Guilherme Dietrich

Sylvie Recous
email: sylvie.recous@inra.fr

Patrick Leal Pinheiro

Adams Douglas

Adriane Luiza Weiler

Mathias Schu

Leite Roberto

Sandro José Rambo

Giacomini

Patrick Leal Pinheiro

Douglas Adams Weiler

Adriane Luiza Schu

Mathias Roberto

Leite Rambo

Sandro José Giacomini

Gradient of decomposition in sugarcane mulches of various thicknesses

Keywords: Straw litter layer, carbon decomposition, nitrogen, crop residue water properties, soilresidue contact

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

No-till in cropping systems are increasingly used worldwide for protecting soil from physical degradation and enhancing soil fertility [START_REF] Tiecher | Soil fertility and nutrient budget after 23-years of different soil tillage systems and winter cover crops in a subtropical Oxisol[END_REF], and the development of perennial or semi-perennial crops (miscanthus, sugarcane, etc.) dedicated to bioenergy [START_REF] Peyrard | Management practices of Miscanthus × giganteus strongly influence soil properties and N2O emissions over the long term[END_REF][START_REF] Bordonal | Sustainability of sugarcane production in Brazil. A review[END_REF] leads to the generalization of agricultural situations with crop residue layers, i.e., mulch, decomposing on the soil surface. Predicting the decomposition rates of crop residue mulches according to system characteristics (crop and rotation types, climatic conditions) is important because of the crucial services these mulches provide, such as soil cover and protection against erosion and weed growth, water regulation, nutrient recycling, and habitats and resources for soil communities. Moreover, a better understanding of the decomposition of straw at the soil surface is important because this process determines the fluxes of carbon (C) and nitrogen (N) in the soil and to the atmosphere via the emission of greenhouse gases [START_REF] Chen | Soil nitrous oxide emissions following crop residue addition: A meta-analysis[END_REF]Pinheiro et al., accepted;[START_REF] Vasconcelos | Greenhouse gas emission responses to sugarcane straw removal[END_REF].

Many studies have shown that the decomposition rate of straw at the soil surface is lower than that of residues incorporated into the soil [START_REF] Allmaras | Wheat straw composition and placement effects on decomposition in dryland agriculture of the Pacific Northwest[END_REF][START_REF] Schomberg | Decomposition and nitrogen dynamics of crop residues: residue quality and water effects[END_REF][START_REF] Coppens | Soil moisture, carbon and nitrogen dynamics following incorporation and surface application of labeled crop residues in soil columns[END_REF][START_REF] Curtin | Decomposition rate of cereal straw as affected by soil placement[END_REF]. This difference was attributed to the straw undergoing greater and more frequent fluctuations in moisture at the soil surface than within the soil, particularly frequent drying between rains, combined with a reduced diffusion of nutrients and water from the soil to decomposition sites in the mulch [START_REF] Christensen | Barley straw decomposition under field conditions -effect of placement and initial nitrogen-content on weight-loss and nitrogen dynamics[END_REF]. The soil-residue contact determined by the characteristics of crop residues (morphology and particle size, mass) and the characteristics of the underlying soil layer (texture, water content, mineral N content) are considered important in driving decomposition (e.g., [START_REF] Angers | Decomposition of wheat straw and rye residues as affected by particle size[END_REF]Henriksen and Bleland, 2002;[START_REF] Iqbal | Tissue density determines the water storage characteristics of crop residues[END_REF]. The straw mass, together with the residue morphology, is the main factor that determines the proportion of the mulch that is in contact with the soil. For a given crop residue type, as the residue amount on the soil surface increases, the thickness of the produced mulch increases [START_REF] Putuhena | Estimation of interception capacity of the forest floor[END_REF], and the proportion of the residues that are in contact with the soil decreases. This was reflected in the fact that under a number of experimental conditions the rate of decomposition decreased with an increase in the mass of crop residues at the soil surface (e. g. [START_REF] Steiner | Crop residue decomposition in notillage small-grain fields[END_REF]. These results had led models that simulate the decomposition of straw at the soil surface to consider the degradation of the mulch to occur from a decomposing layer in contact with the soil fed by an unaltered layer of overlying residues (APSIM [START_REF] Thorburn | Modelling decomposition of sugar cane surface residues with APSIM-Residue[END_REF]CANTIS Findeling et al., 2007). However, a number of recent studies have changed this perception by showing that mulch decomposition can be proportional to the initial straw mass, within a range of 3 to 12 Mg ha -1 for cover crop mulches [START_REF] Acosta | Fitomass decomposition and nitrogen release of cover crops in function of the level of residue input to soil under no-tillage system[END_REF]Halde and Entz, 2015), sugarcane [START_REF] Dietrich | Sugarcane mulch C and N dynamics during decomposition under different rates of trash removal[END_REF], or 5 to 15 Mg ha -1 for rye [START_REF] Williams | Establishing the relationship of soil nitrogen immobilization to cereal rye residues in a mulched system[END_REF]. These results are questioning the hypothesis that the undecomposed proportion of the top layer increases with increasing residue mass. Therefore, it is necessary to improve our understanding of the decomposition processes of residue mulch, particularly how these decomposition processes are affected by the mass and thickness of mulch.

To answer these questions, the objective of this work was to investigate the vertical heterogeneity of the decomposition process within sugarcane straw mulches with different initial masses, thicknesses, and N availabilities. We investigated the following aspects in field conditions over one year after sugarcane harvest: i) how decomposition develops at different locations within mulches of varied initial masses, ii) the consequences of mulch features on mulch C and N dynamics and on the conditions in the soil layer below the mulch, and iii) the factors driving mulch decomposition and the role of the contact between soil and mulch in this decomposition.

Materials and methods

Study site

The study was located at the Federal University of Santa Maria (29°42'54" S, 53°42'23" W, approximately 90 m elevation) in the state of Rio Grande do Sul, Brazil. The climate, according to the Köppen classification system, is a type Cfa subtropical humid climate. The mean annual temperature is 19.3°C; June is the coldest month, with a mean minimum temperature of 9.3°C, and January is the warmest month, with a mean maximum temperature of 30.4°C. The mean annual precipitation is 1660 mm, without a dry season. The soil is classified as Typic Paleudalf according to USDA taxonomy (Soil Survey Staff 2010). Selected soil characteristics (0-20 cm layer) at the beginning of the experiment were 110 g kg -1 clay, 260 g kg -1 silt, 630 g kg -1 sand, soil bulk density = 1.63 g dm -3 , pH (water) = 5.4; Mehlich-I P = 33.4 mg dm -3 , total C = 4.1 g kg -1 and total N = 0.32 g kg -1 .

Main experimental design

The experiment was carried out at the first sugarcane ratoon from November 2015 to November 2016 and began after the harvest of cane plants on November 20, 2015. The sugarcane variety was RB 956911, and the harvest was manual, removing all aboveground biomass (stalk + leaves + tops) and leaving the soil uncovered. An area was demarcated with 32 plots in 4 lines with 1.4 m of spacing between rows and length of 4 m. Each plot had an area of 22.4 m 2 . We used a randomized complete block design in a factorial scheme of 4x2 with four replicates. The first treatment was the straw quantity treatment, with four amounts of straw on the soil surface: 0, 4, 8 and 12 Mg ha -1 of dry matter (DM). The second treatment was the fertilizer N treatment, with 0 or 100 kg N ha -1 added as urea. The straw was homogeneously distributed by hand on the soil surface, resulting in 8 treatments (0S, 0S+N, 4S, 4S+N, 8S, 8S+N, 12S, 12S+N). The urea N was applied as granules broadcasted in a single dose on the straw on day 52 after the plant cane harvest.

Straw decomposition microplots

added into 0.16-m 2 wooden frames (40-cm length x 40-cm width x 8-cm height) between rows of sugarcane, as described by [START_REF] Dietrich | Sugarcane mulch C and N dynamics during decomposition under different rates of trash removal[END_REF]. To evaluate the gradient of decomposition within the mulch, the mulch was formed by straw layers of 4 Mg ha -1 , and we stacked a single layer, 2 layers or 3 layers inside the litter box to reconstitute the 3 mulch quantities. The layers were separated by a malleable 10-mm synthetic mesh with a wide aperture so each layer could subsequently be sampled separately for analysis. The 4 Mg ha -1 treatment (4S) had a single layer, which was, in essence, simultaneously in contact with the soil (low position: L) and in contact with the atmosphere (top position: T), the layer in this treatment is referred to as 4S L/T. The 8 Mg ha -1 treatment was composed of two layers, a top layer (8S T) and a low layer (8S L); the 12 Mg ha -1 treatment was composed of three layers, a top layer (12S T), a middle layer (12S M) and a low layer (12S L).

Destructive sampling of litter boxes was performed on days 0, 15,36,51,72,103,168,238,296 and 360 after straw application from November 2015 to November 2016. At each sampling date, the straw from each layer in each destroyed litter box was collected separately, weighed wet, stored in paper bags and dried in an oven at 40°C. The straw was then separated from adhering soil by dry cleaning, and a subsample was dried in an oven at 65°C to determine the remaining straw DM. Straw ashes were measured by incinerating 0.4 g in a muffle furnace at 550°C for 3 h. The total C and N concentrations of the straw were determined using an elemental analyser (FlashEA 1112 model, ThermoFinnigan, Milan, Italy). Straw soluble (SOL), cellulose (CEL), hemicellulose (HEM) and lignin (LIG) fractions were determined in residues pre-dried at 40°C using the method described by [START_REF] Van Soest | Use of detergents in the analysis of fibrous feeds I: preparation of fiber residues of low nitrogen content[END_REF]. The water-soluble organic C (Csw) and water-soluble total N (Nsw) were extracted using 2 g of chopped plant residue (1 cm) pre-dried at 40°C in 80 mL of water following the methodology described by [START_REF] Schmatz | Crop residue quality and soil type influence the priming effect but not the fate of crop residue C[END_REF]. The Csw and Nsw contents were analysed using a total organic carbon analyser equipped with a kit for N analysis (Shimadzu TOC-LCPH, Shimadzu, Kyoto, Japan). The chemical characteristics of the straw at the beginning of the experiment were as follows: 228 ±7.5 g kg -1 SOL, 372 ±2.7 g kg -1 CEL, 335 ±5.2 g kg -1 HEM, 65 ±1.6 g kg -1 LIG, 425 Mulch thickness was evaluated in four litter boxes randomly selected by treatment (4S, 8S, 12S). In each box, nine non-destructive thickness measurements were performed using a graduate ruler. The measurements were performed at 0, 29,54,72,103,168,238,296 and 360 days after straw addition.

Soil water content and soil temperature

Sensors with two 30-cm-long stainless steel rods (FDR CS 616-L, Campbell Scientific, Logan, Utah, US) inserted at 2.5 and 7.5 cm soil depth were used to measure soil moisture. The type T copper-constantan thermocouple was installed at the same depths to measure soil temperature. Both sensors were installed between sugarcane rows and coupled to a datalogger (CR1000, Campbell Scientific, Logan, Utah, USA), and measurements were performed at a 10 min sampling interval. The air temperature at 30-cm depth inside of the sugarcane plantation was also evaluated using the same sensor model. The sensor was covered by a polyvinyl chloride pipe wrapped by asphalt adhesive tape with holes on the sides. Daily air temperature and rainfall data were obtained from an automatic weather station located 1.7 km away from the experiment site.

Calculations and statistical analysis

The remaining C and N of the straw at each sampling date were calculated as the remaining mass of each straw layer multiplied by its C and N concentrations (deduced from the ash content). In treatments 8S and 12S (with and without N), the amounts of C and N remaining were obtained by summing the amounts of C and N in each straw layer. The C remaining in each layer and in the total mulch over time was described with a simple one-compartment equation according to Plante and Parton (2007):

C remaining = C0e -kt
where C remaining is the mass of the remaining C (Mg C ha -1 ), C0 is the straw C pool at time 0 (Mg C ha -1 ), k is the degradation rate constant (day -1 ), and t is the time (days).

The mean residence time (t1/2) of the straw, which corresponds to the time necessary to degrade half of the initial quantity of mulch, was calculated from k according to Plante and Parton (2007). The LCI (lignocellulose index) was calculated from the CEL, HEM and LIG fractions of sugarcane straw and corresponds to the relative enrichment of LIG in the plant cell walls [START_REF] Whittinghill | Anthropogenic N Deposition Increases Soil C Storage by Decreasing the Extent of Litter Decay: Analysis of Field Observations with an Ecosystem Model[END_REF]. This index was taken as an indicator of the degradation process of the lignocellulose substrates.

All data were subjected to analysis of variance (ANOVA). The effect of straw quantity or straw layer, N fertilization and time, and their interaction on mulch decomposition (C and N remaining, SOL, CEL, HEM and LIG remaining and LCI index) and physical (thickness and water content) variables, were tested. Similarly, we tested the effect of straw quantity or straw layer and N fertilization, and their interaction on decomposition model parameters (C0, k and t1/2). In the straw layer data analysis, data from the 4S treatment were included in the statistical analysis. This treatment was considered to have a straw layer representing both L and T straw because the layer was in direct contact with the soil and the atmosphere simultaneously. The effect of straw quantity, soil depth and time, and their interaction on soil moisture and soil temperature also were tested. When ANOVA showed significant (P<0.05), the differences between means of treatments were compared by the Tukey test (P<0.05). Relationships between remaining C or N measured at the end of experiment and the average water content of straw layers in the mulch (T/L, L, M and T) for the three treatments (4S, 8S and 12S) over the 360 days of the experiment was fitted.

Results

Climatic and soil environmental conditions

The total annual rainfall was 1519 mm, distributed over 107 events, and the mean air temperature was 18.4 °C, similar to the historical average (Fig. 2a). The relative air humidity ranged from 39 to 99%, with most values in the 70-90% range (Fig. 2b). As the temperature and moisture at the 2.5 and 7.5 cm soil depths varied in the same way over time and between treatments, only the 2.5 cm soil depth data are shown here (Fig. 2c and2d). The effect of straw amount on soil temperature was observed only in the first three months, when the sugarcane canopy had still not completely covered the soil surface (Fig. 2c). The mean soil temperature during this period at 2.5 cm depth decreased with increasing amounts of straw, as 0S (28.2°C)>4S (27.1°C)>8S (26.4°C) = 12S (26.5°C), indicating an approximately 2°C difference in mean temperature between the highest (12S) and the lowest (0S) amount of straw. The soil water content responded to daily precipitation and was affected by straw quantity (Fig. 2d). Averaged over the experimental year, the soil volumetric water content was 0.17 for 0S, 0.19 for 4S, 0.21 for 8S and 0.23 g cm -3 for 12S. The soil water content was markedly lower in the treatments with lower straw quantities in the soil surface during periods with a small amount of rainfall or a short dry period, e.g., from day 46 to 67, day 190 to 217, and day 297 to 320.

Dynamic of thickness and water retention of mulch

The initial mulch thickness varied significantly with straw quantity, and, as expected, had a ranking of 12S (4.7 cm)>8S (3.6 cm)>4S (1.9 cm) at day 0 (Fig. 3a). The thickness was not strictly proportional to the mass, suggesting a layer settling with the larger mass and therefore a higher mulch density for 12S treatment. Then the evolution of mulch thickness was proportional to the initial thickness in the three straw treatments, with a 32% decrease in thickness at day 29, 50% decrease in thickness at day 103 and 68% decrease in thickness after 360 days. At that time, the mulch was 0.6, 1.2 and 1.5 cm thick for 4S, 8S and 12S, respectively.

The water content of the sampled mulches varied greatly over the course of the year, ranging between 10 and 80 g H2O 100 g -1 DM. The water content varied significantly as a function of straw quantity at some sampling dates (Fig. 3b) and differed according to straw layer position (Fig. 3c and Fig. S1). Comparing the layer positions for a given treatment, we observed that the water contents in the L layers of 8S and 12S were higher than those in the T layers of 8S and 12S, except for the samplings performed at days 51, 103 and 238. Comparing different mulch treatments for a given layer position, we observed that generally speaking, the thicker the mulch, the higher the moisture content of a layer in a given position was. On average over the year, the moisture content of the whole mulch was finally law or negligible for the three quantities of mulch (Fig. 3b), suggesting compensation between the different layers according to thickness.

Dynamic of mulch decomposition

The straw C and N remaining as mulch, expressed as the percentage of the initial straw C or N added, decreased quickly over time but were not influenced by straw quantity, N fertilization or their interaction (Table S1); therefore, the mean data of the 0N and 100N treatments for each straw treatment are presented here (Fig. 4a and4b). Notably, approximately half of the straw C and twothirds of straw N lost during the 360-day period were lost in the first 36 days, regardless of the initial straw quantity. The straw C and N lost by day 360, averaged for the 12S, 8S and 4S treatments, amounted to 75 and 46% of the initial C and N, respectively. The C:N ratio of the remaining straw decreased from an initial C:N of 53 to a final C:N of 29, with no significant differences between straw treatments or between N treatments (Fig. S2).

Straw Csw decreased very quickly, and by day 36, only 14, 12 and 19% of the initial Csw remained in the 4S, 8S and 12S treatments, respectively, with significant transient differences between treatments (Fig. 4c). Straw Nsw decreased more slowly until day 72 and then remained stable until the end of the experiment (Fig. 4d). The chemical composition of the straw particles changed over time, with, at day 360, a relative increase in HEM and LIG and a strong decrease in CEL, while the SOL fraction remained stable (Fig. S2). This final composition was not influenced by straw quantity, N application or their interaction (P >0.05) (Table S1). The C0 straw pools calculated with the onecompartment equation was, as expected, close to the initial straw quantities, representing 1.47 Mg ha -1 (4S), 2.99 Mg ha -1 (8S) and 4.45 Mg ha -1 (12S) (Table 1). The straw quantity and N addition did not change the k decomposition rates, with on average k = 0.0044 and t1/2 = 158 days.

Effect of straw layer position on decomposition

C and N evolution in the mulch layers

The dynamics of straw C and N in the various mulch layers were significantly influenced by the position of the straw layers within the mulch (P<0.001) but were not significantly influenced by N fertilization or the interaction between straw layer position and N fertilization (P>0.05) (Table S2); therefore, the means of the two N treatments are presented here (Fig. 5). In treatment 12S (Fig. 5a), the dynamics were characterized by a rapid decrease in straw C that was similar among all layer positions (0-36 days), followed by a slower decrease, during which the decomposition of 12 L>12M>12T. The same pattern of decomposition was also observed in the 8S treatment, with layer 8 L>layer 8T (Fig. 5c). Comparing the C loss of the top layers across the three straw treatments (4S L/T vs. 8S T vs. 12S T), we observed that the dynamics were very similar, except for at two dates, which demonstrated a transient difference that subsequently disappeared (Fig. 5e). The same comparison among low layers (4S L/T vs. 8S L vs. 12S L) showed that the 4S L/T layer decomposed less rapidly and had significantly less decomposition over the final two sampling dates than the 8S L and 12S L layers (Fig. 5g). As expected, the decomposition parameters of the adjusted model were significantly different only for mulch layer position (Table 1): the highest and lowest decomposition k rates were observed in the L and T layers, respectively. The straw N kinetics followed the same pattern for each layer position (Fig. 5b,d,f,h). The dynamics were characterized by a rapid decrease in straw N (0-36 days) that was similar among all layer positions, followed by a slower decrease up to day 103, after which the remaining N values were relatively stable until day 360. Differences in the remaining N among the straw layers were observed mainly from day 238 in the 12S (T>M>L) and 8S (T>L) mulch. The N remaining in the mulch at the end of the experiment also differed between the low layers, with 4S L/T>12S L.

Changes in chemical composition

The chemical composition of the straw layers, i.e., the straw C:N ratio and chemical composition expressed by the LCI, taken as an indicator of the decomposition process, varied over time for all layer positions and depended on the position of layers in the mulch (Fig. 6 and Fig. S3).

Considering the 12S treatment as an example, we observed that the pattern of evolution of the C:N ratio was similar among the three layers, but in general, the C:N decrease was more pronounced for low layers, with 12S L<12S M<12ST (Fig. 6a). The LCI increased from 0.08 to 0.17 over time for all layers and was significantly different between layers at different positions until day 168 (e.g., 12L = 12 M>12T), but no difference persisted at day 360 (Fig. 6b).

Relationships with straw water content

The C and N remaining at day 360 in the L, M, T or L/T straw layers were significantly and linearly related to the average moisture content of these layers sampled over the year, regardless of the position of the layer within the mulch and the mulch quantity (Fig. 7). This relationship was also found when using the degradation constant k, which reflects all decomposition kinetics, with k= 0.0001x -0.0005 (R 2 =0.882, P<0.05).

Discussion

Effect of mulch mass and thickness on mulch decomposition

A main hypothesis, based on recent published work, was that increasing the mass of mulch residues at the soil surface does not affect the decomposition rate of these mulches [START_REF] Acosta | Fitomass decomposition and nitrogen release of cover crops in function of the level of residue input to soil under no-tillage system[END_REF][START_REF] Halde | Plant species and mulch application rate affected decomposition of cover crop mulches used in organic rotational no-till systems[END_REF][START_REF] Dietrich | Sugarcane mulch C and N dynamics during decomposition under different rates of trash removal[END_REF]. This situation was reflected in the present study with 3 amounts of sugarcane straw initially placed on the soil surface, which increased the initial mulch thickness (12S>8S>4S). We confirmed that the straw C and -N dynamics and calculated decomposition rates of the mulches were similar, regardless of the initial quantity of straw and whether they received 100 kg of fertilizer N 52 days after straw application. This result signifies that, although the proportion of the total mass in contact with the soil decreased as the mass increased, the decomposition was not modified. In fact, the initial thickness of the mulch varied from 1.9 cm to 4.7 cm, which is not a great difference. This may be due to our experimental system, which, although using hand-cut straw particles of a realistic decimetre length, minimized the thickness by "stacking" the straw in layers. In the field, mulches can be thicker but less dense. [START_REF] Meier | The availability of nitrogen from sugarcane trash on contrasting soils in the wet tropics of North Queensland[END_REF] observed an initial sugarcane mulch thickness of about 11 cm for a 10 Mg ha -1 mulch mass, decreasing rapidly in one year. [START_REF] Stott | Wheat residue loss from fields under no-till management[END_REF] measured that the initial wheat straw layer thickness ranged from 12 and 20 cm. Although mulch thickness is a parameter important for both the interpretation and the modelling of mulch decomposition, data are still very rare. These results suggest that neither microbial colonization nor N availability were limiting factors of decomposition for the thickest mulches, which invalidates the hypothesis that soil-straw contact plays a major role as a key driver of mulch decomposition [START_REF] Coppens | Decomposition of mulched versus incorporated crop residues: Modelling with PASTIS clarifies interactions between residue quality and location[END_REF][START_REF] Thorburn | Modelling decomposition of sugar cane surface residues with APSIM-Residue[END_REF]. The initial thickness of the mulch was not proportional to the mass, due to a settling of the mulch with increasing weight of the residues. It should also be noted that the decrease was not linear over time, probably due to the effect of two combined factors, an initial possible and rather rapid settling under the influence of rainfall, combined with a more regular decrease under the effect of decomposition. It is noteworthy that the decrease in thickness was proportional to the initial thickness, reflecting the observed proportionality of the decomposition rate with the initial mass.

Strikingly, the N dynamics in mulch differed substantially from the C dynamics in mulch.

After a synchronized, rapid C and N net loss at the beginning of decomposition, the N in the mulch remained stable while decomposition continued, and adding N onto the mulch in fertilized plots did not modify this dynamic. Sousa Jr et al. ( 2017) also observed little and transient effect of adding vinasse onto sugarcane mulch on decomposition rate. These results suggest the efficient use of straw N and recycling of microbial N by decomposers growing on the straw and that the availability of N to decomposers was not a limiting factor of decomposition, even with increased mulch masses and thicknesses and a lack of mineral N in the soil under the mulches in non-fertilized plot (Pinheiro et al., accepted). Some work has also highlighted the possibility of a significant contribution of asymbiotic N-fixation during decomposition of sugarcane litter. [START_REF] Lincoln | Nitrogen fixataion during decomposition of sugarcane (Saccharum officinarum) is an important contribution to nutrient supply in traditional agricultural systems of Hawai'i[END_REF] calculated that the asymbiotic N fixed represented 17 to 40% of the N present in the litter. They pointed out that the potential N-fixation per unit litter depended on the thickness of the litter with an optimum for a 5cmthick litter. This amount of fixed N could represent from 5 to 39 kg N ha -1 year -1 depending on the conditions tested. [START_REF] Patriquin | Nitrogen fixation in sugar cane litter[END_REF] earlier found that 10 to 30 kg N ha -1 were potentially fixed during sugarcane litter decomposition. The possible contribution of non-symbiotic fixation to mulch N dynamics would merit further investigation, as it appears to be quantitatively important. The relatively high C:N ratio of sugarcane straw suggests also that fungi may be the main group of microorganisms involved in decomposition, particularly for the upper layers of the mulch, due to their ability to translocate N from the soil through the hypha network to substrates that are recalcitrant or poor in N [START_REF] Frey | Fungal translocation as a mechanism for soil nitrogen inputs to surface residue decomposition in a no-tillage agroecosystem[END_REF].

The amount of straw on the soil surface also affected soil temperature and moisture and had a substantial effect on the straw water content. Previous studies demonstrated that mulch moisture is driven by the water retention characteristics of residue particles and by rain and evaporation regimes [START_REF] Lee | Soil moisture and soil-litter mixing effects on surface litter decomposition: a controlled environment assessment[END_REF][START_REF] Iqbal | Rain regime and soil type affect the C and N dynamics in soil columns that are covered with mixed-species mulches[END_REF]. Few studies have evaluated the interaction between different amounts of straw and soil moisture [START_REF] Cook | Mulch effects on rainfall interception, soil physical characteristics and temperature under Zea mays L[END_REF]. Our results clearly showed an increase in soil moisture with increasing straw masses on the soil surface, mainly in periods with low rainfall, when thicker mulch prevented soil water evaporation more effectively, confirming other studies [START_REF] Sousa | Sugar cane straw left in the field during harvest: decomposition dynamics and composition changes[END_REF]. This is because the straw on the surface acts as a physical barrier to the transfer of heat and water to the atmosphere [START_REF] Fuchs | Mulch resistance to water vapor transport[END_REF]. Despite rain water being retained by mulches, soil without mulch remained drier than soils covered by mulches. This is due to the fact that the increased evaporation of soil not covered by mulch (0S) was not offset by the higher amount of water reaching soil after the rains in such situation. It is understandable that this effect depends on several factors, not only on the intensity and frequency of rainfall, but also on the proportion of mulch-covered area and the characteristics of the mulch, and may vary according to agricultural and climatic conditions.

Water and decomposition gradients within the mulch

The amount of straw on the soil surface not only affected the overall soil and straw moisture but also created a moisture gradient within the mulch. Sampling straw layers in different positions in the mulch showed that the layers in contact with the soil in the thicker mulch (8S L and 12S L) were almost always wetter during the year than the more distant layers (8S T, 12S M and 12S T), which were exposed to solar radiation and wind. The higher moisture of the layers in contact with the soil was mainly due to the lower water loss by evaporation but may also be due to the absorption of water from soil by the straw in a phenomenon defined as the "sponge effect" [START_REF] Kravchenko | Hotspots of soil N2O emission enhanced through water absorption by plant residue[END_REF]).

These moisture conditions favour high microbial activities and faster decomposition at the soil-mulch interface.

The kinetics of straw C and N for each layer position and each mulch mass treatment indicated that all layers decomposed, even at the top of the thicker mulch (12S T), which was the most distant layer from the soil and that the decomposition of the top layers started as quickly as the decomposition of the other layers. This suggests that part of the initial fast C and N losses could be due to the leaching of soluble compounds under rain. However, the change in the chemical composition of mulch over time, expressed here by the LCI index, occurred in all straw layers, regardless of position. The LCI corresponds to the relative enrichment of LIG in the plant cell walls of the residues [START_REF] Whittinghill | Anthropogenic N Deposition Increases Soil C Storage by Decreasing the Extent of Litter Decay: Analysis of Field Observations with an Ecosystem Model[END_REF] because structural components such as CEL and HEM are more easily decomposed than LIG [START_REF] Berg | Plant Litter: Decomposition, Humus Formation, Carbon Sequestration[END_REF]. This confirms that the C or N loss of the top layers was not simply due to the possible physical transport of straw particles from an upper layer to a lower layer during experimental handling or by rain leaching; rather, remaining particles of each layer underwent decomposition. The high LCI value of the low 12S layer (12S L) also confirms that decomposition advanced more in this layer than in the other straw layers.

We observed that the top layers behaved similarly (12S T= 8S T= 4S L/T), while the decomposition speed of the low layers increased as the overlying mulch thickness increased (12S L>8S L>4S L/T). This pattern indicates that the top layers were under similar control of humectationdesiccation cycles, driven by climate, while the low layers differentiated from each other by the thickness of the overlying mulch, which in turn influenced the low-layer moisture. These findings are very well supported by the strong relationships between the amount of straw C or N remaining after 360 days (or the k degradation constant) and the mean straw moisture in each layer, irrespective of layer position and the mulch thickness. We conclude that a mulch system of various thicknesses is driven primarily by water dynamics, which is under the control of water retention-evaporation by the straw particles. The moisture gradient in the mulch depended on mulch thickness and determined decomposition gradient, which finally defined the decomposition rate of the whole mulch. This raises the question of understanding how the observed differences between layers perfectly balance in order not to be able to find any difference in decomposition rate between mulches of different thicknesses.

Our assumption is that, in our experimental conditions (climate, range of DM), the higher decomposition rates of the low layers in thick mulches compensated for the large mass of these thick mulches not in direct contact with the soil. This "compensation" hypothesis is also supported by our results of water content of whole mulches (measured 9 times during the year), showing that although a gradient was established within the mulches, and transient differences appeared between mulches of increasing thickness, on average there were little or no differences in water content of the mulches regardless of their initial mass. These results confirm the hypothesis of [START_REF] Dietrich | Sugarcane mulch C and N dynamics during decomposition under different rates of trash removal[END_REF] that increasing mulch quantity has an antagonistic effect, increasing the proportion of straw less wet and not in direct contact with the soil but improving the environmental conditions of decomposition of the mulch at the soil-mulch interface. [START_REF] Williams | Establishing the relationship of soil nitrogen immobilization to cereal rye residues in a mulched system[END_REF], measuring the decomposition of varied quantities of surface rye straw, also obtained similar decomposition rates in the range of 5 to 15 Mg ha -1 . Conversely, [START_REF] Vasconcelos | Greenhouse gas emission responses to sugarcane straw removal[END_REF][START_REF] Sousa | Sugar cane straw left in the field during harvest: decomposition dynamics and composition changes[END_REF], investigating sugarcane, showed that the decomposition rate increased with the amount of straw in the range 3-14 Mg DM ha -1 , suggesting that the positive effect of increased moisture prevailed in thicker mulches. However, the thickness and density of the mulches were not available in these studies; therefore, it is not possible to interpret the differences with our results. Thus, further research is needed to establish the mulch thickness range and rain patterns for which these results are valid and, in particular, better understand how the dynamics of mulch layers that are thicker than those explored here and in the literature could modify the above conclusions. It is expected that these results will strongly depend on climatic conditions and will particularly be affected by interactions between mulch properties and thickness and water dynamics in dry periods.

Consequences for much conceptual modelling

Many modules developed in several simulation models (i.e., APSIM, [START_REF] Thorburn | Modelling decomposition of sugar cane surface residues with APSIM-Residue[END_REF]PASTIS, Findeling et al., 2007) represent residue mulch composed of two layers; one of these layers is in contact with the soil, undergoing decomposition with moisture, available N and microbial biomass processes influenced by the underlying soil layer, and the other layer is undecomposed straw, which "feeds" the lower layer as it empties. Moreover, in the model proposed by [START_REF] Findeling | Modelling water, carbon and nitrogen dynamics in soil covered with decomposing mulch[END_REF], the water dynamics of mulch are simulated by considering the whole mass of straw that forms the mulch to present the same moisture content. Our results demonstrated that upper mulch layers, which are not in direct contact with the soil (exposed to air), also decompose and that there is a gradient of moisture and decomposition (and probably temperature, although we did not measure this factor) throughout the mulch volume. These findings suggest the need for more sophisticated models dedicated to the simulation of mulch evolution, by considering the whole straw layer to be decomposing, albeit with a gradient, and with an accurate estimation of the mulch water content over time, including knowledge about water interception, retention and leaching for various crop residues [START_REF] Iqbal | Tissue density determines the water storage characteristics of crop residues[END_REF][START_REF] Sato | Experimental analysis of moisture dynamics of litter layers-the effects of rainfall conditions and leaf shapes[END_REF].

Conclusions

By varying mulch thickness under similar climatic conditions, we demonstrated that not only the mulch in contact with the soil but also the whole mulch decomposed, starting quickly after the residues were deposited on the soil, with a vertical gradient of decomposition closely linked to the moisture gradient. The moisture gradient of the mulch, which is under the control of humectation (rain) and desiccation (evaporation) and depends on thickness, drives decomposition; therefore, it is important to correctly model water properties of crop residues and water transport between soil, mulch and atmosphere. Our results indicated the importance of soil-residue layer contact properties in driving mulch decomposition. We also showed that, despite heterogeneity within the mulch, decomposition was proportional to the initial amount of mulch mass for the studied crop. For the present range of straw masses, as mulch thickness increased, the moisture and decomposition rate of the underlying layers increased compensating for the increase in the fraction of mulch more distant from the soil and drier. Considering the importance of these results for the dynamics of nutrient recycling and release in non-tilled agrosystems, the ability of mulch decomposers to efficiently recycle N and the dynamics of microbial colonization within the mulch should receive more attention. Finally, the accurate prediction of water dynamics in soil-mulch systems is crucial because these systems act as hot spots for many processes, particularly gas emissions (NH3, N2 and N2O), which are under the control of moisture conditions. Values are the mean of 8 replicates per treatment and per date.

Figure Captions

Table 1

Decomposition rate (k) and mean residence time (t½) for the mulch with three straw quantity (4, 8 and 12 Mg ha -1 ) and two N rate (0 and 100 kg ha -1 ) and mulch straw layers (L: low, M: middle, T: top).

Parameters

Initial straw quantity (Mg ha -1 ) N rate (kg ha Values followed by the same letters are not significantly different (Tukey at P<0.05).

a C0 is straw C pool.

b k is the degradation rate constant.

c t1/2 is the half-life of the mulch.

Fig. 1 .

 1 Fig. 1. Schematic layout of the field plots demonstrating the locations of soil water content and soil

Fig. 2 .

 2 Fig. 2. Precipitation and air temperature (a), relative humidity (b), soil temperature (c) and soil

Fig. 3 .

 3 Fig. 3. Evolution of the thickness (a) and water content of the mulch over the 360 days of the

Fig. 4 .

 4 Fig. 4. Dynamics of total C (a), total N (b), water-soluble C (c) and water-soluble N (d) in the remaining mulch particles for three quantities of straw (4S: Mg ha -1 , 8S: Mg ha -1 and 12S: Mg ha -1 ). Data are the average of the treatments with 0 (0N) and 100 (100N) kg N ha -1 . The vertical bars in the

Fig. 5 .

 5 Fig. 5. C and N remaining in the mulch straw layers: (a,b) and (c,d) show different positions in the

  25

Fig. 6 .

 6 Fig. 6. Evolution of the C:N ratio (a) and of the lignocellulose index (LCI) (b) of the straw for the three layer positions in the mulch (L: low, M: middle, T: top) for the 12S treatment (12 Mg ha -1 ). The

Fig. 7 .Fig

 7 Fig. 7. Relationships between straw C or straw N and the average water content of straw layers with different positions in the mulch (L: low, M: middle, T: top) for the three treatments (4S: Mg ha -1 , 8S: Mg ha -1 and 12S: Mg ha -1 ) over the 360 days of the experiment. Values are the mean of 4 replicates

Fig

  Fig. 6.

Acknowledgements

This work was supported by the Brazil government through the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). The bilateral Brazil-France collaboration was funded under Program CNPq-Ciência sem Fronteiras, Process Number 208415/2017-3 and by the Institut National de la Recherche Agronomique (INRA) -Environment & Agronomy Division. Declarations of interest: none


Dr Arnaud Viarouge
email: arnaud.viarouge@parisdescartes.fr

Olivier Houde

Grégoire Borst

Olivier Houdé

The progressive 6-year-old conserver: Numerical saliency and sensitivity as core mechanisms of numerical abstraction in a Piaget-like estimation task

Keywords: Numerical Estimation, Spontaneous Orientation to Magnitudes, Individual differences, Heuristics

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In Piaget's theory of numerical development, children do not reach a proper understanding of the concept of number until they are able to perceive the conservation of the numerical properties of an array of objects despite visual transformations [START_REF] Piaget | The Child's Conception of Number[END_REF]. Indeed, in the seminal number-conservation task, until approximately 6 or 7 years of age, children fail to perceive the numerical equivalence between two rows of tokens after the length of one row is transformed (i.e., by spreading the tokens apart), although they are able to do so when the two rows of tokens have the same length (i.e., before the transformation). Several studies provided converging evidence that success in the number-conservation task in children and adults relies in part on the executive ability to solve the interference between irrelevant nonnumerical dimensions of magnitude (the length or density of the rows of tokens) and the pertinent numerical dimension [START_REF] Houdé | Negative priming effect after inhibition of number/length interference in a Piaget-like task[END_REF][START_REF] Houdé | Functional magnetic resonance imaging study of Piaget's conservation-of-number task in preschool and school-age children: a neo-Piagetian approach[END_REF]. The authors suggest that the participants do so by inhibiting an automatic but misleading 'length equals number' strategy. Since Piaget's work, research in the field of numerical cognition has revealed the existence of numerical abilities long before children's success in Piaget's number-conservation task.

Children's development of the concept of number is thought to build, at least partly, upon preverbal representations of quantities that are present very early in infancy (Dehaene, 2001;[START_REF] Izard | Newborn infants perceive abstract numbers[END_REF]). One of these numerical intuitions consists in the ability to perceive the approximate quantity represented by a set of objects (e.g., an array of dots). Echoing Piaget's work, many studies investigating these nonsymbolic numerical representations have reported the impact of visuospatial cues on numerical estimation [START_REF] Gebuis | The interplay between nonsymbolic number and its continuous visual properties[END_REF]. For instance, congruency effects have been observed throughout development in the context of numerical estimation tasks, showing that the participants' performances are influenced by irrelevant spatial dimensions of the stimuli, such as the size of the dots, or the extent of the space occupied by the array (i.e., field area, often measured by the convex hull surrounding the dot array). In particular, field area has been shown to influence numerosity judgments in preschool and school-aged children (e.g., [START_REF] Gilmore | Congruency effects in dot comparison tasks: convex hull is more important than dot area[END_REF]. In numerical comparison tasks, number/field area incongruent trials (i.e., where the most numerous array of dots is the one occupying less field area) are performed less accurately and more slowly than congruent trials (i.e., where the most numerous array is also the one occupying more field area). Recent studies have observed that the developmental trajectories of these congruency effects can differ depending on the interfering spatial dimension [START_REF] Gilmore | Congruency effects in dot comparison tasks: convex hull is more important than dot area[END_REF]. In the case of field area, its impact on numerical representation can be observed all the way into adulthood [START_REF] Clayton | Dot comparison stimuli are not all alike: the effect of different visual controls on ANS measurement[END_REF][START_REF] Gilmore | Congruency effects in dot comparison tasks: convex hull is more important than dot area[END_REF]. As in the number-conservation task (whereby field area is equivalent to the length of the rows of tokens), inhibition seems to play a critical role in blocking the influence of irrelevant spatial information to make judgments based on the numerical aspects of the stimuli in numerical estimation tasks [START_REF] Leibovich | From 'sense of number' to 'sense of magnitude' -The role of continuous magnitudes in numerical cognition[END_REF]. These observations support the idea that, beyond children's precision of their numerical representations, their ability to abstract number from other nonnumerical dimensions of magnitude is a strong predictor of math achievement [START_REF] Gilmore | Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement[END_REF]but see [START_REF] Keller | Inhibitory control may not explain the link between approximation and math abilities in kindergarteners from middle class families[END_REF]. This is further supported by studies demonstrating that conservation abilities in pre-and early-school years are related to children's mathematical abilities such as math fluency [START_REF] Cooper | Closing the achievement gap: Examining the role of cognitive developmental level in academic achievement[END_REF][START_REF] Ramos-Christian | Math fluency: Accuracy versus speed in preocupational and concrete operational first and second grade children[END_REF]).

Children's numerical estimation is not only related to their ability to inhibit irrelevant spatial information. For instance, individual differences in children's spontaneous orientation to number are predictive of math achievement (e.g., [START_REF] Hannula-Sormunen | Preschool Children's Spontaneous Focusing on Numerosity, Subitizing, and Counting Skills as Predictors of Their Mathematical Performance Seven Years Later at School[END_REF]. Moreover, children's spontaneous orientation towards numerosity is related to their MECHANISMS OF NUMERICAL ABSTRACTION ability to perform number/size incongruent trials in a nonsymbolic numerical comparison task [START_REF] Viarouge | Spontaneous orientation towards irrelevant dimensions of magnitude and numerical acuity[END_REF]. In the study of [START_REF] Viarouge | Spontaneous orientation towards irrelevant dimensions of magnitude and numerical acuity[END_REF], the authors assessed children's relative spontaneous orientation towards three dimensions of magnitude by asking them to sort cards representing arrays of dots. No further instruction was provided on the sorting criterion the children should use. The cards could be sorted on the basis of three dimensions of magnitude: number, size, or spacing. Children who sorted the cards more frequently by number in the card sorting task (hereafter referred to as the SOMAG task for "Spontaneous Orientation to MAGnitudes") performed better on number/size incongruent trials in a numerical comparison task. This indicated that children's spontaneous orientation towards different dimensions of magnitude is related to their ability to solve interferences due to irrelevant visual cues in numerical estimation. Consistent with these results, a study demonstrated that kindergarteners who succeed in the number-conservation task focus more spontaneously on number than kindergarteners who fail the number-conservation task, and vice versa for length [START_REF] Miller | Facilitation of attention to number and conservation of number[END_REF].

Taken together, these findings suggested that the development of children's numerical representations relies on a variety of mechanisms allowing them to abstract the numerical aspects of their environment from other nonnumerical dimensions of magnitude. Hence, what had been described by Piaget as a shift between two stages of numerical development, from a nonconserving to a conserving response in contexts such as the number conservation task now appears as a progressive ability to focus on the numerical aspects of one's environment.

However, so far, the mechanisms behind this progressive abstraction remain underspecified.

Our study is thus aimed to better characterize the mechanisms involved in numerical abstraction, which is crucial to understand the predictive value of early numerical representations for math achievement.

Both the ability to inhibit irrelevant dimensions and the spontaneous orientation to these dimensions are likely to contribute to this development. From this perspective, children's performance in the seminal number conservation task should depend both on the strength of the interference between the numerical and nonnumerical dimensions and on children's sensitivity to the numerical dimension of the task. In classic nonsymbolic numerical comparison tasks using arrays of dots, isolating nonnumerical dimensions of magnitude to study their impact on performance is particularly tedious. Indeed, all the dimensions of magnitude cannot be controlled for simultaneously. This is why we will use a Piaget-like number conservation task, whereby a same-different numerical judgement is performed on two horizontally arranged rows of tokens. This context allows investigating the impact of length/field area on numerical representation, while keeping other dimensions of magnitude such as size and total surface area constant. To date, no study has manipulated this level of saliency in a systematic manner, while conjointly taking into account individual measures of spontaneous orientation towards magnitudes.

In the current study, we asked 6-year-old children to perform the SOMAG task used in [START_REF] Viarouge | Spontaneous orientation towards irrelevant dimensions of magnitude and numerical acuity[END_REF] to assess their relative spontaneous orientation towards different magnitudes (number, size, spacing). The participants were also asked to perform a computerized "same/different" numerical estimation task with horizontally arranged dots (as in Piaget's number-conservation task after the length of one row is transformed). The spacing of the dots in one row (and thus the length ratio between the two rows) was systematically varied. Using a Piaget-like same-different task will allow isolating for the first time the selective impact of the field area dimension on numerical equivalence perception by manipulating the strength of the interference between field area and number in a systematic manner at a crucial age of numerical development in Piaget's theory. We investigated whether MECHANISMS OF NUMERICAL ABSTRACTION children's ability to perceive the numerical equivalence between the two rows of dots (a) varies systematically with the degree of interference between the nonnumerical and numerical dimensions of magnitude and (b) is related to children's spontaneous orientation towards the various dimensions of magnitude present in the tasks. We reasoned that, if succeeding in the same-different numerical estimation task requires inhibition of irrelevant nonnumerical dimensions, then children should make more errors as the difference in dot spacing increases.

Indeed, as the difference in dot spacing increases, the nonnumerical dimensions of magnitude will interfere to a greater extent with numerical processing. Additionally, children who spontaneously focus more on number over spacing in the SOMAG task should make less errors in the same-different numerical estimation task, because spacing is less salient for these children and thus interferes to a lesser extent with their numerical processing. Note that, since item size is maintained constant across the trials of the same-different task, children's performances should only correlate with their spontaneous orientation towards number over spacing, but not with their orientation towards number over size.

Materials and Methods

Participants

We collected data on the SOMAG task and a numerical same-different task described below in ninety-three typically developing children (47 girls, mean age = 6.1 ± 0.62 years) from kindergarten and first grade classes at three public schools. Sensitivity analyses were performed using G*Power 3.1.9.2 [START_REF] Faul | G*Power 3: a flexible statistical power analysis program for the social, behavioral, and biomedical sciences[END_REF], revealing that, given our sample size, an alpha-level of .05 and a level of power of .80, minimum effect sizes that can be significantly detected were f = .12 for the repeated-measures ANOVA and r = .23 for the correlation analyses described below. MECHANISMS OF NUMERICAL ABSTRACTION

The same group of children also performed a nonsymbolic numerical comparison task, and the results were reported in [START_REF] Viarouge | Spontaneous orientation towards irrelevant dimensions of magnitude and numerical acuity[END_REF]. Participation was voluntary after obtaining signed informed consent from the children's parents. Since the SOMAG task assesses children's spontaneous orientation towards either numerical or nonnumerical dimensions of magnitude, this task was always administered first for all children. The numerical same-different and comparison tasks were counterbalanced across the participants.

The results described below remained identical when controlling for the order of the two tasks. Children were tested individually by groups of three or four in a quiet room. Apart from encouraging comments, no feedback on accuracy in the numerical same-different task or on the chosen sorting criteria in the SOMAG task was given. Each child received a certificate with stickers for their participation in the study. The teachers and parents were instructed not to talk with the children about the specific content and goals of the study. The study was carried out in accordance with national and international norms that govern the use of human research participants.

Material and design

SOMAG task.

In each trial of this computerized task, children were presented with three images introduced to them as cards showing arrays of dots. Two cards (targets) were horizontally aligned at the top of the screen, while the third card (reference) was centered at the bottom of the screen. Children were asked to spontaneously pick one target card to go with the reference card. The instructions were as follows: "Which card goes best with that one (pointing at the bottom card)? This one (pointing at the card on the left) or that one (pointing at the card on the right)? Give me your answer without thinking too much!" The experimenters were instructed not to mention any of the dimensions of magnitude tested in this task (number, size and spacing). Children responded using the same keys as in the MECHANISMS OF NUMERICAL ABSTRACTION numerical same-different task. They were instructed to press the right key (green sticker) when they wanted to pick the top right card and the left key (red sticker) when they wanted to pick the top left card.

The collections of dots shown on the cards could be sorted according to the number of dots, aspects of the dot size or of the spacing of the dots (the Size and Spacing dimensions were defined following DeWind, Adams, Platt, & Brannon, 2015). The task included three experimental conditions, depending on the two competing dimensions of magnitude matching with the bottom card (see Figure 1): in one condition, the top cards matched with the bottom card in either number or Size, while Spacing was maintained constant across the three sets of dots (later referred to as the "NumSize" condition); in one condition, the top cards matched the bottom one in Number or Spacing, while Size was maintained constant ("NumSpace" condition); and in one condition, the top cards matched the bottom one in either Size or Spacing, while the number of dots was maintained constant ("SizeSpace" condition). MECHANISMS OF NUMERICAL ABSTRACTION For each dimension, two levels of magnitude were used, which were selected so that they would be easily discriminable by children. A ratio of 1:2 was used for variations in Size and Spacing, while a ratio of 2:3 was used for the number of dots (sets of either six or nine dots).

Across all trials, we counterbalanced the location of the higher level of magnitude (the larger Size, Spacing or number could either be on the two top cards or on the bottom card), and the location of the two competing dimensions of magnitude (either on the top left or top right card). Each trial started with the presentation of a centered fixation cross for 1500 ms, followed by the presentation of the three cards for a maximum duration of 4500 ms. The next trial started when the child gave his/her response, either before or after the end of the Numerical same-different task. In each trial of the numerical same-different task, children were presented with two horizontal rows of evenly spaced blue dots on a white background and were asked to perform a "same/different" numerical judgment. The two rows were left aligned with each other and were located respectively on the top and bottom half of the screen (at a 1.7° visual angle from the center). The spacing between the dots of each row was varied so that the length of the two rows differed by one of four possible length ratios (1:2, 2:3, 6:7, and 9:10). The location (top or bottom) of the longer row was counterbalanced across trials. In 24 test trials, both rows had an equal number of dots (6, 7 or 8). The length of the longest row represented either 12.4°, 14.6°, or 16.7° of visual angle, depending on the number of dots presented, and the length of the shortest row varied according to the length ratio tested. To prevent children from learning that the number of dots was the same in both rows while maintaining a short duration of the experiment, 16 additional trials had a different number of dots (6 vs. 7, or 7 vs. 8 dots), leading to a total of 40 trials. These numerical ratios were shown to be perceivable by 6-year-olds with an accuracy level of approximately 75% [START_REF] Halberda | Developmental change in the acuity of the 'Number Sense': The Approximate Number System in 3-, 4-, 5-, and 6-year-olds and adults[END_REF]. The task started with four training trials to make sure that the children understood the instructions correctly. All four training trials showed rows with a length ratio of 1:2, two "equal number" trials (6 tokens in each row), and two "unequal number" trials (6 vs. 7 tokens), with counterbalanced location of the longest row. Each trial started with the presentation of a centered fixation cross for a duration of 1500 ms, followed by the presentation of the two rows of dots for a maximum duration of 1200 ms. This presentation time was chosen to prevent the use of counting during the task. Children were asked the following: "Do the two rows have the same number of dots, or a different number of dots?" MECHANISMS OF NUMERICAL ABSTRACTION Children were instructed to respond using the K and D keys on an AZERTY keyboard. One green and one red sticker were placed on each of the two keys, and children were instructed to press the green key ("K" on the right) when they thought the number of dots was the same in both rows and the red key ("D" on the left) when they thought the number of dots in each row was different. Due to the tendency of some of the children to spontaneously give their responses verbally or by pointing at the screen, the experimenters then pressed the corresponding response key themselves. For this reason, reaction times were not included in the analyses reported below. When children responded that the number of dots in each row was different, the experimenter followed by asking, "Which row had more?" The child gave a verbal response, which was coded by the experimenter by pressing the up or down arrow key, depending on the location of the chosen row on the screen. The next trial started as soon as a "same number" response or the location of the row judged as containing more dots was provided. The 40 trials were presented in a different random order for each child, and the total duration of the task was approximately 5 minutes.

Results

Effect of the strength of the number/length interference

Three children systematically alternated the right and left responses throughout the task without following the instructions. Their data were removed from the analyses described below. One child's performance fell over the average ± 2.5 standard deviations on both accuracy levels for numerically equal and different trials and was not further analyzed, which led to a total of 89 children included in the analyses. All the results of the statistical analyses reported below were similar when adding the data from this child. For each child, we computed the average accuracy score on the 24 test trials (same number of dots in each row) MECHANISMS OF NUMERICAL ABSTRACTION for each length ratio. Children judged the two rows as containing a different number of dots 89% of the time across all trials. However, a repeated-measures ANOVA using length ratio as a within-subject factor showed a significant effect of the ratio on the children's performances, F(3, 264) = 24.36, p < .0001, η = .22, with accuracy scores increasing with the length ratio (1:2 = 4.31%; 2:3 = 4.12%; 6:7 = 12.92%; 9:10 = 22.85%). Post hoc Bonferroni tests confirmed a significant difference in performance between the 2:3 and 6:7 length ratio (p < .01) and between the 6:7 and 9:10 length ratio (p < .001) but not between the first two length ratios (p = 1, Figure 2).

It is worth noting that 86 of the 89 children who judged two numerically equal rows as different subsequently stated that there were more dots in the longer row (significant Binomial tests, p < .012), confirming the use of a misleading "length equals number" rather than a "density equals number" strategy. We computed, for each participant, the slope of the linear regression of the accuracy scores against the level of length ratio. This measure allowed us to evaluate individual differences in the impact of the length ratio in the numerical samedifferent task. We observed a significant correlation between this slope and the slope obtained

Effect of the spontaneous orientation to numerical and nonnumerical dimensions of magnitude

Despite the same instructions being repeated throughout the SOMAG task, some children had a hard time choosing sorting criteria. Of the 89 children included in the analyses of the numerical same-different task, 13 children did not follow the instructions of the SOMAG task and systematically alternated between the left and right response keys. These children might have been confused by the absence of a clear rule to follow and decided to follow a reassuring strategy. Their data were removed from the following analyses, which were run on a total of 76 children. We verified that the effect of length ratio on the performance in the samedifferent task was still present in this sample (F(3, 225) = 23.6, p < .001, η = .22), with similar increasing accuracy scores (1:2 = 4.39%; 2:3 = 3.73%; 6:7 = 13.16%; 9:10 = 24.34%).

The SOMAG task provided individual scores corresponding to the total number of times each dimension of magnitude (numerosity, Size, Spacing) was spontaneously chosen as a sorting criterion by the child in the three experimental conditions (NumSize, NumSpace, and SizeSpace). We ran correlational analyses between the SOMAG task scores and children's average performances in the numerical same-different task (see Table 1 for a full description of the correlations). The average performance in the test trials of the numerical same-different task (i.e., trials in which there was a conflict between number and length/field area) was associated with the score in the NumSpace condition of the SOMAG task, r(74) = .30, p < .01, but not with the NumSize or the SizeSpace condition, p > .49. This correlation remained significant after controlling for the exact age of the participants, r(74) = .35, p < .01. This MECHANISMS OF NUMERICAL ABSTRACTION association showed that children who chose to sort the cards according to their numerosity rather than the spacing of the dots in the SOMAG task were also more likely to make correct numerical judgments regardless of the difference in the spacing of the dots between the two rows in the numerical same-different task (Figure 3). Note that, while there was a significant correlation between the two SOMAG conditions involving Size (r(74) = .75, p < .001) we found no significant relation between the two conditions of the SOMAG involving the Number dimension (r(74) = .03, p = .77, Table 1). Thus, choosing Number as a sorting criterion over Spacing was not systematically related to a general tendency of orienting towards numerical over nonnumerical dimensions of magnitude.

A multiple regression analysis confirmed that adding the NumSize and SizeSpace conditions to the model did not significantly change the observed relation between NumSpace and our same-different task (∆R 2 = .01, F(2, 72) < 1).

Discussion

By conjointly taking into account the impact of the strength of the interference between numerical and nonnumerical dimensions of magnitude and individual differences in spontaneous orientation to number, the current study shed new light on the mechanisms underlying the development of children's ability to abstract number. We showed that both factors significantly contribute to children's ability to handle the number/length interference in a Piaget-like same-different numerical task.

First, we provided evidence that a systematic manipulation of the length ratio between the two rows of dots in a Piaget-like same/different numerical estimation task affects children's performance. Specifically, we found that children's ability to make a "same" judgment on two rows containing an equal number of dots decreased progressively as the difference in the lengths of the two rows increased. Thus, children's performance on this task decreased as the interference between the nonnumerical and numerical magnitudes increased. This result adds further support to the idea that the performance observed in numerical judgement tasks are 1: Table of correlations between the performance on the test trials of the same-different task and the different SOMAG conditions. For each of the SOMAG conditions, the correlations were computed based on the dimension between parentheses. MECHANISMS OF NUMERICAL ABSTRACTION impacted by nonnumerical magnitudes, even in a same-different task thought to tap more directly into magnitude representation [START_REF] Van Opstal | The origins of the numerical distance effect: The samedifferent task[END_REF].

One of the key advantages of the Piaget-like same/different numerical estimation task designed in the present study is it allows systematic manipulation of the strength of the interference between number and a nonnumerical dimension, with a fixed level of numerical difficulty. Indeed, in our test trials (where the two rows contained the same number of tokens), the evolution of the performance cannot be explained in terms of numerical difficulty and can only be attributed to an influence of the nonnumerical aspects of the task. We argue that manipulating the length ratio modified the strength of the interference between length and number, and concurrently the probability to rely on the "length equals number" misleading heuristic. In turn, this manipulation varied the level of inhibitory control required to block such a misleading strategy in order to perceive the numerical equivalence in our samedifferent task. This is in line with previous findings showing that the inhibition of the 'length equals number' strategy is needed to overcome systematic errors in a Piaget-like numberconservation task [START_REF] Houdé | Negative priming effect after inhibition of number/length interference in a Piaget-like task[END_REF][START_REF] Houdé | Functional magnetic resonance imaging study of Piaget's conservation-of-number task in preschool and school-age children: a neo-Piagetian approach[END_REF].

Second, we found that children tended to make fewer errors in the numerical same-different task when they sorted the cards in the SOMAG task more frequently by number than by spacing. This correlation suggests that the prepotency to rely on the 'length equals number' misleading strategy in the number conservation task is also dependent on individual differences in children's spontaneous orientation towards relevant or irrelevant dimensions of the stimuli.

Note that the children's performance in our numerical same-different task was not related to their spontaneous orientation towards number over size or towards size over spacing, as assessed by the SOMAG task. This is consistent with the material of our same-different task, whereby the size of the dots is maintained as a constant and only dimensions related to MECHANISMS OF NUMERICAL ABSTRACTION density and field area vary. Thus, our results suggest a specificity of spontaneous orientation towards magnitudes, with different predictive values of orientation towards number over size v. over spacing, depending on the experimental context and/or possibly age. This hypothesis is further supported by the lack of correlation between the NumSize and NumSpace conditions in the SOMAG task. This specificity is interesting to put in perspective with the results reported in the study of [START_REF] Viarouge | Spontaneous orientation towards irrelevant dimensions of magnitude and numerical acuity[END_REF] based on data collected for the same group of children. These results

showed that both the NumSize and the NumSpace SOMAG conditions explained significant and distinct parts of the variance in the performance on Size/Number incongruent trials of a nonsymbolic numerical comparison task.

The different patterns of correlation with the SOMAG reported in the current study might be explained by differences between the numerical same-different and comparison tasks. As mentioned above, one important aspect of the current study is that the same-different task allows separating the Size and Spacing dimensions more clearly than in the numerical comparison tasks, such as the one used by [START_REF] Viarouge | Spontaneous orientation towards irrelevant dimensions of magnitude and numerical acuity[END_REF]. Indeed, in the latter task, given that the Spacing dimension is related both to field area and sparsity [START_REF] Dewind | Modeling the approximate number system to quantify the contribution of visual stimulus features[END_REF], variations in these dimensions could have occurred, which in turn might explain why the NumSpace condition was correlated with performance on Size/number incongruent trials.

Importantly, in the numerically equal trials of our same-different task, both the size and total surface area of the two rows of dots are maintained as constant, and the impact of different levels of dots spacing in the context of a fixed numerical difficulty is fully isolated. Another A limitation of our study is the lack of a separate measure of children's inhibitory control ability. This would have allowed us to determine the respective contribution of inhibitory control ability and spontaneous orientation towards number over spacing to the performance observed in our Piaget-like same-different task. Further studies are needed to better understand the relation between the ability to process number in contexts in which irrelevant non-numerical dimensions interfere with it, the spontaneous orientation towards relevant and irrelevant magnitudes, and the ability to inhibit these irrelevant dimensions.

Finally, our study provides new insights into the possible mediating factors of the relation between early numerical abilities and formal mathematics. In particular, these findings help specify the mechanisms involved in the abstraction of number by suggesting to take into account individual differences in spontaneous orientation to numerical and nonnumerical magnitudes. Indeed, recent studies have questioned the predictive value of numerical intuitions for math achievement. In particular, some authors have suggested that inhibitory control acts as a mediating factor of this relation, with mixed results. According to this view, children's capacity to grasp the invariance of number with regard to other dimensions of magnitude would be crucial for their true understanding of the concept of number, beyond the precision of their early numerical representations. This capacity to abstract number from other quantities might be particularly important in early school years, when the covariance of number with dimensions such as length might be unintentionally reinforced through education. For instance, the use of nonsymbolic representations of integers via rows of evenly spaced images might strengthen the misleading "number equals length" heuristic. MECHANISMS OF NUMERICAL ABSTRACTION

Conclusion

Altogether, these results indicated that the development of numerical representations in children relies on a variety of factors contributing to children's ability to abstract number from other irrelevant, nonnumerical dimensions of magnitude. In particular, we showed that conserving responses in a Piaget-like numerical estimation task appear progressively dependent on the saliency of the numerical dimension. This saliency depended both on the degree of interference with nonnumerical magnitudes and on individual differences in spontaneous orientation towards these dimensions.

Our results also provided additional evidence that cognitive development does not rely exclusively on the acquisition of knowledge of increasing complexity [START_REF] Piaget | The Child's Conception of Number[END_REF] but also on children's ability to inhibit misleading strategies and previously acquired knowledge [START_REF] Bjorklund | The resources construct in cognitive development: Diverse sources of evidence and a theory of inefficient inhibition[END_REF][START_REF] Houdé | Inhibition and cognitive development: Object, number, categorization, and reasoning[END_REF] and ability that can be expressed more or less easily by the children depending on the context (i.e., the level of interference) and their spontaneous orientation towards different dimensions of the context.

In light of current debates regarding the predictive value of numerical intuitions for math achievement, our study helps specify the factors involved in numerical development and provides new insights regarding the possible mediating factors of this relation.

Figure 1 :

 1 Figure 1: Examples of the stimuli used in the SOMAG task, for each of the three experimental conditions. In each condition, one dimension (Number, Size, or Spacing) is maintained constant. The bottom reference card can be sorted with one of the two top target cards according to one of the two remaining dimensions (e.g. NumSize condition pits the Number and Size dimensions against each other).

Figure 2 :

 2 Figure 2: Performance on the numerically equal trials of the numerical estimation task, for each level of length ratios between the two rows of dots. Error bars represent ± one standard error of the mean.

Figure 3 :

 3 Figure 3: Correlation between the spontaneous orientation to Magnitudes in the three experimental conditions of the SOMAG task and the performance in the numerical estimation task.

  potential explanation for this different pattern of correlations might have to do with what the different conditions of the SOMAG are actually capturing. The correlation between the two conditions involving Size suggests that children's choices were strongly influenced by this dimension. Added to the lack of correlation between the NumSpace and the other two MECHANISMS OF NUMERICAL ABSTRACTION conditions, this suggests that the NumSpace condition might be capturing children's spontaneous orientation to Number more specifically. This could then explain why this condition relates to both tasks involving interferences with number, regardless of the competing nonnumerical dimension.