall and about its role in the very efficient strategy of reorientation observed in this species. In current fibre mechanics models, maturation strain associated to the end of the lignification process is assumed to occur simultaneously in the whole S 2 or G-layer. Moreover, it is known that the final prestressing values in a multilayered composite depend on the history of layer deposition [START_REF] Yamamoto | Origin of the Biomechanical properties of wood related to the fine structure of the multi-layered cell wall[END_REF]. This effect of successive step construction in a material on field of stresses is also true for various natural or man-made structures such as trees, bridges, etc. [START_REF] Guitard | Growth stress generation: a new mechanical model of the dimensional change of wood cells during maturation[END_REF][START_REF] Malzbender | Mechanical and thermal stresses in multilayered materials[END_REF]).It may be important to know whether each thick layer and associated thin layer is fully lignified successively or if the lignification process occurs after the deposition of all the layers. This raises questions about the rythm of secondary wall development. [START_REF] Hosoo | Diurnal difference in the amount of immunogold-labeled glucomannans detected with field emission scanning electron microscopy at the innermost surface of developing secondary walls of differentiating conifer tracheids[END_REF]2003) showed diurnal periodicity in the deposition of cell wall components on the innermost surface of developing cells. This seems to indicate that there is diurnal periodicity in cell wall formation, corresponding to the 24-h light-dark cycle. Question about relation between thick and thin layers alternance and circadian rythm should be investigated, and young trees artificially inclined in controlled conditions can be a good material for such study, knowing that tension wood will be rapidly induced on upper part of the young tree [START_REF] Jourez | Anatomical characteristics of tension wood and opposite wood in young inclined stems of poplar (Populus euramericana cv "Ghoy")[END_REF].

The multilayered tension wood has been, until now, only found in others genus of the Flacourtiaceae Family [START_REF] Daniel | Polylaminate concentric cell wall layering in fibres of Homalium foetidum and its effect on degradation by microfungi[END_REF]Clair et al. 2006b). It should be interesting to investigate whether this feature occurs in the whole family.

FIGURES LEGENDS

Fig. 1 Fig. 2 Fig. 3 Fig. 4

 1234 Fig. 1 Localization of specimen used for the experiments

Fig. 7

 7 Fig. 7 Longitudinal section of Laetia procera opposite wood. (a) Observation of the

Fig.Fig. 9

 9 Fig. Result of the Wiesner reaction on transversal sections of tension (a) and opposite (b)

  Fig. 2

Fig. 9

 9 Fig. 9

Fig. 10

 10 Fig. 10

  Fig. 11

Table 1

 1 Diameter at breast height (DBH, in cm), Growth Strain (microstrains) means value, standard deviation and number of positions used for upper and lower side for each tree

	Tension wood (TW)	Opposite wood (NW)
	DBH
	Trees
	(cm)

Mean of the upper side Number of positions Mean of the lower side Number of positions Upper -lower side means

	Lp1	19	2714 ± 700	3	666 ± 293	3	2246
	Lp2	23	2035 ± 625	2	515 ± 23	3	1520
	Lp3	28	2701 ± 318	3	666 ± 247	3	2035
	Lp4	22	1590 ± 681	3	582 ± 105	3	1008
	Lp5	26	2912 ± 404	3	416 ± 200	3	2496

Table 2

 2 Values and standard deviation of the various measurements on fibre secondary wall of tension wood specimen from Lp1 tree

Thin layer number Tree specimen GS (µstrain) Thick layer thickness (µm) Thickness (µm) min mean max Thin layer thickness / thick layer thickness

	Lp1	1	3504	1.38 ± 0.26	0.08 ± 0.03	4	5.30	8	0.06
		2	2170	1.31 ± 0.15	0.13 ± 0.03	5	6.00	7	0.10
		3	2467	1.44 ± 0.52	0.15 ± 0.03	4	5.13	7	0.11

Table 3

 3 MFA and diameter of cellulose aggregates (mean ± SD) for each sample from Lp1 tree. TW: tension wood, OW: opposite wood, LW: lateral wood

						Average	Average
		Type			Average MFA	diameter of	diameter of
			GS	Average MFA (°)
	Tree	of	Specimen		(°)	cellulose	cellulose
			(µstrains)	/ specimen
		wood			/ type of wood	aggregates	aggregates
						(nm)	(nm) / type
						/ specimen	of wood
	Lp1	TW	3504	3.1 ± 1.9		22.8 ± 3.0
			2170	8.6 ± 5.7	5.2 ± 3.1	21.7 ± 2.2	21.9 ± 0.8
			2467	3.8 ± 2.3		21.3 ± 3.3
		LW	-115	16.6 ± 4.8		18.3 ± 1.6
		OW	346	20.7 ± 5.3		20.4 ± 2.2
		OW	922	16.2 ± 6.2	17.5 ± 2.8	17.2 ± 1.9	18.4 ± 1.6
		OW	730	13.9 ± 4.8		16.4 ± 2.1
		LW	461	19.9 ± 5.2		19.6 ± 1.6

Table 4

 4 Absorbance at wavelengths 270 and 280 nm and ratio of absorbance A280/A260 of the various layers of specimen of Laetia procera n°1

			GS
		Tree specimen
			(µstrains)
					Thick layers			Thin layers
				A280	A270 A280/A260 A280	A270 A280/A260
	Tension	Lp1	3504	0.15	0.14	1.15	0.18	0.18	1.08
	wood		2170	0.15	0.14	1.19	0.29	0.27	1.17
	specimen		2467	0.24	0.22	1.20	0.38	0.36	1.13
					S 2 layers
				A280	A270 A280/A260
	Opposite		-115	0.45	0.40	1.27
	and		346	0.44	0.40	1.31
	lateral		922	0.37	0.33	1.25
	wood		730	0.34	0.31	1.25
	specimen		461	0.53	0.47	1.30

ACKNOWLEDGMENTS

Many thanks to Tancrède Almeras for his critical review of this paper and Ivan Scotti for English corrections.


domain: math shs

text:
Julia Pilet
email: julia.pilet@u-pec.fr

Julie Horoks
email: julie.horoks@u-pec.fr

Assessment in mathematics as a lever to promote students' learning and teachers' professional development

Keywords: Assessment, algebra, teachers' practice, teachers' professional development, collaborative work

In this paper, we are presenting our analytical tools to characterize assessment activities as part of teachers' practice, on a specific mathematical content (algebra). We are also presenting the principles of our collaboration with high school teachers, inside a particular workgroup (LéA), to explain why we came to consider assessment as a potential lever to enhance both the students' learning in mathematics, and the teachers' development. We are presenting a few results on the effects of this collaborative work on teachers' practice when assessing students' learning, and on our means to analyze the students' results throughout the process.

Assessment as a framework to characterize teachers' practice

In this first part, we are presenting the framework that we have built to analyze teachers' assessment practice, leaning on previous studies about teachers' practice, and on the teaching of algebra in particular.

Defining assessment practice

Assessment can be found in many aspects of the teachers' activity, and it would be easy to call assessment any interaction between the teacher and the students. To restrain our observations, we draw upon De Ketele's definition (1989), and call assessment any gathering of information by the teacher on the students' activity and knowledge, and the interpretation and use of this information.

By activity, we mean everything the students do, say, think, (or do not do). Of course, not everything is accessible to neither the researcher nor the teacher, but we consider that students' learning happens through their mathematical activities, at least partially (cf. [START_REF] Rogalski | Theory of activity and developmental frameworks for an analysis of teachers' practices and students' learning[END_REF], about the use of Activity Theory as a framework for research in the Math Education field). These activities may consist on participating in a debate around a task or listening to a mathematical discourse in class. But in many occasion, they result from the tasks proposed by the teacher, and on the choices that the teacher makes to manage the solving of the task, which are elements that we consider when analyzing the teachers' practice in class.

Assessment is easier to pinpoint for the researcher, when it is formal, for example a written summative test at the end of a teaching sequence, or short diagnostic tests happening at the beginning of every session. Informal assessment, on the other hand, is more difficult to identify, but can happen in many occasions during the class, through the interactions between the teacher and the students, giving information on the students state of knowledge, to the teacher or to the students themselves. To be able to characterize assessment practice in any case, we have drawn a list of criteria, whether the assessment is formal or not, and at any point in the teaching sequence.

Characterizing assessment practice

One of the elements that we take into consideration to characterize assessment practice, is the distance between each assessment task and the similar tasks previously given by the teachers on the same mathematical content [START_REF] Horoks | Les triangles semblables en classe de 2nde: Des enseignements aux apprentissages Etude de cas[END_REF]. For example, when a teacher is assessing the students' knowledge on a mathematical content through a final test, we can question the choices of tasks made by this teacher and their link with the tasks that were actually worked on before the test. A certain gap between the test's and previous tasks (or between tasks from a diagnostic test and all the possible prerequisite tasks) can be interpreted in different manners: it could be explained by the function given to this test, (rewarding or challenging the students for example) or maybe by a lack of pedagogical content knowledge [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] for the teacher. The epistemological and didactical analyses of the mathematical contents are crucial here to make the comparison between the two sets of tasks. In the case of informal assessment, each new task given by the teacher has to be considered among a set of previous tasks, depending on the "study moments" [START_REF] Chevallard | L'analyse des pratiques enseignantes en théorie anthropologique du didactique[END_REF][START_REF] Barbé | Didactic restrictions on the teacher's practice: The case of limits of functions in Spanish high schools[END_REF], related to the distance from the first encounter with the mathematical notion. The distance between the tasks allows us to measure the complexity for the students compared to the tasks that they have already worked on. More globally, the whole range of tasks proposed by the teacher on a mathematical content, with the absence of particular tasks in the related assessments, can tell us some of the intentions of the teacher for assessment, in relation with teaching.

Another element that we take into account, is the "depth" of the information: indeed, the process of "taking / interpreting / exploiting information on students' activity" in mathematics, can take its roots in the solution that the students produce (the result of a task), the way they solve the task (procedure to achieve a result), or the knowledge that is put into action to complete this procedure. Linking the activity to the student's knowledge requires, from the teacher, an understanding of the conceptualization of the mathematical contents behind the procedure, which is usually specific to the particular content. Regarding algebra in particular, it leads us to consider some specific elements, such as the form of writing for calculations or the type of reasoning. Indeed, some forms of writing for the calculation for example (cf. figure 1) can inform the teacher on the meaning of the equal sign for each student (computation, equivalence), or on the student's structural / procedural view of numerical or algebraic expressions. The interpretation of the information can also differ, depending on the reference taken for this interpretation: a comparison with what is expected by the institution (curricula, external assessment), or what could be expected by the teacher, considering all the previous work and the teacher's knowledge of the didactics of the mathematical notion at stake (errors, obstacles and breaches, steps in the conceptualization, etc). It can also be a comparison between the students' different procedures or a comparison in time for one student, to appraise his or her progress. These comparisons can be made explicit or not to the students. Here again, it can be linked to the various possible functions of the assessment.

Finally, the exploitation of the information differs depending on the moment when it occurs: whether it leads to immediate feedbacks related to students' result, procedure or knowledge or, in a more or less short term, when it influences the planning for the next activities.

Before giving an example of teachers' formal assessment practice, we will first describe our working context with a group of teachers.

Description of the collaborative work inside the Léa

A "Lieu d'Education Associé (Léa)" is an instance created by the French Institute for Education (IFE) to promote research with people who play an acting part in education. For 3 years, they are associated with a team of researchers to investigate questions about education and to build realistic resources for teachers or educators. Our Léa takes place, since May 2014, in a high school (students from 11 to 15 year old) in an Educational Priority Area, with 9 teachers (4 at the beginning) and 7 researchers, who meet every month to work together, to build teaching materials for algebra and to discuss assessment practice that could promote students' learning. The Léa can give us access to those teachers' evaluation practice in the long term.

An example of a comparison of teachers' formal assessment practice

We asked Léa teachers to design a diagnostic test at the beginning of the year for their 7th-grade students, to assess their numerical and pre-algebraic knowledge before introducing algebra. The tasks of the tests that they individually proposed were not covering all the range of the required knowledge for the introduction of algebra [START_REF] Carraher | Early algebra and algebraic reasoning[END_REF][START_REF] Kieran | Learning and Teaching Algebra At the Middle School Through College Levels. Building Meaning for Symbols and Their Manipulation[END_REF]. The teachers justified their choices by giving institutional or social reasons, rather than epistemological or didactical ones.

-(3 + 4) x 2 -8 = 6 -3 + 4 = 7 x 2 = 14 -8 = 6 + 4 x2 -8 -3 ------> 7 ------> 14 ------> 6 -3 + 4 = 7 7 x 2 = 14 We also conducted interviews with these teachers to find out about their views about assessment, after they had proposed their first summative test of the year. They were asked questions about their choices of tasks and the feedbacks they gave to the students afterwards (cf. table 1 for two of the teachers). Their answers showed a great variety in the tasks they proposed, regarding the distance with previous tasks, and probably resulting from different views on the functions given to formal assessment, despite the fact that these teachers often worked together. What was common to all the teachers on the other hand, is that they did not usually give many feedbacks to their students. Indeed, those teachers gave a mark without informing the students with the necessary elements to understand their mistakes and the limitations of their reasoning. Another comparison, related to informal assessment, for one teacher at different moments, will be made in the last part of this paper.

Assessment as a tool to enhance students' learning in mathematics Definition of formative assessment

For [START_REF] Black | Assessment and Classroom learning[END_REF], an assessment can be formative when a teacher uses the information on the students to help them engage in the work on a task, or to help each of them auto-evaluate their knowledge:

The term 'assessment refers to all those activities undertaken by teachers, and by their students in assessing themselves, which provide information to be used as feedback to modify the teaching and learning activities in which they are engaged. Such assessment becomes "formative assessment" when the evidence is actually used to adapt the teaching work to meet the needs. (page 2)

In terms of gathering/interpreting/exploiting information, formal assessment can play a more or less formative function, depending on the chosen tasks (if the tasks are way too complex or too simple; the students' productions might not reveal many useful information for the teacher). It depends also on the feedbacks made to the students. These facts can both be witnessed and analyzed by the researcher.

But when in comes to informal formative assessment in class, even if it is possible to see a teacher going around in the class when the students are working on a task, we can only witness the information actually gathered if the teacher is using it right away to guide the students' work.

Deciding not to use the information right away, but reorganizing the plan of the next sessions, for the entire class or a particular student, could also be an exploitation of the information to promote the students' learning, but the researcher would then hardly acknowledge it. In any case, the research time allowed to the students to work on the task will probably have an influence on their mathematical activity, and on the information that the teacher will be able to take on this activity, depending on the task.

A key moment: sharing the students' productions after letting them work on a task

The moment of pooling of students' procedures, after letting them work on a task, alone or in groups, seems to us like a good opportunity for informal formative assessment, where students could compare their solution with others' and know if they are close to what was expected. It depends of course on how the teachers choose to manage this moment of the session, and on the use they will make of the students' productions. This is why we will look more closely at those moments in the classrooms to analyze the use that the teachers make of the students' productions: how is the students' work taken into consideration? What kind of productions do the teachers choose to share with the entire class? Is there a variety in these productions, regarding the result of the task or the possible procedures? Are there errors, typical or not, showed to the students? These elements are indicative of the information probably gathered by the teacher on the students' work while they were working, but depends also on an a priori analysis of the task, strongly linked to the mathematical contents to be mobilized, in order for the teacher to anticipate the possible outcomes.

We also analyze the exploitation of these productions. However, the interpretation of the information by the teachers remains mostly invisible to the researcher, except when the teachers explicitly mention the reference they use to compare (with what is expected at the end of the year, with what the students already did before, between students…). We note if the teachers organize a comparison of the results or of the procedures. Do they rank them to show the relevance and limits of each solution? How is organized the (in)validation of the solutions? Who is (in)validating them? With which arguments? And which conclusion? These elements can inform us on the role given to the students in the validation and institutionalization process and in the assessment process in general. We will give an example of this type of analysis for one teacher, in the last part of this paper.

We have hypotheses on the conditions that we consider more favorable towards student's learning, for example by making use of various students' procedures and errors and by implicating them in the validation, using mathematical arguments. We will confront these hypotheses with the results in algebra of a hundred of high school students, whose teachers' assessment practice was analyzed in this study,. In order to do that, we will analyze each task given by each of the teachers participating in the Léa, as part of the formal assessment process in algebra during 3 years, in terms of kinds of tasks [START_REF] Chevallard | L'analyse des pratiques enseignantes en théorie anthropologique du didactique[END_REF] and adaptations (cf. [START_REF] Robert | Tâches mathématiques et activités des élèves: une discussion sur le jeu des adaptations introduites au démarrage des exercices cherchés en classe[END_REF], to determine their variety and complexity. We will collect the students' productions, analyze their answers and characterize them according to the different degrees of algebraic competencies defined by [START_REF] Grugeon | Diagnostic et parcours différenciés d'enseignement en algèbre élémentaire[END_REF] and [START_REF] Chenevotot-Quentin | The diagnostic assessment Pepite and the question of its transfer à different school levels[END_REF] for the design of a diagnostic assessment in algebra. This analysis is now in progress, as we are beginning the third year of our work inside the Léa.

Assessment as a lever for professional development

Our tools to analyze professional development

For our research, we already have collected a wide range of data inside our Léa, that we still have to fully process: to document the teachers' assessment practice, we gathered their personal documents for the class and asked them to film themselves regularly in their own class; to measure the possible effect of this practice on the students' learning in algebra, we collected many students' productions; and at last, to try to estimate the effects of our collaborative work on the teachers' practice, we recorded the discussions during our meetings, and kept the reports of these meetings, when written by the teachers.

The analysis of the teachers' assessment practice takes into account, as explained before, the list of tasks proposed to the students in algebra and the management of the resolution of these tasks in class (informal assessment) or after a test (formal assessment). Their point of view about assessment is also visible through the interviews we conducted at the beginning of the project, or through the discourse of the teachers during the meetings of the Léa. More specifically, we are interested in the ways they argument their choices for their class, through the type of reasons they give for their choices of tasks or management (institutional, social, didactical, mathematical, etc). The moments when the teachers disagree with the researchers, or try to convince new teachers who joined the collaborative workgroup, are particularly interesting for us for that matter; to gather information on a possible evolution on the teachers' point of view on assessment, and more globally, on the teaching of algebra.

Some results about the changes in the teachers practice and arguments

To illustrate our analyses of the teachers' practice and their development, we are giving here an example of two sessions, for the same teacher, filmed one year apart. We analyzed the moment of sharing the students' productions, after working on similar tasks. These tasks both involve testing a calculation program with several numbers to notice an unchanging result or property and proving it with algebra. Our analyses are based on the indicators that we have already listed (the variety of the productions chosen to be displayed, the exploitation of students' errors, the initiatives in the validation, the arguments for justification).

Our analyses after the first year of collaboration (see table 2 "year 1") tend to find that teachers' assessment practice are very settled and stable. It seems that our didactical contributions about the teaching and the learning of algebra have helped teachers' practice to evolve [START_REF] Horoks | Etudier et faire évoluer les pratiques d'évaluation des enseignants de mathématiques en algèbre au collège dans le cadre d'un Léa[END_REF]: indeed they have better indicators to select the students' productions that they will use for the discussion after a task. But the exploitation that they make of these productions hasn't really changed after the first year: when sharing them with the class, the Léa's teachers don't usually organize a comparison between the students' productions nor give a validation based on mathematical reasons. We also found that these teachers usually leave no initiative for students when working on the more complex algebraic tasks, which leads to the impossibility to rely on their production (see table 2, "year 1").

After the second year, where we decided to share some of our tools to analyze the sessions in class with the teachers, we can notice some evolution in the exploitation of the information (see table 2, "year 2"). Even though the second part of the task is more complex, this teacher still relies on the students' productions, even if they are not mathematically correct, to build, along with the students, the reasoning that will allow the class to invalidate the proposed solution.

However, even if the tasks are similar between year 1 and year 2, students have a higher grade in year 2 which may also explain the different choices made by the teacher. We should go on studying practice for a longer time, to identify its stability, and this is what we plan to do in the Léa project.

Figure 1 :

 1 Figure 1: Different writings for the calculation, giving information on the students' knowledge

-

  To be able to give marks for the institution -To work on the basics -To learn by adapting to a different situation -To adapt the teaching plan ahead

Table 1 : Formal assessment (summative test)

 1

Proceedings of CERME10

Both the cognitive (contents and tasks) and mediative (organization of the sessions in class) elements of the teacher activity play a part in the assessment process that we are trying to analyze here. But, as emphasized by [START_REF] Robert | A cross-analysis of the mathematics teacher's activity. An example in a French 10th-grade class[END_REF] there are other constraints of this professional occupation to be taken into account: the social (type of school), the institutional (curricula) and personal (carrier and education) components, playing a significant part when interpreting teachers' practice, including for us their choices in terms of assessment. We analyze teacher's practice through all these components, at different levels: locally in the classroom or globally within all the teaching plan, and we believe that it can give us access more deeply into the teachers' consistency and explain their stability. Yet, after the second year, we noticed some changes in the arguments that the teachers are giving to justify their choices, shifting a little from social and institutional reasons to mathematical or didactical ones, that we would hope to link to our work together, that is still going on..


text:
Nadine Grapin
email: nadine.grapin@u-pec.fr

Nathalie Sayac
email: nathalie.sayac@u-pec.fr

Using external assessments for improving assessment practice of primary school teachers: A first study and some methodological questions

Keywords: Assessment practices, external and internal assessment, validity in assessment

 but also the notions of "evaluative episode" and "professional judgement in assessment", developed by Sayac (in progress) for defining the didactic paradigm of assessment. In our paper, we present our methodology and focus on one aspect of this research: the use of external assessments as tools for improving assessment practices.

In France, recent modifications in curricula and institutional directions encourage teachers to assess competencies (and not only knowledge), but also to develop, in their classes, "assessment for learning": teachers should consider that assessment is a part of the didactic process and use assessment information for adjusting their teaching strategies. A recent study by [START_REF] Sayac | Nature et diversité des tâches mathématiques proposées en évaluation sommative par des professeurs des écoles, en France[END_REF] with some primary school teachers, has showed that assessments in mathematics are mainly summative and that teachers use the results principally for the end of term report. Moreover, assessment tasks are not complex and previously studied before taking the test. Furthermore, few assessment tools are made available to primary teachers. Until 2012, national diagnostic assessments were organized at the beginning of Grades 3 and 6 and teachers could use them for a diagnostic purpose, but these tests do not exist anymore. In the same time, the number of large scale assessments in primary school has increased in France, but items are not free and only results are published (Brun & Pastor, 2009, Dalibard & Pastor, 2014;Lescure & Pastor, 2008,). So, teachers cannot use items or results directly for their classes and educators can only exploit general results for providing an overview of trends in students' mathematical knowledge or difficulties.

Since 2016, a bank of exercises (with scoring procedure, explanation of the difficulties and propositions of teaching strategies) has been created for helping teachers to assess students at the beginning of Grade 3. It is a commendable initiative, but it raises a lot of questions about these exercises and their use: even if each task is relevant with regard to its assessment aim, teacher must select many of them to elaborate a complete test. How is this selection done and which competencies are finally assessed? Is scoring guide used or not by teachers? How? Etc. Like other external assessment tools, this could improve teaching and assessment practices, but merely delivering assessment tasks seems not to be sufficient: for designing tests and using results for a better regulation in classes, we think that, in addition to assessment tools, teachers must have knowledge about mathematical notions, teaching them.

Finally, these observations about the bank of exercises have led us to conduct a study to analyze and improve teachers' assessment practice; we suppose that training teachers in assessment will also impact their teaching as a whole. We have chosen to conduct our research in a specific mathematical domain (whole numbers) at the beginning of the elementary school (Grade 1 to 3), in a collaborative environment. After specifying our research and training aims, we explain our theorical frameworks and methodology for studying this environment and the professional development of any actor and conclude with perspectives.

Research aims and context

This research aims to continuing to explore the assessment practices primary school teachers in mathematics but also to improve them as part of a collaborative research-training. It is undertake in a special network "AeDeP" at FIE (standing for Associated educational Design-experiment Places at French Institute for Education); this type of project is initially based on an educational question shared by different actors (teachers, researchers, school directors, local authorities) and is built for sharing experiences and designing common tools. Our two key issues are:

1. How do primary school teachers assess their students? We focus on test content (which type of tasks do they propose? What techniques are necessary to solve them? Etc.), but also on how teachers design their assessments (what kind of resources they use? What do they do with the results? Etc.) 2. How can such a collaborative research-training improve teachers' assessment practice, and more generally mathematics teaching?

We present in this paper one part of our three-year project, called "EvalNumC2" and we focus on the development of assessment practices in mathematics at primary school. For this part of the project, regular meetings are planned (one per month) with all the actors (ten primary school teachers and us, two educators/researchers) and with different aims depending on the timing of the research. At the beginning, researchers will only collect the tests produced by teachers (without training) and information about teachers' practice; after, external assessments and didactic tools designed by researchers will be introduced.

Theoretical framework

Activity theory for studying professional development in a collaborative environment

We consider that the Activity Theory expanded by [START_REF] Engeström | Expansive learning at work: Toward an activity theoretical reconceptualization[END_REF] is a good framework to study this kind of collaborative project of research and training through the objects and the tools used by the subjects, i.e. the teachers involved and us, as mathematics teacher educators/researchers (MTE-Rs). In this framework, the activity of teachers in which we are interested is their assessment practice and the activity of MTE-Rs is to explore and develop these assessment practices within the collaborative environment. We look at the activity of the different subjects in order to produce results concerning professional development of each one, promoted through this collaborative device [START_REF] Jaworski | Inquiry community in an activity theory frame[END_REF].

Among the tools used in the AeDeP, we have chosen to focus on two specific ones: a list of criterions for studying the validity of test items and tests designed by researchers. In the following we explain why.

Didactic paradigm of assessment

To study the assessment practices of teachers, we adopt the didactic paradigm of assessment developed by Sayac (in progress). In this framework, teachers' assessment practices are studied through the evaluative episodes they propose during the learning process, but also from the evaluative logic of teachers that becomes apparent in the design of the episodes (resources, method, nature of the tests provided), through their professional judgment in assessment and their grading practices.

A number of researchers draw on the notion of professional judgment when considering learning assessment by teachers [START_REF] Wyatt-Smith | The centrality of teachers' judgement practice in assessment: A study of standards in moderation[END_REF][START_REF] Laveault | Le jugement professionnel : foyer de tensions et de synergies nouvelles en évaluation scolaire[END_REF]Wyatt-Smith, Morgan & Watson, 2002). For them, professional judgment includes both cognitive process and social practice [START_REF] Mottier Lopez | Le jugement professionnel en évaluation : un acte cognitif et une pratique sociale située[END_REF], which is not same as a "mechanical gesture of measurement" [START_REF] Wyatt-Smith | The centrality of teachers' judgement practice in assessment: A study of standards in moderation[END_REF], but must be considered as a "flexible dynamic process comprised of middle and final judgments" [START_REF] Tourmen | Evaluators' Decision Making The Relationship Between Theory, Practice, and Experience[END_REF]. The professional judgment of teachers could be viewed as an act of discernment and as the ability to build intelligibility of the phenomenon of assessment, while taking into account the epistemic, technical, social, ethical and paradigmatic dimensions of classroom assessment practices [START_REF] Tessaro | Améliorer la qualité des pratiques évaluatives: une articulation entre formation initiale et formation continue[END_REF]. In the didactic paradigm of assessment, the professional judgment is considered as a kind of "didactic vigilance" (Pézard, 2010) specifically applied to the assessment activity of teachers. This allows them to on the one hand give a valid verdict (Chevallard, 1989) about students' mathematical knowledge, individually and collectively, from data collected during the different evaluative episodes. On the other hand this allows them to mutually articulate the different moments of the learning process (especially to connect evaluative episodes to the other moments of the learning process), based on data collected during the different evaluative episodes. This professional judgment in assessment is related to teachers' mathematical and didactical knowledge and assessment skills. It also depends on individual factors as beliefs on learning and assessment as well as professional and personal experiences on assessment (Brady & Bowd, 2006;[START_REF] Martino | Attitude towards mathematics: a bridge between beliefs and emotions[END_REF][START_REF] Jong | Assessing attitudes toward mathematics across teacher education contexts[END_REF].

Validity of tests

Researchers and teachers, and more generally, all assessment designers, have a same preoccupation about the test: they want to be sure that their tests assess what they should assess and only that. In previous work, we have described a methodology and listed didactic criterions for analysing the validity of an external assessment in mathematics [START_REF] Grapin | Étude de la validité de dispositifs d'évaluation et conception d'un modèle d'analyse multidimensionnelle des connaissances numériques des élèves de fin d'école[END_REF]2016). We transfer and adapt these principles with two different aims: as researchers, for analyzing the content of internal tests designed by teachers (classroom assessment), and, as educators, for helping teachers to construct their own assessments.

For studying the validity of a test of a mathematical domain, we consider two levels: locally (exercise by exercise) and globally (the test as a whole). From a didactical point of view, the a priori analysis of each item is crucial because it gives indicators to guarantee that a task is relevant for achieving its assessment aim. For each item, we realize such an analysis specifying the tool or object aspect, the registers implicated with their possible congruence [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF], the types of tasks involved in the resolution, the different techniques (adequate and inadequate through curricula) for solving the problem, the arithmetic problem classes [START_REF] Vergnaud | The theory of conceptual fields[END_REF], the complexity levels [START_REF] Sayac | Évaluation externe et didactique des mathématiques : un regard croisé[END_REF]. We also take into account the techniques involved in the resolution; if an item can be solved with a technique or a strategy different from the ones expected relatively to his assessment' aims, we consider the item as inconsistent.

For studying and ranking items according to their complexity, we have developed a tool [START_REF] Sayac | Évaluation externe et didactique des mathématiques : un regard croisé[END_REF] which takes into account three factors. In the first one, the wording and the task context are considered (what is difficult to understand the question?), in the second one, the mathematical knowledge involved in the solving process is studied, and finally in the third one, concerns the level of competency (is the task usual or not, does the student have to take initiative?). For each of these factors, we attribute a degree of complexity between 1 (simple) and 3 (complex). We also observed that discussions between teachers arise during the use of this tool because they do not have the same ideas about the complexity, depending on their teaching or their representations of mathematical notions [START_REF] Sayac | Former à l'évaluation à partir d'un outil de chercheur : enjeux et perspectives[END_REF]. So, this tool seems particularly appropriate to use with the teachers in our project.

On a global level, we study whether the items are representative for the curriculum: have all types of tasks been represented? What are the complexity levels (defined a priori) of the items? Are they different or similar? Which registers of representation are involved? When a same type of task is represented by different items, are the effective techniques similar? Etc.

Methodological elements and preliminary results

The Engeström triangle [START_REF] Engeström | Expansive learning at work: Toward an activity theoretical reconceptualization[END_REF] allows identifying, for each subject, objects that will evolve during the collaborative project of the research and training, through the mediation proposed via the tools, the rules and the division of labor and the communities.

For the teachers involved in the project, the main object is to assess their students. It comes to study specifically, from the collaborative environment, tests designed by researcher as one of the tools used. We will study how these tests could:

1-Enhance assessment tasks proposed by the teachers in terms of diversity, complexity and coverage of the mathematical domain.

2-Develop teacher's professional judgment through the study of students' answers to these tests and the confrontation between all the teachers during the meetings.

3-Work on the coding of students' answers and therefore, on grading.

Studying or using external tests could foster the teachers' professional development, because they will be validly designed from the epistemological and didactical point of view (Grapin & Grugeon, 2015). The evidence of validity will allow us to show how these tests could be relevant for the three points above.

We consider as Johnson, Severance, Penuel and Leary (2016) that: Professional development organized around the analysis of mathematical tasks has potential to prepare teachers for standards implementation by helping them develop common understandings of standards and how to help students meet ambitious new learning goals. (p. 173) Therefore, we believe that the contribution of assessment tasks from external tests, in a collaborative context, could develop teachers' skills on assessment tasks design and contribute to enhance their professional judgment in evaluation (Gueudet, Pépin & Trouche, 2013).

Methodology

At the start of the school year, we collected tests designed by teachers involved in our project in order to analyze the assessment tasks with the tool developed in previous research [START_REF] Sayac | Évaluation externe et didactique des mathématiques : un regard croisé[END_REF]. We will also collect new tests designed by the teachers at the end of the school year, after the collaborative group work in the meetings. Each assessment will be analyzed in terms of its validity with the criterions listed above; we will principally observe the change in content between the beginning and the end of the project (variety of type of tasks, complexity of tasks and the coding of students' answers).

Each teacher will also fill in a questionnaire about his or her assessment practices (How does he or she design tests? Which resources does he or she use? What are the periods of assessment in his/her classes? How does he or she use the results?). The results of these questionnaires will be used to compare teachers but also as an element for analysing the evolution of their practice.

Lastly, for relying assessment and teaching in classes, each teacher involved in the project will have to keep a "daily book" (journal) in which he or she explains briefly the aim and the content of each mathematical course (in the numerical domain). He or she will also have to identify, according to his/her own representations, the evaluative episodes and describe these more specifically. We will analyze the content of the tasks proposed in tests and during teaching to study their correlations and their evolution during the project. As observed by Grugeon and Bedja (2016), we suppose that training teachers in assessment will also improve teaching: teachers should propose a wider variety of types of tasks, but they also should be able to have a better interpretation of students' errors and propose adapted instruction to upgrade students' level of understanding.

Preliminary results

At the time of writing, we cannot present full results because this project started in September 2016 and we are in the process of collecting first data; we show however two example of tasks, extracted from the same test, one of the teachers in our study used in Grade 3.

In a first task (Figure 1), the five questions are similar and aim to assess the decomposition of written numbers in canonical expressions. In all examples, the underlying structure is regular. We observe that such a task is not complex (we quote level 1 on each factor of complexity) and assesses five times the same knowledge: only the positional aspect of numeration (and not the decimal aspect). The second exercise of the test (Figure 2) looks like the first one and assesses the same type of knowledge. Throughout the full test, there isn't any exercise for assessing the decimal aspect of numeration. We conclude that such a test is not valid and in the project we are going to elaborate other questions with the teachers to fill this lack.

We have not yet achieved the analysis of all classroom assessments designed by teachers involved in the project, but it seems that, as we can observe in the two previous examples, tasks are repetitive, having low complexity and for the numeration, assess principally positional aspect of the numeration. Such observations led us to propose external assessments with other types of tasks: exercises designed to assess numerical aspect of the numeration but also complex situations intended to develop students' abilities.

Conclusion and perspectives

We have focused in the paper on the original and theorical notion of "didactic paradigm" designed recently by Sayac (in progress). What we can tell currently, from the first data collected (tests, questionnaires, interviews) is that the teachers, participating in our research, design tests with low levels of complexity and have invested very little in assessment as a professional gesture. They assess their students as they can, with very subjective practices. So, it seems that our research, with its training dimension, will make possible a real professional development concerning the assessment tasks proposed in mathematics by these teachers and their professional judgment in assessment.

So, besides studying and providing primary school assessment practice in mathematics and designing assessment tools, our research aims also to develop these theorical elements. At the end of the project, it will be possible to show the impact on the teachers' practice and interests, and the limitations of this methodology, according to our theorical framework. It would then be possible to extend such studies in other mathematical domains (as geometry) or in other levels, for example at secondary school.

Figure 1 :

 1 Figure 1: First exercise and students' answer extracted from a classroom assessment designed by a Grade 3 teacher.

Figure 2 :

 2 Figure 2: Second exercise and students' answer extracted from a classroom assessment designed by a Grade 3 teachers.

Proceedings of CERME10


text:
Annalisa Cusi
email: annalisa.cusi@unito.it

Francesca Morselli
email: morselli@dima.unige.it

Cristina Sabena
email: cristina.sabena@unito.it

Enhancing formative assessment in mathematical class discussion: a matter of feedback

Keywords: formative assessment, technology, feedback, teacher, peers

This contribution is framed in a European project on the use of technology to foster formative assessment strategies (FaSMEd project) and addresses the crucial issue of feedback therein. The theoretical framework refers to formative assessment, with specific focus on different levels of feedback. By analyzing data from teaching experiments in grades 5 and 7, we identify strategies employed by the teacher to provide feedback during class discussion and investigate the effect of such strategies on the enactment of formative assessment.

Introduction and theoretical background Formative assessment and feedback

This contribution is framed within the European Project FaSMEd ("Improving progress for lower achievers through Formative Assessment in Science and Mathematics Education"), aimed at investigating the use of technology to promote formative assessment (FA) practices in the mathematics and science classroom. FA is conceived as a method of teaching where "[...] evidence about student achievement is elicited, interpreted, and used by teachers, learners, or their peers, to make decisions about the next steps in instruction that are likely to be better, or better founded, than the decisions they would have taken in the absence of the evidence that was elicited" (Black & Wiliam, 2009, p. 7). [START_REF] Wiliam | Integrating assessment with instruction: What will it take to make it work[END_REF] describe five key FA strategies: (A) Clarifying and sharing learning intentions and criteria for success; (B) Engineering effective classroom discussions and other learning tasks that elicit evidence of student understanding; (C) Providing feedback that moves learners forward; (D) Activating students as instructional resources for one another; (E) Activating students as the owners of their own learning.

Feedback is a crucial issue in FA. Hattie and Temperley (2007) define feedback as "information provided by an agent (e.g., teacher, peer, book, parent, self, experience) regarding aspects of one's performance or understanding" (p.81) and identify four major levels of feedback: (1) feedback about the task (concerning how well a task is being accomplished or performed); (2) feedback about the processing of the task (concerning the processes underlying tasks or relating and extending tasks); (3) feedback about self-regulation (concerning the way students monitor, direct, and regulate actions toward the learning goal); (4) feedback about the self as a person (consisting in positive (and sometimes negative) evaluations and affect about the student). Hattie and Temperley also point out that feedback is a consequence of specific actions and that can also be sought by students, peers, and so on, and detected by a learner without it being intentionally sought." (p.82).

Enhancing formative assessment: technology and tasks

Specific theoretical and methodological assumptions of the Italian team (the three authors) within FaSMEd concern the importance of fostering students' development of ongoing reflections on the teaching-learning processes, and helping students to make their thinking visible (Collins, Brown and Newmann, 1989), sharing their ideas with the teacher and the classmates. These basic assumptions entail specific choices concerning the technology and the tasks.

Concerning technology, each class is equipped with a Connected Classroom Technology (CCT) through which the students' tablets and the teacher's laptop are connected. In order to foster collaboration and sharing of ideas, students are asked to work in pairs or in small groups on the same tablet. By means of the CCT equipment, students are able to receive worksheets from the teacher, send back their written answers, and answer to instant polls proposed by the teacher; the teacher can easily collect the students' opinions and reflections during or at the end of an activity, as well as the written answers, and receive the statistics concerning the answers to the polls. The teacher's computer is connected to an Interactive White Board (IWB) or a projector, so that it is possible to select and display written productions and the results of polls.

Concerning the tasks, students are asked to work on sequences of activities with a strong argumentative component: they are required to provide their answer and explain it in a written text. In this way, they are encouraged to make their thinking visible and to provide the teacher and the peers with a written text that will be shared and analysed during mathematical discussions [START_REF] Bartolini Bussi | Verbal interaction in mathematics classroom: A Vygotskian analysis[END_REF]. The mathematical content at issue is relationships and functions, and their different representations (symbolic, tabular, graphic). Activities are adapted from the ArAl project [START_REF] Cusi | Early Algebra: Theoretical Issues and Educational Strategies for Bringing the Teachers to Promote a Linguistic and Metacognitive approach to it[END_REF] and The Mathematics Assessment Program (http://map.mathshell.org).

Summing up, the typical lesson starts with a peer activity on one worksheet. After having collected all the students' written answers, the teacher promotes a class discussion, starting from the analysis of some written answers (selected and displayed on the IWB). The discussion concerns the task level (correct answers and typical mistakes), the task processing level (effective ways of approaching the task) and the communicative level (effective ways of communicating the answer and the explanation). Comparison between different solutions is especially promoted. For further details on the organization of the lessons, see [START_REF] Cusi | Enhancing formative assessment strategies in mathematics through classroom connected technology[END_REF].

Previous results and the current research questions

In former studies [START_REF] Cusi | Enhancing formative assessment strategies in mathematics through classroom connected technology[END_REF] we analysed classroom discussions performed within the FaSMEd teaching experiments, and highlighted that CCT may support the activation of FA strategies by the teacher, by the peers (peer assessment), and by the student himself (selfassessment). In this contribution we focus on strategy C (providing feedback that moves learners forward) and we investigate: what are teacher's strategies that may foster FA strategy C; which level of feedback is provided; what are the effects of this strategy (in terms of activation of other strategies such as D and E).

The context

In Italy the FaSMEd project involved 20 teachers, from three different clusters of schools located in the North-West of Italy (from grade 4 to grade 7). During all the teaching experiments, one of the authors was present in the classes with the teachers, acting as a participant observer. The analysis is based on video-recordings of the classroom discussions, with the help of written transcripts and field notes by the participant observer (one of the authors).

Analysis

By analysing several episodes from the teaching experiments, we came to a first characterization of teacher's feedback strategies, that is the ways in which she gives feedback to students. In the subsequent part, we provide a short example for each kind of feedback strategy, highlighting the level of feedback provided. Moreover, we will discuss the effects of each feedback strategy in terms of activation of FA strategies. All the examples come from the third lesson on time-distance graphs, performed in grade 7. The lesson sequence on time-distance graphs (about 20 hours, 8 worksheets) was adapted from the Mathematics Assessment Program (http://map.mathshell.org) and was introduced by an experience with a motion sensor, which provided instantaneous graphical representation of a linear motion performed by the students. This lesson was chosen because it contains all the typical teacher's feedback strategies that recurred in different classes and grades in our teaching experiments. Here we refer to worksheet 6, where a graph and three possible stories are presented:

Story A: Tommaso took his dog for a walk to the park. He set off slowly and then increased his pace. At the park Tommaso turned around and walked slowly back home.

Story B: Tom rode his bike east from his home up a steep hill. After a while the slope eased off. At the top he raced down the other side.

Story C: Tommaso went for a jog. At the end of his road he bumped into a friend and his pace slowed. When Tommaso left his friend he walked quickly back home.

The students are asked to answer to the following question: "What is the story that this graph represents? Justify you answer." Students work in pairs and send their written answers to the teacher's computer, as soon as they feel ready. The teacher, together with the participant observer, reads the answers as they arrive at her laptop and selects some of them for the discussion. The first selected answer is the one by the group of Mil and Pon: "For us the answer is B for two reasons:

1. You cannot do 1600 meters by foot in half an hour 2. The graph represents precisely the information given by the story. Then Tommaso climbs the hills, the first trait is the climb, the second is still a climb but less steep. When he comes to the top, then Tommaso climbs down and goes back home".

We may observe that Mil and Pon highlight two reasons for the choice of story B: the first one is based on everyday life experience (they draw from the graph the information that 800+800 meters are walked, and they point out that it is not possible to walk for 1600 meters in half an hour; since it is actually possible to walk 1600 meters in half an hour, this argument is wrong), the second one is based on a wrong interpretation of the graph: they interpret the graph as a picture of the hill, that Tommaso first climbs up and then descends down. For the teacher, the discussion of students' production is the occasion for giving feedback on two levels: about the task (clarifying that the graph represents the relation between distance from home and time, and is not a picture of the hill, so it does not share with it any resemblance, in principle) and about the way of processing the task (pointing out that the justification must be based on a careful analysis of the information provided by the text and the graph). To this aim, the teacher promotes a discussion (strategy B). Mario is asked to read the production of Mil and Pon, then the discussion starts.

Transcript Analysis

217. Teacher: Then, answer B for two reasons. Ok, Lollo?

The teacher encourages the students to activate themselves as resources for Mil and Pon (strategy D). Somebody else was talking about the second motivation, motivation B, the fact that the graph explains us that Tommaso climbs the hill and so on. Lollo said: "No, because when we did the experience with the sensor we went on a oblique line, but the path we were doing was not on a hill, it was not steep".

Lollo

The teacher synthetizes the interventions of Lollo, Mario and Rob, stressing that the justification 1 is not correct. Then she shifts the focus on justification 2, focusing on the correct interpretation of oblique lines within a time-distance graph. In this way, she activates strategy C, giving Mil and Pon a feedback about the task (it is a mistake to interpret the task as the picture of a hill) and the processing of the task (focusing on the ways in which the time-distance graphs should be interpreted). Here we may see instances of both rephrasing (the teacher reformulates some arguments so as to make them more intelligible to the mates) and revoicing (the teacher revoices some of the students' interventions, so as to draw the attention on specific effective parts of the given arguments). […]

Ur

and pointing out that something doesn't work in what Mil and Pon wrote: in their interpretation of the graph as a picture the first trait is the steeper part of the hill, but in the interpretation of the graph in terms of speed (as in the previous experience with the motion sensors) the segment is steeper when the speed increases. Mark expresses his own doubts about the two contrasting interpretations: in reference to everyday experience, it is not so common to walk faster in the steeper trait of a hill. Mark's intervention is an instance of strategy E, but his intervention could also act as feedback for Mil and Pon (strategies C, D).

Teacher:

But I… this answer really tells that the first segment, the first two parts of segment that go up describe the hill, the steep climb, the less steep climb, the top and after the descent…

The teacher goes back to Mil and Pon's written answer, so as to foster the comparison between their answer and the intervention of Mark. By contrasting in this way the two answers, the teacher is implicitly giving a feedback to Mil and Pon (strategy C) and turning Mark as instructional resource for them (strategy D).

235. Student: That is wrong. This intervention confirms that the contrasting was efficient in fostering the comparison between the different positions of Mark and Mil and Pon.

236. Teacher: Then the idea that the segments, as Rob said… "the graph is different from the drawing of a hill", or Lollo said "when we did it with the sensors we saw this kind of segments but we were not climbing, it meant that we changed the speed"… Let's remember always that the y axis describes what? The distance from home in meters.

The teacher intervenes with a rephrasing: she teacher reformulates and synthetizes the interventions of the students, so as to give a feedback to Mil and Pon. The activated strategy is C (providing feedback). In this way she is efficient in turning the feedback about the task into a feedback about the processing of the task (she draws the attention on the meaning of the two axes). We call this kind of intervention a rephrasing with scaffolding, since the teacher, besides rephrasing, adds some elements to guide the work on the graph.

Results and discussion

Within the FaSMEd project, we performed several teaching experiments in grades 5 to 7, setting up task sequences and proposing them in a CCT environment. As a first result [START_REF] Cusi | Enhancing formative assessment strategies in mathematics through classroom connected technology[END_REF], we showed how technology may support the activation of several FA strategies. In the current paper we focused on FA strategy C (providing feedback) and explored the ways in which the teachers may intentionally provide feedback during class discussions, the kind of feedback that is provided and the possible links with FA strategies.

The analysis of several class discussions performed during the teaching experiments led us to identify typical strategies employed by the teacher to provide feedback. Such strategies are exemplified in this paper through the analysis of a class discussion in grade 7. Here we summarize the strategies and discuss further developments of our study. The first strategy is revoicing, that occurs when the teacher mirrors one student's intervention so as to draw the attention on it. Often, during the revoicing, the teacher, stresses with voice intonation some crucial words of the sentence she is mirroring. Rephrasing takes place when the teacher reformulates the intervention of one student, with the double aim of drawing the attention of the class and making the intervention more intelligible to everybody. Rephrasing is applied when the teacher feels that the intervention could be useful but needs to be communicated in a better way so as to become a resource for the others. We also found special instances of rephrasing, when the teacher, besides rephrasing, adds some elements to guide the students' work. Drawing from [START_REF] Wood | The role of tutoring in problem solving[END_REF] the term "scaffolding", we call this special strategy a rephrasing with scaffolding. The revoicing and rephrasing strategies are used to activate strategy D, since they turn one student (the author of the intervention) into a resource for the class. Moreover, we observed that often revoicing and rephrasing (and rephrasing with scaffolding) are efficient in promoting the evolution of the kind of feedback, for instance (as in the reported example) from a feedback on the task to a feedback on the processing of the task. Relaunching occurs when the teacher reacts to a student's intervention, which (s)he considers interesting for the class, not giving a direct feedback, but posing a connected question. In this way, by relaunching the teacher provides an implicit feedback (strategy C) on the student's intervention, suggesting that the issue is interesting and worth to be deepened or, conversely, has some problematic points and should be reworked on. Contrasting takes place when the teacher draws the attention on two or more interventions, representing two different positions, so as to promote a comparison. By contrasting, FA strategy D and E are activated (the authors of the two positions may be resource for the class as well as responsible of their own learning).

The aforementioned strategies, besides being efficient ways to boost the discussion, are powerful formative assessment tools, since they foster the activation of formative assessment strategies. When addressing one student's statement, the teacher gives an implicit feedback on it (strategy C), suggesting the intervention deserves further attention. Moreover, in this way strategies D and E are activated and the feedback may evolve from feedback on the task to feedback on the processing of the task. We deem that this kind of classification may shed light into the crucial role of the teacher in enhancing FA within class discussions. All the documented strategies seem to be intentionally applied by the teacher. Anyway, the given feedback is implicit, since the teacher does not address directly the correctness of the student's intervention. As a consequence, the feedback is not always sought by the students. We are aware of the fact that we were able to single out and discuss only some effects of a given feedback, namely when a student explicitly refers to a previous intervention or changes his mind immediately after an intervention by a peer or by the teacher. Other effects of a given feedback are less visible during a class discussion: in order to study them, it will be necessary to analyse further activities of the students or collect a-posteriori interviews.

For the moment we focused on class discussions around the analysis and comparison of students' written productions. In the future we plan to go on with our analysis, focusing on other crucial moments of the teaching experiments, such as the discussion after a poll, or the discussion on specific helping worksheets. As a further development, we plan to compare the strategies we outlined with Bartolini Bussi (1998)'s classification of teacher's interventions during a mathematical discussion. Moreover, we aim at complementing the present study, concerning the way feedback is given (feedback strategies), with a study on the content of feedback. To this aim, we plan to deepen the categorization of levels of feedback provided by Hattie & Temperley (2007), so as to take into account the specific features of the proposed mathematical tasks.

Proceedings of CERME10

Acknowledgment Research funded by the European Community's Seventh Framework Programme fp7/2007-2013 under grant agreement No [612337].