Elisabeth Marie Victoire

Véronique Bouteiller

Myriam Bouichou

Mirah Rakarabo

Victor Da Silva

Amandine Bonnet

Lucas Adelaide

Philippe Turcry

Pauline Barthelemy

Jonathan Mai-Nhu

Elisabeth Marie-Victoire
email: elisabeth.marie-victoire@culture.gouv.fr

Victor Da-Silva

Lucas Adelaïde

François Cussigh

Concrete of tomorrow: corrosion performances in marine environment

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

Within the frame of the French national project PERDUB [START_REF] Rozière | Cussigh: A performancebased approach for durability of concrete exposed to carbonation[END_REF], an intensive series of studies mixing lab scale, natural ageing and durability modelling was launched in 2015. The aims of the project are multiple. One of these is to provide innovative concrete mix designs to improve durability performances and lower carbon footprint. Another one is to characterize the performances of these concretes, especially against carbonation and chloride ingress. For the latest, a vast database of performances is being produced at the lab scale in combination with life cycle modelling. A last part of the project is dedicated to the evaluation of the performances of 11 of the concrete mixes designed against corrosion induced by natural carbonation or exposure to marine environment, for a period of 20 years. To follow up the initiation and propagation phases of corrosion, an approach combining non-destructive testing, embedded sensors and corrosion modelling was adopted. In the present paper, the experimental program and the first non-destructive electrochemical tests performed on 11 walls exposed for a little more than a year to natural ageing on marine environment, in the harbor of La Rochelle (French Atlantic ocean shore) in tidal zone are presented.

Experimental program 2.1 Specimen

A series of 11 walls T-shaped, 0.16m-large, with two 1m² faces (1mx1m), and a foot (0.4x0.15m) to ensure their stability against the tide were cast (Fig . 1a). Two types of steel bars were embedded: smooth round shaped (RL) representative of ancient constructions (and which surface can be determined accurately for electrochemical calculations) and high-bond ribbed (HA) representative of modern ones. These rebars are isolated which means not in electrical contact with the others in the specimen. Two concrete covers were considered: 10mm and 20mm, to evaluate the progression of aggressive agents. Additional bars were added on the sides and foot of the walls for stability issues (Fig. 1b).

Eleven concrete mix designs with different binders were selected, from standard CEM I to binders containing mineral additions such as fly ashes (V), limestone filler (L), slag (S), silica fumes (FS), siliceous addition (AS), or metakaolin (MK). Due to these additions, the content of clinker was quite diverse with values ranging between 101 and 346 kg/m 3 (Table 2). Different types of aggregates were chosen: two types of alluvial silico-limestones, one soft limestone and one hard limestone.

For each concrete mix design, water to binder ratios varying from 0.35 to 0.6 were considered (Table 2).

Compressive strengths measured on 3 cylinders of the same concrete after 28 days of wet curing (stored in water) were ranging between 29 and 103 MPa, while open porosities measured after 90 days of wet curing were varying between 10 and 20% (Table 3).

Measuring techniques

The first aim of the monitoring implemented in this study is to detect the corrosion initiation and afterwards to evaluate the corrosion activity in the propagation phase.

The second one is to improve the accuracy of the corrosion diagnosis. Standard corrosion diagnosis is based on instant non-destructive electrochemical measurements. However, previous studies have shown the impact of environmental conditions and specially rain events on the data collected, evidencing the need for continuous monitoring [START_REF] Andrade | Evolution of reinforcement corrosion due to climatic variations[END_REF] [3] [START_REF] Marie-Victoire | Parameters affecting on site corrosion rate evaluation[END_REF]. Nevertheless, those monitoring techniques have rarely been installed on real structures in natural ageing conditions [START_REF] Marie-Victoire | Corrosion at low moisture content in both carbonated and chloride polluted concrete -Villa E-1027, a case study[END_REF] and still need to be validated. Therefore, the two types of technologies were considered and compared: embedded sensors and non-destructive testing.

Concerning the embedded sensors, for economic reasons, only three concretes B01, B04 and B31 were instrumented, and three additional walls were specifically cast. On those three walls, a complementary rebars network was introduced for the non-destructive measurements, and a series of sensors was embedded to monitor versus time: resistivity (Multi Ring Electrodes ® ); potential (ERE 20 ® ); resistance, potential and corrosion current (Anode Ladder ® ). In a dedicated reservation, additional temperature and relative humidity sensors (HMP110 ® ) were introduced from outside. Finally, all the sensors were connected to a multiplexer, itself connected to an analyzer, wired linked to a computer to provide data gathered in an automatic database [START_REF] Bouteiller | Evaluation de la durabilité du béton armé vis-à-vis des ions chlorure à l'aide de capteurs noyés dans le béton versus des auscultations en parement[END_REF]. Most of the sensors can deliver data at different depths allowing following the gradual ingress of aggressive agents into the concrete. For the non-destructive testing, two types of techniques were considered: electrochemical tools [START_REF] Bouteiller | Non Destructive Testing and Corrosion Health Monitoring of reinforced concrete slabs submitted to chloride ions diffusion during five years[END_REF] and evanescent field dielectrometry [START_REF] Bouichou | Measurement of water content and salinity index in concrete by evanescent field dielectrometry[END_REF] [START_REF] Marie-Victoire | On site electrochemistry vs. Evanescent Field Dielectrometry[END_REF]. Resistivity was measured with a four-point Wenner probe (Resipod ® ) [START_REF] Polder | Test methods for onsite measurement of resistivity of concrete[END_REF] and with the disc method (Gecor10 ® ) [START_REF] Feliu | A new method for In-situ measurement of electrical resistivity of reinforced concrete[END_REF], potential mappings were performed with a copper/copper sulphate electrode (Canin ® ) [START_REF] Elsener | Half-cell potential measurements -Potential mapping on reinforced concrete structures[END_REF] and corrosion rate was evaluated using a Gecor10 ® [START_REF] Andrade | Test methods for on-site corrosion rate measurement of steel reinforcement in concrete by means of the polarization resistance method[END_REF] Only the first series of non-destructive electrochemical measurements on the eleven noninstrumented walls are presented in this paper.

Experimental protocol

The reinforced concrete walls were cast from December 2019 to January 2020. After almost one year of exposure to natural ageing in semi-rural environment, 60km southwest from Paris, they were implanted in December 2020 in marine environment in the harbor of La Rochelle.

The specimen are exposed to natural ageing in tidal zone (Fig. 3a), which means around 6 hours immersed and 6 hours in the open air for every tide. As a consequence, the measurements can only be performed at low tide (Fig 3b) which means that to achieve the measurements on the eleven walls without embedded sensors (B01, B02, B04, B05, B07, B31, B36, B37, B38, BM40, B41), 10 days are necessary. The first set of experiments took place in October 2021. The following protocol was adopted for the electrochemical measurements for each rebar (Fig. 4) of each side of the walls (Fig. 5): -Potential mapping [START_REF] Elsener | Half-cell potential measurements -Potential mapping on reinforced concrete structures[END_REF], with 3 vertical lines spaced 5cm apart, with a measurement every 5cm, -Resistivity measurements on point 8 (40cm from the top of the wall) and point 14 (70cm from the top of the wall), first with a 4-points (5cm spacing electrodes) Wenner probe (4 measurements per point according to the RILEM recommendation) [START_REF] Polder | Test methods for onsite measurement of resistivity of concrete[END_REF] and second with the disc method (5 measurements per point 5cm spaced left and right from the rebar) [START_REF] Feliu | A new method for In-situ measurement of electrical resistivity of reinforced concrete[END_REF], -And one potential and LPR measurements on point 8 [START_REF] Andrade | Test methods for on-site corrosion rate measurement of steel reinforcement in concrete by means of the polarization resistance method[END_REF]. 3 First results and discussions

Potential measurements

First, potential mapping clearly evidenced a difference between the two concrete covers, with less electronegative values of potential with the 20mm concrete cover (between -367 and -200mV, Fig. 678) than with 10 mm of concrete cover (between -560 and -317mV, Fig. 678). For example for wall B37, the shift is averagely varying from 100 to 200 mV between the two concrete covers (Fig. 6). This could be linked to the moisture and salt contents, which are probably higher at lower depth. Distinct results are also noted between the two rebars shapes, for both concrete covers, but no real trend emerges.

Another phenomenon can be observed on the potential maps, which exhibit an electronegative gradient from the top to the bottom of the walls on both faces, but clearer with the lowest concrete cover (RL10 and HA10, Fig. 6). The explanation could come from the duration of the tide's descent on the one hand, as the bottom of the walls stays wet a longer time (Fig. 8). On the other hand, when the tidal coefficient is low, the walls are not totally immersed (Fig. 9), which means that the top of the walls remains in the open air and can dry, while the bottom is totally under water.

When comparing the results of the 11 walls, B04, B36 and B40 exhibit the most electronegative values, while B31, B38, B41 lead to the less electronegative ones. For B04, the explanation could come from its W/B, and its high carbonation potential as the concrete is composed of a CEM III with low clinker content (Table 2). So that it is possible that the carbonation has progressed towards the lowest concrete cover of 10mm during the one year of exposure to natural ageing in semi-rural environment, before being implanted in marine environment. This carbonation could have favored the chlorides penetration, For B40, its quite high porosity and its high sensitivity to chloride migration (Table 3) may be the cause of these quite electronegative values. The same observation can be done for B01 and B07, which exhibit quite electronegative values. Finally, for wall B36, its behavior is probably linked to its porous limestone as aggregates, its very high open porosity, and its quite high sensitivity to chloride penetration (Table 3).

Resistivity measurements

Resistivity evaluated with the Wenner technique (Fig. 11) is consistent with the potential mapping, with the lowest values obtained for walls B40, B36 and B04. In addition, the walls B01 and B07 also exhibited quite low values, all indicative of corrosion favorable conditions, according to the RILEM recommendation. Contrarily, B31, B38, B41 showed the highest resistivity, which is also consistent with the potential mapping. As for the clear variations observed between the two concrete covers.

The results obtained with the Disc method (Fig. 12) are in line with that of the Wenner technique, but the measured values are generally lower (Fig. 13). This result has probably to do with the depth of concrete affected by the measurement, which is smaller for the Disc method than for the Wenner technique. Feliu et al. [START_REF] Feliu | A new method for In-situ measurement of electrical resistivity of reinforced concrete[END_REF] demonstrated that at a depth equivalent to the disc diameter (2cm), the percentage of ohmic drop is already of 71%, reaching 81% at a depth of 3cm. As the Wenner probe used in this study had an electrode spacing of 5cm, the depth concerned by the measurement might be higher [START_REF] Du Plooy | Development of a multi-ring resistivity cell and multi-electrode resistivity probe for investigation of cover concrete condition[END_REF] than that of the Disc method. As it is probable that the concrete is wetter on the surface than in depth, this could explain why the resistivity measured with the 4points method was generally higher than with the disc method. It has to be noted that with both techniques, slight differences can be observed between the two rebars shapes whatever the concrete cover. However, no trend emerges, as the variations are more or less in the range of the standard deviation for both measurement techniques.

Corrosion current density

The corrosion current densities are also consistent with the previous resistivity and potential measurements, with the highest values observed for designs B04, B36, B40 and B07 designs, and the lowest for B31, B38 and B41 (Fig. 14).

When referring to the RILEM recommendation [START_REF] Andrade | Test methods for on-site corrosion rate measurement of steel reinforcement in concrete by means of the polarization resistance method[END_REF] almost all the recorded values are above the first threshold of negligible corrosion (0.1µA/cm²). Walls B04, B07, B36, B40, and punctually B02, lead to corrosion current densities higher than 0.5µA/cm², which is the threshold of moderate corrosion. The threshold of 1µA/cm² was exceeded only for the smallest concrete cover with designs B04 and B36. At this early stage of exposure to marine environment, except for the higher values (B04 and B36) the question of real rebars corrosion can be raised. Several hypothesis can be issued: is there enough oxygen at the rebars depth even at low tide? Is slag oxidation interfering with the rebars corrosion measurements? Is the pH of these new concrete designs high enough to protect the rebars? Are the existing RILEM threshold still adapted to the concretes containing mineral additions?

When comparing the values obtained with the two concrete covers, as for the preceding measurements, a tendency towards higher values for the lowest concrete cover is observed, for the same probable reasons.

Finally, has to be noted that for the highest concrete cover (20mm), the values are in the same range for both rebars shapes, except for walls B04 and B40. Contrarily, for the lowest concrete cover (10mm), generally higher values are observed for the high-bond shaped of rebars, except for B36. This might have to do with the calculation of the surface of rebar, which is more approximate for the high-bond shaped rebars.

Conclusions and outlooks

An intensive project PN-PERFDUB dedicated to the production and evaluation of several reinforced concrete, less polluting and more durable, was initiated in 2015 in France. Among the aims of the project, one is to produce a database on corrosion performances of 11 of the concretes in either conditions favorable for carbonation or marine environments, and a second one is to try to improve the accuracy of the corrosion diagnosis, by mixing instant non-destructive testing and permanent monitoring using sensors.

The first results of the non-destructive electrochemical measurements obtained after almost a year of natural ageing in marine environment showed noticeable differences between the concrete mix designs, but also a clear influence of the concrete cover and of the tide. Surprisingly due to the early stage of exposure, some of the concretes exhibited quite high corrosion current densities. At this step of the project, several questions can be raised: is the durability of some mix designs questionable? Is oxygen sufficiently available at low tide at the rebars depth? What is the influence of the mineral additions in the binders, especially the slag content? Are the RILEM threshold still adequate for concretes with high mineral additions content (limestone, metakaolin, slag…) which questions the adequacy of their pH value to protect the steel rebars?

The next phases of the project will help to clarify these points, as they include a comparison with embedded sensors and with complementary non-electrochemical non-destructive techniques (evanescent field dielectrometry). In addition, lab tests will be performed to evaluate, among a series of performances (open porosity, transfer coefficients, chlorides diffusion coefficients…), the chloride ingress and the carbonation depth. Therefore, sampling campaigns are scheduled in parallel to the nondestructive measurements and detailed observation of the law-walls, for the next 20 years. Finally, the data will be compared to that of the extensive laboratory tests performed on these concretes in another part of the project, prior to the manufacture if the walls, and with the service life models.

Fig. 1 .

 1 Fig. 1. a) Reinforced concrete wall specimens, b) Rebar network.

Fig. 2 .

 2 Fig. 2. Walls formwork and embedded sensors.

-Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. a) Specimen in the tidal zone of La Rochelle harbor. b) Wenner resistivity measurement at low tide.

Fig. 6 .

 6 Fig. 6. Potential mapping on the face with a 10mm concrete cover.

Fig. 7 .

 7 Fig. 7. Potential mapping on the face with a 20mm concrete cover.

Fig. 8 .

 8 Fig. 8. Potential measured with the Gecor10 ® on point 8 of each rebar and concrete cover.

Fig. 9 .

 9 Fig. 9. During the tide's descent, the bottom of the walls remains a longer time immersed.

Fig. 10 .

 10 Fig. 10.. At high tide of low coefficient, the top of the walls remains in the open air.

Fig. 11 .

 11 Fig. 11. Average resistivity value evaluated with the Wenner probe on point 8 and 14 for each rebar and concrete cover. RILEM thresholds in dashed lines.

Fig. 12 .

 12 Fig. 12. Average resistivity evaluated with the Disc method probe on point 8 and 14 for each rebar and concrete cover. RILEM thresholds in dashed lines.

Fig. 13 .

 13 Fig. 13. Comparison (Wenner vs Disc) of the average resistivity evaluated on point 8 and 14 of each rebar with a 10mm concrete cover. RILEM thresholds in dashed lines.

Fig. 14 .

 14 Fig. 14. Corrosion current density evaluated with a Gecor10 ® on point 8 for each rebar and concrete cover. RILEM thresholds in dashed lines.

Table 1 .

 1 Concrete binders and aggregates.

	Concrete	Binder	Aggregates
		(kg/m 3 )	Type	Water
				absorption (%)
	B01	CEM I (289)	2	2.7
	B02	CEM I (231) + fly ash (98)	2	2.7
	B04	CEM III A (287)	2	2.7
	B05	CEM I (122) + Slag (184)	2	2.7
	B07	CEM I (267) +	2
		Limestone		2.7
		addition (188)
	B31	CEM III A (383)	1	1.0
	B36	CEM V (363)	4	4.0
	B37	CEM V (374)	3	0.6
	B38	CEM I (354) +	1
		Silica fume		1.0
		(30)
	B40	CEM I (261) +	2
		siliceous		2.7
		addition (112)
	B41	CEM I (302) +
		metakaolin	1	1.0
		(76)
	Type 1 aggregate: Silica-limestone Alluvial, mixed rounded-
	crushed.
	Type 2 aggregate: Silica-limestone Alluvial, round + mixed.
	Type 3 aggregate: Dense limestone, crushed.
	Type 4 aggregate: Limestone, crushed.

Table 2 .

 2 Clinker content, water to binder ratio and water to cement ratio.

	Concrete Clinker	W/B	W/C
		content
		(kg/m 3 )
	B01	272	0.60	0.60
	B02	217	0.54	0.77
	B04	101	0.61	0.61
	B05	115	0.57	1.44
	B07	262	0.41	0.71
	B31	137	0.4	0.4
	B36	207	0.46	0.46
	B37	207	0.45	0.45
	B38	346	0.35	0.38
	B40	245	0.47	0.67
	B41	293	0.35	0.44

Table 3 .

 3 Concrete preliminary laboratory characterization.

Concrete Open porosity after 90 days of wet curing (%) Compressive strength after 28 days of wet curing (MPa) Chloride migration coefficient after 90±5 days of wet curing (x10 -12 m²/s)

	B01	16.9	28.8	19*
	B02	17.4	31.9	5.7*
	B04	16.7	49.9	2.5**
	B05	18.6	44.5	2.0
	B07	16.8	42.6	38
	B31	12.1	61.6	0.4*
	B36	19.6	49.9	11.7
	B37	12.8	74.4	0.4
	B38	10.3	103.5	0.2
	B40	16.6	39.9	24.8
	B41	10,8	88.5	0,8

*/** Test performed after 145±5 days (*) or 130 days (**) of wet curing

2.2 Monitoring

The authors would like to thank the financial support of IREX and all the contributors of the PN-PERFDUB. They also would like to thank the City of La Rochelle, the La Rochelle Harbor, the Centre National des Monuments Nationaux, and the Yacht Club Classique for their logistical support and access authorization.


Eric Iacono
email: eric.iacono@cyu.fr

Hippolyte Gros
email: hippolyte.gros@cyu.fr

Evelyne Clément
email: evelyne.clement@cyu.fr

Training flexible categorization to improve arithmetic problem solving: A school-based intervention with 5th graders

Keywords: cognitive flexibility, school-based intervention, categorization, evidence-based education, arithmetic word problems

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

Introduction

Can a school intervention based on flexible categorization successfully target the expression of cognitive flexibility in arithmetic problem solving? By making it possible to adapt to a constantly changing environment, to discover creative solutions, to transfer knowledge from one situation to another, to infer the meanings of new words, or more generally to switch from one behavior to another depending on the environmental constraints, cognitive flexibility is considered a hallmark of human cognition (e.g., [START_REF] Deák | The development of cognitive flexibility and language abilities[END_REF][START_REF] Ionescu | Exploring the nature of cognitive flexibility[END_REF][START_REF] Ionescu | The variability-stability-flexibility pattern: A possible key to understanding the flexibility of the human mind[END_REF]. While there is no unified definition in the literature, it is widely accepted that flexibility plays a key role in the development of thinking, language, reasoning, and knowledge acquisition (e.g., [START_REF] Blaye | Thematic and taxonomic relations in preschoolers: The development of flexibility in categorization choices[END_REF][START_REF] Clément | La résolution de problème. A la découverte de la flexibilité cognitive[END_REF][START_REF] Clément | Successful solution discovery and cognitive flexibility[END_REF][START_REF] Deák | The development of cognitive flexibility and language abilities[END_REF][START_REF] Sloutsky | Attentional learning and flexible induction: How mundane mechanisms give rise to smart behaviors[END_REF]. Cognitive flexibility is often conceived of as the ability to cleverly change behavior in an appropriate manner depending on what a situation requires. However, upon closer scrutiny, this general definition of flexibility appears almost indistinguishable from a broad definition of intelligence. To overcome this limitation, cognitive scientists have striven to describe the specific cognitive processes underlying the flexible expression of our behavior and thoughts in terms of cognitive processes.

In a canonical sense, cognitive flexibility (also called set shifting, see [START_REF] Diamond | Why improving and assessing executive functions early in life is critical[END_REF] is described in the literature as an executive function consisting in the ability to shift attention from one stimulus, rule, or task to another (e.g., [START_REF] Diamond | Executive functions[END_REF][START_REF] Lehto | Dimensions of executive functioning: Evidence from children[END_REF][START_REF] Miyake | The unity and diversity of executive functions and their contributions to complex "frontal lobe" tasks: A latent variable analysis[END_REF][START_REF] Meiran | Component processes in task switching[END_REF][START_REF] Monsell | Task switching[END_REF]. However, it appears that in complex activities such as problem solving, cognitive flexibility also entails the ability to flexibly consider the same problem from different perspectives, to flexibly categorize it at different levels of abstraction. In the following sections, we present the empirical studies that investigate these two fundamental processes (i.e., attentional shifting and flexible categorization).

Attentional shifting

Since the pioneer work of [START_REF] Miyake | The unity and diversity of executive functions and their contributions to complex "frontal lobe" tasks: A latent variable analysis[END_REF], cognitive flexibility, inhibition, and working memory are considered as the three core functions of executive control. In this approach, flexibility is often described as a set shifting (for a developmental approach, see for example [START_REF] Davidson | Development of cognitive control and executive functions from 4 to 13 years: Evidence from manipulations of memory, inhibition, and task switching[END_REF]. In that sense, the cognitive processes underlying set shifting are mainly related to attentional flexibility [START_REF] Ionescu | The variability-stability-flexibility pattern: A possible key to understanding the flexibility of the human mind[END_REF]. In the following, we will refer to this process as attentional flexibility.

In a large body of research, attentional flexibility has been operationalized within the well-known task-switching paradigm (see [START_REF] Monsell | Task switching[END_REF]. A series of tasks have been designed according to this paradigm, among which the widely used rule-based switching tasks. These consist in presenting multidimensional stimuli that first need to be categorized according to a specific dimension (e.g., size) before changing the rule and switching to another dimension (e.g., number of items). The Wisconsin test [START_REF] Heaton | Wisconsin Card Sorting Test Manual[END_REF] for adolescents and adults, or the Dimensional Change Card Sort (DCCS) for preschoolers [START_REF] Frye | Theory of mind and rule-based reasoning[END_REF][START_REF] Zelazo | The Dimensional Change Card Sort (DCCS): a method of assessing executive function in children[END_REF] are some of the most typical examples of rule-based switching tasks. For instance, in the standard DCCS, children have to sort bi-dimensional cards (fish and boats that are either red or blue) according to a first rule (e.g., color) and then, according to a second rule (e.g., shape).

Developmental research reports two robust results regarding attentional shifting. First, factor analyses with children show that attentional shifting emerges later than inhibition and working memory (e.g., [START_REF] Carroll | Cognitive flexibility in young children: Beyond perseveration[END_REF]. Second, before 3 years old, children present a perseverative behavior when asked to perform rulebased switching tasks: after sorting the stimulus according to one dimension (e.g., shape), they experience considerable difficulty sorting it according to another dimension (e.g., color) (e.g., [START_REF] Deák | Cognitive flexibility in young children: General or task-specific capacity[END_REF][START_REF] Hofstadter | Children's task-switching efficiency: missing our cue?[END_REF][START_REF] Legare | Cultural variation in cognitive flexibility reveals diversity in the development of executive functions[END_REF][START_REF] Zelazo | The Dimensional Change Card Sort (DCCS): a method of assessing executive function in children[END_REF].

Although the task switching paradigm is well suited to assess attentional flexibility, its ability to assess other components of flexibility is limited for at least two reasons. First, in several situations involving flexibility, such as, for instance, problem solving, there is no imposed rule change and one needs to spontaneously adopt a new perspective without any outside prompt. Second, most of the time, achieving the goal requires to flexibly re-interpret (rerepresent) the situation, which goes beyond a mere switch of attention. Thus, representational flexibility relies both on the attentional processes required to take into account environmental cues (for instance, noticing when the strategy used is not optimal), and on the ability to flexibly categorize the problem by re-interpreting the situation it depicts.

Flexible categorization in problem solving

The idea that a solver's ability to adequately categorize a problem is related to their ability to find its solution is perfectly exemplified by the seminal work of [START_REF] Chi | Categorization and representation of physics problems by experts and novices[END_REF] on the nature of expertise. In their study, they asked both novices and experts to sort a set of physics problems into as many categories as they liked. Their results showed that while novices tend to categorize problems based on their surface features (e.g., sorting together all the problems mentioning a spring), experts base their categories on deeper abstract principles (e.g., putting together problems that can be solved by applying the Law of Conservation of Energy). The process by which one may ignore the surface features of a problem and flexibly adopt a new point of view to find its solution is a notoriously difficult endeavor.

In fact, an increasing amount of literature has shown that disregarding the initial interpretation of a problem and flexibly categorizing it according to a more abstract perspective is a key step in solving counter-intuitive problems [START_REF] Brissiaud | Arithmetic word problem solving: a Situation Strategy First framework[END_REF][START_REF] Ragni | On selecting evidence to test hypotheses: A Theory of Selection Tasks[END_REF]. Notably, interferences between one's experience and knowledge about the world and one's reasoning have been shown to greatly impact solving performance (e.g., [START_REF] Bassok | Judging a book by its cover: Interpretative effects of content on problemsolving transfer[END_REF][START_REF] Bassok | Adding apples and oranges: Alignment of semantic and formal knowledge[END_REF][START_REF] Cosmides | Cognitive adaptation for social exchange[END_REF][START_REF] Gros | What we count dictates how we count: A tale of two encodings[END_REF]. For instance, it has been demonstrated that differences of difficulty between isomorphic problems (problems sharing the same solution) may be explained by the influence of prior knowledge on the way the situation is interpreted. When prior knowledge leads participants to initially think that an action is not possible, they will take a longer time to get around this self-imposed limit and they will make more mistakes while doing so [START_REF] Clément | Knowledge of domain effects in problem representation: The case of Tower of Hanoi isomorphs[END_REF]. In a similar fashion, getting rid of the common misconception that a subtraction necessarily describes the removal of a part from the whole is an arduous task [START_REF] Brissiaud | Arithmetic word problem solving: a Situation Strategy First framework[END_REF]. Adopting the more accurate conception according to which a subtraction describes the distance between two numerical values requires a form of conceptual flexibility relying on flexible categorization of the very concept of subtraction [START_REF] Hofstadter | Children's task-switching efficiency: missing our cue?[END_REF].

In sum, in problem solving, it has been demonstrated that semantic recoding and re-categorization of the situation are the core cognitive processes underlying flexible responses (e.g., [START_REF] Clément | La résolution de problème. A la découverte de la flexibilité cognitive[END_REF][START_REF] Clément | Successful solution discovery and cognitive flexibility[END_REF][START_REF] Clerc | Metacognition and cognitive flexibility in transfer of learning[END_REF][START_REF] Gamo | Transfer of strategies by semantic recoding in arithmetic problem solving[END_REF][START_REF] Gros | Context, content effects and flexibility[END_REF][START_REF] Gros | Semantic congruence in arithmetic: A new conceptual model for word problem solving[END_REF][START_REF] Scheibling-Sève | Assessing conceptual knowledge through solving arithmetic word problems[END_REF][START_REF] Scheibling-Sève | Developing cognitive flexibility in solving arithmetic word problems[END_REF]. In fact, finding the solution to a problem usually means looking at the problem from different perspectives. By adopting a new, usually more abstract point of view on the situation, one may be able to identify the deep structure of the problem and, in doing so, find a previously hidden path to the solution [START_REF] Clément | La résolution de problème. A la découverte de la flexibilité cognitive[END_REF][START_REF] Gros | Semantic congruence in arithmetic: A new conceptual model for word problem solving[END_REF].

Flexibility and academic achievement

Whether it be scientific research in education, psychology, or didactics, a growing body of evidence points towards executive functions (i.e., inhibition, working memory, and set shifting) as crucial contributors to school achievements (e.g., [START_REF] Agostino | Executive functions underlying multiplicative reasoning: Problem type matters[END_REF][START_REF] Monette | The role of the executive functions in school achievement at the end of Grade 1[END_REF][START_REF] St Clair-Thompson | Executive functions and achievements in school: Shifting, updating, inhibition, and working memory[END_REF][START_REF] Stad | Prediction school achievement: Differential effects of dynamic testing measures and cognitive flexibility for math performance[END_REF][START_REF] Yeniad | Shifting ability predicts math and reading performance in children: A meta-analytical study[END_REF]. For instance, after controlling for inhibition, working memory, planning, and fluid intelligence, [START_REF] Magalhães | Executive functions predict literacy and mathematics achievements: The unique contribution of cognitive flexibility in grades 2, 4, and 6[END_REF] found that cognitive flexibility accounts for a significant amount of variance in literacy and mathematics outcomes across Grades 2, 4, and 6.

More specifically, regarding math-achievement, in a recent study [START_REF] Hästö | Relationship between mathematical flexibility and success in national examinations[END_REF] demonstrated that cognitive flexibility predicts students' performance on finals, notably in mathematics. Cognitive flexibility has been argued to be a major contributor to math-performance, since, when one way of solving a mathematical problem isn't working, students need to switch between different strategies, and re-interpret the situation [START_REF] Gros | When masters of abstraction run into a concrete wall: Experts failing arithmetic word problems[END_REF][START_REF] Gros | Semantic congruence in arithmetic: A new conceptual model for word problem solving[END_REF][START_REF] Sander | Analogy and transfer: encoding the problem at the right level of abstraction[END_REF][START_REF] Vicente | Influence of situational and conceptual rewording on word problem solving[END_REF].

Considered as determinant factors in school achievement, a body of research focused on interventions for improving executive functions (e.g., [START_REF] Blakey | A short executive function training program improves preschoolers' working memory[END_REF][START_REF] Diamond | Conclusions about interventions, programs, and approaches for improving executive functions that appear justified and those that, despite much hype, do not[END_REF][START_REF] Mennetrey | Cognitive flexibility training in three-year-old children[END_REF]. Surprisingly, very few studies have been conducted on cognitive flexibility as the ability to consider the problems from different perspectives. In fact, most studies are often devoted to training working memory, inhibitory control, or attentional shifting, and to evaluate the effectiveness of such training in different areas of daily life (e.g., [START_REF] Diamond | Executive functions[END_REF][START_REF] Diamond | Conclusions about interventions, programs, and approaches for improving executive functions that appear justified and those that, despite much hype, do not[END_REF]. To date, only a limited number of studies have targeted the stimulation of flexibility as it manifests itself in complex cognitive activities, such as problem solving. Therefore, the present study focuses on the representational flexibility involved in proportional math problem solving.

The present study

Because of the crucial role of flexibility in school achievement, fostering it in the classroom is a major educational question. The aim of the present study is to promote flexible categorization in mathematics with 5th graders, and more particularly in arithmetic proportion word problems. In order to encourage students to identify, beyond their superficial similarities or dissimilarities, the abstract and deep structure of problems, we conceived an educational intervention based on comparison between proportion arithmetic word problems. In fact, in the analogical transfer literature, it is well established that comparing two analogous problems (sharing the same solution path) leads to a more abstract representation of which category the problems belong to [START_REF] Catrambone | Overcoming contextual limitations on problem-solving transfer[END_REF][START_REF] George | Breaking past the surface: Remote analogical transfer as creative insight[END_REF][START_REF] Holyoak | Analogy[END_REF][START_REF] Loewenstein | Analogical encoding facilitates knowledge transfer in negotiation[END_REF]. In the same way, fostering students to compare two strategies to find the solution to the same problem allows them to flexibly re-interpret the problem in such a way that they encode the problem in a higher degree of abstraction [START_REF] Brissiaud | Teaching and Development: Solving "Missing Addend" Problems Using Substraction[END_REF][START_REF] Gamo | Transfer of strategies by semantic recoding in arithmetic problem solving[END_REF][START_REF] Scheibling-Sève | Developing cognitive flexibility in solving arithmetic word problems[END_REF].

In accordance with these works, we postulate that encouraging children to compare problems and to identify their superficial and deep similarities will encourage them to perceive the abstract mathematical structure underlying proportion arithmetic word problems. This should help promote flexibility in the way students perceive, categorize, and solve the problems.

Hypotheses

We therefore hypothesized that the trained group (experimental group) would present a higher progression than the control group, both on categorization and resolution tasks. More precisely, at the post-test, the trained group should more frequently spontaneously categorize the problems according to their structural similarities rather than to their superficial resemblance, compared to the control group (Hypothesis 1). Second, the experimental group should display better post-test solving performance than the control group (Hypothesis 2).

Method Participants

The experiment was conducted with 10 classes of 5th graders belonging to five elementary schools located in highpriority education networks in the Paris region. A total of 147 5 th graders took part in the study (mean age = 11.4 years; SD = 0.45). The experimental group included 77 students (40 girls). The control group included 70 students (39 girls). The 5 classes of each group were drawn from the same highpriority education network, thus indicating comparable socioeconomic status and overall diversity. Written consent to take part in the experiment was obtained from all the participants' parents. The experiment was conducted during regular school hours for both groups.

Design

To assess the effectiveness of our educational intervention, the experiment was designed in three phases; a pretest, followed by 8 learning sessions on multiplicative problems (the content of which differed between the experimental and control groups), and a post-test. The learning sessions were of identical duration and frequency between the two groups. The pre-and post-test were identical, the only difference consisting in a supplementary task at the end of the post-test, in which students were asked to create word problems corresponding to the four problem structures that had been taught (see Table 1).

In an effort to maximize the ecological validity of the study, the training sessions in both groups were conducted by the classes' regular teachers. Prior to the study, the teachers had been taught how to use the experiment's educational materials to teach their classes. The first author was in charge of training the teachers, due to his background as an educational advisor specialized in teacher training. The teachers from the experimental group were introduced to the online platform that they would have to use to conduct the learning sessions. They were also given instructions to mainly focus training on a better understanding of the quaternary problems as defined by [START_REF] Vergnaud | Multiplicative structures[END_REF]. It should be noted that the teachers were completely blind to the hypotheses of our study, they did not know whether they were in the control or in the experimental group, and they had to follow the set timeline for each session.

Materials

Pre-and post-tests A series of 16 problems was constructed according to the four multiplicative structures defined by [START_REF] Vergnaud | Multiplicative structures[END_REF] : multiplication (e.g., In 1 box there are 107 sweets. I have 253 identical boxes. How many sweets do I have in total ?), firsttype division (e.g., I have 1 box of sweets to share between 24 people. It contains 72 pieces. How many sweets will each person get?), second-type division (e.g., In 1 box I have 107 identical sweets. I have 749 sweets in total. How many boxes can I fill?), and direct proportion (e.g., I fill 2 boxes with 30 sweets. I have 48 boxes. How many sweets do I need to fill them?). Each of these four multiplicative structures was presented in four different contexts describing either sweets, flowers, notebooks, or photo albums (see Table 1). The length of the problems was controlled, each problem statement shared the same number of sentences and presented the three numerical values in the same order. In the categorization task, each problem was printed on a card, following Chi et al.'s (1981) experimental design. The participants were instructed to classify the problems as they saw fit, creating as many categories as they wished. Then, in the problem-solving task, students were presented with an 8page booklet displaying 8 of these problems (one per page). Students were instructed to try to solve the problems. Problem order was randomized between participants. For every 3 pupils, I hand out 24 notebooks. There are 336 pupils in the school. How many notebooks do I need in total?

Learning sessions

Learning sessions took place in 8 sessions over 4 weeks (two 55 minutes sessions each week). Teachers of the experimental group were given direct online access to each lesson so they could display the educational materials onto the interactive whiteboard. This made it possible to ensure that the teachers could each teach the exact same lesson in their own way. Furthermore, we had printed out a sheet for each teacher reminding them of how to conduct the session. This was done so that the experiment was as close as possible to real-life conditions; the type of training and instructions that teachers received was similar to the guidelines they regularly receive following educational reforms. In the control group, the learning sessions followed the regular curriculum, using the teachers' usual Math textbook (e.g., [START_REF] Charnay | Cap Maths CM2, Manuel de l'élève[END_REF]. These sessions were built around the proportionality table and the "rule of three". In the experimental group, learning sessions were based on a general principle of comparison between analogous problems instead.

The first 6 sessions were focused on the comparison of problems belonging to two multiplicative structures (i.e., first-type division/second-type division; multiplication/ direct proportion; multiplication/ first-type division; secondtype division/direct proportion; multiplication/second-type division; first-type division/ direct proportion). In the last two lessons, problems belonging to the four multiplicative structures were compared. An equivalent number of problems sharing or not the same surface similarities was shown in each lesson. The aim of the experimental intervention was to get the children to ignore the surface similarities between the problems and identify their deep structure.

Results

Categorization task

As in [START_REF] Gros | What we count dictates how we count: A tale of two encodings[END_REF], we used similarity analysis to investigate the categories created by the participants. We coded the categories created by each participant at pre-test and post-test with a co-occurrence matrix describing how many times two problem statements were grouped together within the same category. We used these co-occurrence matrices to compute 2 proximity matrices for each group, describing the mean perceived similarity between each problem at pre-test and post-test. These matrices provide, for each pair of problem statements, a numerical estimation of the frequency at which the problems were sorted together.

We calculated the mean proximity score of the 24 pairs of problems sharing the same structure (e.g., multiplication problem mentioning sweets with multiplication problem mentioning notebooks) for each group and each test. At pretest, the mean proximity score for problems sharing the same structure in the control group (m = 0.554, SD = 0.013) was not significantly different from that of the experimental group (m = 0.555, SD = 0.017): t(23) = 0.25, p = .805, paired t-test. On the other hand, at post-test, the mean proximity score for structural pairings was significantly higher in the experimental group (m = 0.601, SD = 0.018) than in the control group (m = 0.536, SD = 0.012)), as hypothesized (t(23) = 17.19, p < .0001, paired t-test). Thus, the data collected from students' spontaneous categories supports Hypothesis 1: the experimental group was keener to create categories based on the problems' structure than the control group, which relied more heavily on surface similarities in this task.

Solving task

To evaluate the progress of the control and experimental groups, we looked at the increase in performance (number of correctly solved problems) between the pre-test and the posttest (see Fig. 1). We removed from the progress analysis the participants who were unable to participate either to the pretest or to the post-test. The analysis was performed on the remaining 110 students. Results showed that participants in the experimental group went from an average performance score of 33.73% at the pre-test to an average score of 58.13% at the post-test (t(62) =, p < .0001, paired t-test). The participants in the control group also progressed significantly, going from an average performance of 33.24% at pre-test to an average of 44.41% at post-test (t(46) =, p < .01, paired t-test). Crucially, the mean progress between pre and post-test was significantly higher in the experimental group (m = 24.40%) than in the control group (m = 11.17%), t(108) =, p < .01, independent t-test.

These results support Hypothesis 2: while both groups achieved similar performance at pre-test, the experimental group benefited more from the 4-week training. In other words, students who followed the learning sessions based on comparison between analogous problems were more likely to find the solution to the proportion problems than students who followed the textbook-based training focusing on proportionality tables and "rule of three".

Discussion

This study evaluated the effectiveness of an educational intervention targeting the understanding and solving of proportion problems. The intervention based on comparisons between analogous proportion problems sharing or not surface similarities, proved successful in improving performance both in problem categorization (Hypothesis 1) and in problem solving (Hypothesis 2). As predicted, there was no significant difference between the two groups' performance at pre-test, while students from the experimental group improved significantly more at post-test, both in their ability to categorize the problems based on their deep structure, and in their solving performance. In other words, participants in the experimental group developed a higher proficiency in proportion problem solving and they were able to flexibly categorize the problems using a representation with a higher degree of abstraction, moving away from the problems' superficial features.

These findings have important implications for pedagogical design. Indeed, while most French math textbooks teach about proportion problem solving by resorting to procedures (e.g., how to use a proportionality table) and rules (e.g., "the rule of three"), here the training focusing on problem comparison helped students achieve higher performances. Making comparisons helped them to see the structural commonalities between problems sharing the same solution principle and showed them which aspects of the problems were relevant and which could be disregarded. We believe that such an approach could be beneficial in other aspects of mathematical reasoning, as well as in other fields of knowledge. This comparison process makes it possible for students to learn to adopt a different point of view on the problems they encounter, flexibly categorizing the situation to identify the optimal way to broach and solve it. In this perspective, a sequence of comparisons between increasingly dissimilar problems sharing the same solution could result in a deeper understanding of the notions at hand, similarly to how concreteness fading [START_REF] Fyfe | Concreteness fading in mathematics and science instruction: A systematic review[END_REF] uses examples of increasing abstraction to promote transfer.

Another promising direction for future research may lay in the study of the phenomenological components of the process by which students suddenly manage to flexibly recategorize a given problem. Indeed, recent developments in the insight literature have brought forth new venues to coin the subjective experience of learners going through "Aha!" moments (e.g., [START_REF] Creswell | Visceral states call for visceral measures: verbal overshadowing of hunger ratings across assessment modalities[END_REF][START_REF] Laukkonen | How to detect insight moments in problem solving experiments[END_REF][START_REF] Webb | Insight is not in the problem: Investigating insight in problem solving across task types[END_REF]. "Aha!" experiences are thought to play a crucial role in helping students overcome their difficulties in learning mathematics (Liljedahl, 2007), and they are said to improve learning and memorization in general [START_REF] Kizilirmak | Generation and the subjective feeling of "aha!" are independently related to learning from insight[END_REF]. Thus, identifying which training conditions lead to these feelings of insight may bring converging evidence regarding the relevance of similar school interventions based on flexible categorization.

Finally, this study highlights the importance of cognitive flexibility as a central lever of school learning, even outside of task-switching situations. As such, characterizing the different components of cognitive flexibility may be a key step in designing more effective school interventions. Indeed, while it may be argued that attentional flexibility has but a limited influence in class, the oft-forgotten representational flexibility appears to play a major part in the identification of problems' deep structure. This component of cognitive flexibility relying on flexible categorization seems to be a strong contender for school interventions aiming at improving near and far transfer.

Fig 1 .

 1 Fig 1. Students' performance in the solving task, before and after the intervention, depending on the training group.

Table 1 :

 1 Sample problem statements belonging to the 4 relevant structures and presented in 4 different contexts

	Multiplication	First type division	Second type division	Direct proportion
	In 1 box there are 107	I have 1 box of sweets to	In 1 box I have 107 identical	I fill 2 boxes with 30 sweets.
	sweets.	share between 24 people.	sweets.	I have 48 boxes.
	I have 253 identical boxes.	It contains 72 pieces.	I have 749 sweets in total.	How many sweets do I need
	How many sweets do I have	How many sweets will each	How many boxes can I fill?	to fill them?
	in total ?	person get?
	I buy 1 bouquet of roses for	I buy 1 bouquet of 120	1 bouquet contains roses that	I buy 3 roses in a shop.
	each of my 3 children.	roses.	cost €3 each.	It costs me €12.
	The bouquet costs 17 euros.	I pay €276.	I pay €69.	How much do I have to pay
	How much should I pay?	How much does each rose	How many roses are there?	if I want 98 roses?
		cost?
	I have 1 photo album with	I have 1 photo album.	In 1 album, I put 14 photos	I organize my photos in 3
	13 pages.	Across 21 identical pages, I	per page.	albums.
	I can store 18 photos per	have a total of 126 photos.	I have 392 photos.	They can fit 147 photos in
	page.	How many photos are there	How many pages are	total.
	How many photos can I	on each page?	required to store them?	How many albums are
	store?			required for 343 photos?
	In my school, 1 pupil	In my class 1 pupil receives	For 1 row of tables, I hand
	receives 11 notebooks.	4 notebooks.	out 13 notebooks.
	I have 228 pupils.	I hand out a total of 96	I have 91 notebooks.
	How many notebooks will I	notebooks.	How many rows can I
	hand out?	How many pupils do I	complete?
		have?

Acknowledgements

This research was supported in part by a research grant (CYIn-AAP2021-AmbEm-0000000026) attributed by CY Initiative. We are grateful for the stimulating collaboration with schools Pierre Brossolette, Maurice Ravel, Jacqueline Auriol, Jules Ferry, and Paul Raoult, as well as for the helpful contributions of school teachers Frédérique Rouillé, Romain Grosjean, Rémy Boutrand, Sylvain Chebbance, Ivonig Thepot, Myriam Ibodé, Thomas Morel, Arnaud Morandi, François Alexandre Verger, Sandrew Calixto and all their students.


Vincent Cuzuel
email: vincentcuzuel@espci.fr

Audrey Sizun

Guillaume Cognon

Isabelle Rivals

François Heulard

Didier Thiébaut

Jérôme Vial

Human odor and forensics. Optimization of a comprehensive two-dimensional gas chromatography method based on orthogonality: how not to choose between criteria

Keywords: Comprehensive two-dimensional gas chromatography, orthogonality, desirability analysis, forensic, human hand odor

I) Introduction

Profiling human hand odor is a topic of current interest because it would be a considerable breakthrough in forensics, bringing more convincing evidence to magistrates than that of dogs alone. To give more relevance to the information obtained from dogs, there is a need to develop a better understanding of human odor. Indeed, relying only on animals whatever their capabilities is not suitable in courts: identification provided by dogs can merely be used as a corroborative evidence. Thus, the development of an objective analytical strategy to characterize human odor may be a promising complementary tool to the use of trained dogs [START_REF] Curran | Analysis of the uniqueness and persistence of human scent[END_REF][START_REF] Cuzuel | Origin, analytical characterization and use of human odor in forensics[END_REF]. Hundreds of molecules are involved in human hand odor. To our knowledge however, two-dimensional (2D) techniques have not yet been used to characterize living human odor though comprehensive two-dimensional gas chromatography coupled with mass spectrometry (GC×GC-MS) may provide novel information regarding the complexity of the samples.

Like fingerprints, human odor is said to be specific of each individual [START_REF] Cuzuel | Origin, analytical characterization and use of human odor in forensics[END_REF]. In 2014, de Lacy Costello et al. [START_REF] De Lacy Costello | A review of the volatiles from the healthy human body[END_REF] published a comprehensive review of the volatiles emanating from a healthy human body. Different groups of chemical compounds were found in the headspace of biological samples: acids, alcohols, aldehydes, esters, hydrocarbons, ketones, heterocyclic compounds [START_REF] Brown | Applicability of emanating volatile organic compounds from various forensic specimens for individual differentiation[END_REF] and even sulfur-containing compounds in armpit samples [START_REF] Hasegawa | Identification of new odoriferous compounds in human axillary sweat[END_REF]. Considering the nature of the molecules involved, gas chromatography coupled with mass spectrometry (GC-MS) seems to be the most appropriate analytical technique [START_REF] Pandey | Human body-odor components and their determination[END_REF]. However, classic onedimensional (1D) GC-MS showed its limitations in term of peak capacity considering the number of compounds involved: using GC×GC-MS appeared natural to overcome the complexity of the samples.

The first thing to consider when developing a 2D method is the choice of the column set, and orthogonality is often used to characterize the performance of a 2D separation. A system is said to be orthogonal if the first and second dimensions columns are statistically independent. But the independence of the two retention mechanisms might not be sufficient to provide a better separation. Indeed, any change of selectivity between first and second dimension provides always a better separation than that obtained in 1D, but for an efficient separation, the surface coverage of the 2D chromatogram by the spots (compounds) also has to be homogenous. The evaluation of the quality of the separation is still in the center of many debates and there are many criteria for its evaluation, some of them characterizing the absence of correlation or independence, others the occupation of the separation space, and still others trying to combine the two. Schure and Davis recently compared several criteria used for the evaluation of 2D separations [START_REF] Schure | Orthogonal separations: Comparison of orthogonality metrics by statistical analysis[END_REF]: the choice of a criterion appears to be tough, each of them having its advantages and limitations. Though this paper aims neither at reviewing all criteria described in the literature, nor at comparing their performance, we propose here a short overview of the most representative ones.

Correlation coefficients (Pearson's, Kendall's and Spearman's) were used to evaluate the correlation between the retention times of the two dimensions of a 2D separation [START_REF] Al Bakain | Comparison of different statistical approaches to evaluate the orthogonality of chromatographic separations: application to reverse phase systems[END_REF][START_REF] Van Gyseghem | Determining orthogonal and similar chromatographic systems from the injection of mixtures in liquid chromatography-diode array detection and the interpretation of correlation coefficients color maps[END_REF][START_REF] Whelan | Study of the selectivity of reversed-phase columns for the separation of polycarboxylic acids and polyphenol compounds[END_REF][START_REF] Zhu | Effect of column combinations on two-dimensional separation in comprehensive two-dimensional gas chromatography: Estimation of orthogonality and exploring of mechanism[END_REF]. Al Bakain et al. showed that Kendall's coefficient was a good indicator of orthogonality: contrary to Pearson's coefficient, Kendall's is not sensitive to extreme values and provides a good measure of nonlinear monotonous association [START_REF] Al Bakain | Comparison of different statistical approaches to evaluate the orthogonality of chromatographic separations: application to reverse phase systems[END_REF]. However, Gilar et al. stated that correlation coefficients were not well suited for measurement of chromatographic orthogonality, especially for systems with clustered data and outliers [START_REF] Gilar | Comparison of orthogonality estimation methods for the two-dimensional separations of peptides[END_REF]. Slonecker et al. proposed two criteria based on informational theory, the informational similarity and the percentage of synentropy, both based on the mutual information of the system [START_REF] Slonecker | Informational orthogonality of two-dimensional chromatographic separations[END_REF], which is a measure of independence.

Other descriptors rather characterize the distribution of the peaks in the 2D space, using for example nearest neighbor distances (NND) between peaks. One can either sum distances, as discussed by Clark and Evans [START_REF] Clark | Distance to nearest neighbor as a measure of spatial rela-tionships in populations[END_REF], or squared distances, as discussed by Brown and Rothery [START_REF] Brown | Randomness and local regularity of points in a plane[END_REF]. Nowik et al. also used a NND approach in their work [START_REF] Nowik | Assessment of two-dimensional separative systems using nearest-neighbor distances approach. Part 1: Orthogonality aspects[END_REF][START_REF] Nowik | Assessment of two-dimensional separative systems using the nearest neighbor distances approach. Part 2: Separation quality aspects[END_REF]; the arithmetic mean of the NND gives a good indication of peak spreading all over separation space, but is strongly impacted by long distances. The authors also used the geometric and harmonic means which are descriptors of homogeneity and are also sensitive to short distances. They concluded that the harmonic mean of the NND best characterizes orthogonality. Schure et al. proposed a measure of retention time spacing based on a power law fractal distribution [START_REF] Schure | The statistical overlap theory of chromatography using power law (fractal) statistics[END_REF]. The boxcounting dimension is a measure of inter-peak spacing scales and, indirectly, of how effectively the separation space is used [START_REF] Gilar | Comparison of orthogonality estimation methods for the two-dimensional separations of peptides[END_REF][START_REF] Schure | The statistical overlap theory of chromatography using power law (fractal) statistics[END_REF]. The homogeneity of the surface coverage can also be more directly characterized by discretizing the 2D space in bins and evaluating the ratio between number of occupied bins and total number of bins [START_REF] Al Bakain | Comparison of different statistical approaches to evaluate the orthogonality of chromatographic separations: application to reverse phase systems[END_REF][START_REF] Gilar | Comparison of orthogonality estimation methods for the two-dimensional separations of peptides[END_REF][START_REF] Gilar | Orthogonality of separation in twodimensional liquid chromatography[END_REF]. This method can be applied either on the original retention times, or on values normalized between the smallest and the largest one: in the latter case, the unused separation space on the borders is not taken into account. Leonhardt et al. compared the compound distribution in each dimension to the distribution of a fully orthogonal system to evaluate clustering [START_REF] Leonhardt | A new method for the determination of peak distribution across a two-dimensional separation space for the identification of optimal column combinations[END_REF]. Ryan et al. measure directly the ratio between the area occupied by the spots and the unused separation space beneath the second dimension [START_REF] Ryan | Orthogonality considerations in comprehensive twodimensional gas chromatography[END_REF]. The area of occupied space can also be estimated by a rectangular form. However, Semard et al. pointed out that a rectangle is too coarse to describe a complex geometry, and proposed to estimate the percentage of occupied separation space using Delaunay's triangulation algorithms based on the convex hull [START_REF] Semard | Convex hull: A new method to determine the separation space used and to optimize operating conditions for comprehensive two-dimensional gas chromatography[END_REF]. Rutan et al. also used the minimum convex hull to determine a fractional coverage metrics characterizing the effective peak capacity [START_REF] Rutan | Fractional coverage metrics based on ecological home range for calculation of the effective peak capacity in comprehensive two-dimensional separations[END_REF]. Another geometric approach consists in the calculation of the peak spreading angle [START_REF] Zhu | Effect of column combinations on two-dimensional separation in comprehensive two-dimensional gas chromatography: Estimation of orthogonality and exploring of mechanism[END_REF][START_REF] Liu | Geometric Approach to Factor Analysis for the Estimation of Orthogonality and Practical Peak Capacity in Comprehensive TwomDimensional Separations[END_REF]. Gilar et al. concluded that surface coverage and box-counting dimension methods offer useful and intuitively understandable measures of orthogonality. These methods are robust enough for small samples commonly used in chromatographic practice [START_REF] Gilar | Comparison of orthogonality estimation methods for the two-dimensional separations of peptides[END_REF].

In order to take both independence of the retention times and coverage of the separation space into account, Zeng et al. proposed a criterion based both on the bin coverage percentage and Pearson's linear correlation coefficient between the retention times [START_REF] Zeng | A modeling approach for orthogonality of comprehensive two-dimensional separations[END_REF]. In order to account for a large variety of nonlinear relationships, Mani-Varnosfaderani et al. adapted this method by replacing Pearson's correlation by the maximal information coefficient [START_REF] Mani-Varnosfaderani | Assessment of the orthogonality in twodimensional separation systems using criteria defined by the maximal information coefficient[END_REF]. Jacova et al. evaluated different orthogonality assessment tools in GC×GC [START_REF] Jacova | Evaluation of orthogonality assessment tools[END_REF], and using simulation and statistics, they concluded that the geometric approach using a number of bins equal to the number of compounds [START_REF] Al Bakain | Comparison of different statistical approaches to evaluate the orthogonality of chromatographic separations: application to reverse phase systems[END_REF][START_REF] Gilar | Comparison of orthogonality estimation methods for the two-dimensional separations of peptides[END_REF] or the NND approach with the use of the arithmetic and the harmonic means [START_REF] Nowik | Assessment of two-dimensional separative systems using nearest-neighbor distances approach. Part 1: Orthogonality aspects[END_REF][START_REF] Nowik | Assessment of two-dimensional separative systems using the nearest neighbor distances approach. Part 2: Separation quality aspects[END_REF] were the best ones. This short survey shows the extreme diversity of the criteria for the evaluation of 2D separations: considering a single criterion is not an option anymore [START_REF] Cordero | Comprehensive two-dimensional gas chromatography in the analysis of volatile samples of natural origin: A multidisciplinary approach to evaluate the influence of second dimension column coated with mixed stationary phases on system orthogonality[END_REF]. This is the reason why, in this paper, nine criteria including six orthogonality criteria were considered for the evaluation of the separation of a synthetic mixture of 80 representative of human hand odor [START_REF] Cuzuel | Origin, analytical characterization and use of human odor in forensics[END_REF]. The criteria were selected for the complementarity of the information they provide. A chemometric tool called desirability [START_REF] Massart | Handbook of Chemometrics and Qualimetrics: Part A[END_REF][START_REF] Vial | Pharmaceutical quality of docetaxel generics versus originator drug product : a comparative analysis[END_REF] using Derringer functions was used to get an accurate overview of results [START_REF] Derringer | Simultaneous optimization of several response variables[END_REF]. Its optimization led to the best compromise between the criteria and provided an overall quality score, enabling to rank the different column sets objectively.

II) Statistical analysis 1) Criteria

Six orthogonality criteria were selected: Spearman's correlation coefficient, the arithmetic and the harmonic means of the NND, the coefficient of orthogonality calculated with the Camenzuli's Asterisk equations [START_REF] Camenzuli | A new measure of orthogonality for multidimensional chromatography[END_REF], and the calculation of the separation space coverage with both normalized and raw data. The quality of the separation was also evaluated with three additional criteria: the number of coelutions, the number of detected compounds and the run time. The latter criteria and the reason why they were chosen will be detailed in the present section.

Let n be the number of detected compounds of the chromatogram, and 𝑡 ! and 𝑡 " the retention times of the first and second dimension respectively. Let 𝑡

#$% ! , 𝑡 #&' ! , 𝑡 #$%
" and 𝑡

#&' " be the retention times of the first and the last eluted compounds in each dimension. The normalized retention times of the first and second dimension 𝑡

%()# ! and 𝑡 %()#
" are calculated as follows:

𝑡 %()# $ = * ! + * "!# ! * "$% ! + * "!# ! (1)
.

Spearman's correlation coefficient

If the peaks are well-spread all over the 2D separation space, the first and second dimension retention times should not be correlated. Thus, correlation coefficients can be used to evaluate orthogonality [START_REF] Al Bakain | Comparison of different statistical approaches to evaluate the orthogonality of chromatographic separations: application to reverse phase systems[END_REF], with the advantage that they do not require the discretization of the separation space, nor the normalization of the retention times, and can be calculated easily.

Spearman's correlation coefficient ρ provides a measure of monotonous nonlinear correlations between retention times. The compounds are ranked according to their retention time in the first and second dimensions. Let (𝑟 $ *! , 𝑟 $ *" ) denote the ranks of a pair of coordinates of compound i , di = ri t1 -ri t2 , and n the number of compounds, then:

𝜌 = 1 - 6 ∑ 𝑑 $ " % $,! 𝑛(𝑛 " -1) (2)
Spearman's correlation coefficient is preferred to Pearson's which is not suited for the measure of nonlinear associations and is sensitive to extreme values.

Nearest Neighbor distance approach

A recent approach was proposed by Nowik et al. to assess the orthogonality of 2D separative systems using NND [START_REF] Nowik | Assessment of two-dimensional separative systems using nearest-neighbor distances approach. Part 1: Orthogonality aspects[END_REF]. The NND is defined as the shortest of all distances 𝑙 $ of point i to all other points. All calculations were performed with normalized retention time, so that only the time window effectively occupied by the compounds, which is usually narrower than the whole separation space, is considered. Three different descriptors were defined by Nowik et al.: the arithmetic mean 𝐴 ̅ (() , the geometric mean 𝐺 ̅ (() and the harmonic mean 𝐻 5 (() .

𝐴 ̅

(() = ∑ 0 ! #&' !(' %+! (3) 𝐺 ̅ (() = (∏ 𝑙 $ %+! $,! ) ' #&' (4) 𝐻 5 (() = %+! ∑ ' ) ! #&' !(' (5)
The arithmetic and the geometric means are strongly influenced by long distances and give information about peak spreading, while the harmonic mean is more impacted by short distances and therefore gives information about clustering.

Asterisk equations

Camenzuli et al. proposed a new measure of orthogonality for multi-dimensional chromatography [START_REF] Camenzuli | A new measure of orthogonality for multidimensional chromatography[END_REF]. Different equations -known as Asterisk equations -allowed the calculation of 𝐴 1 which gave a measure of orthogonality. All calculations were performed with normalized retention times. Four parameters are calculated to characterize the peak dispersion around four lines. The peak dispersion around each line is compared to that of a uniform distribution. This technique does not require the discretization of the separation space and has the advantage of evaluating the dispersion around four different axes.

Occupied space -geometric criteria

This geometric criterion was used as an indicator of the coverage of the separation space and hence of orthogonality [START_REF] Al Bakain | Comparison of different statistical approaches to evaluate the orthogonality of chromatographic separations: application to reverse phase systems[END_REF]. It is impacted by the number of peaks and requires space discretization into 𝑝 𝑥 𝑝 bins. In the present study, the mixture of standards contains 80 compounds. A value of p=9 was chosen to have a number of bins close to the number of targeted compounds: one compound per bin would lead to a space covering percentage of 98.7%.

This geometric criterion can be calculated using either raw or normalized data. Calculation with normalized data eliminates border areas of the chromatogram, and enables to consider only the relative spatial arrangement of the compounds, whereas working with raw data enables to take the undesired presence of empty areas into account.

Additional separation criteria

Three additional criteria were included for the optimization of the columns set and conditions. Since the developed methods will eventually be used routinely, the run time was taken into account to save time. The number of coelutions as well as the number of detected compounds -compounds of the mixture found on the considered chromatogram -were also evaluated to rank the different columns set. Plotting the number of coelutions versus the number of detected compounds for each analytical setup showed that these two indicators are absolutely not correlated (Pearson correlation coefficient of 6.10 -5 ). Indeed, the modulation system being a dual-jet cryomodulator applied to the second dimension column -on which the modulation is done -some compounds can be subject to an echo phenomenon. A cold-jet default of trapping can lead to the apparition of twin peaks separated by half a modulation time in the second dimension. Thus, an increase in the number of coelutions can be observed while the number of detected compounds remains the same.

2) Desirability parameters

Desirability evaluation is a multicriteria decision-making method using Derringer functions [START_REF] Derringer | Simultaneous optimization of several response variables[END_REF], which necessitates the definition of an individual desirability function for each criterion.

In this way, each measured response of interest is converted into a dimensionless number dk between 0 (criterion not satisfied) and 1 (criterion satisfied). The two bounds of the desirability curve will be described by the letters B0 and B1 (cf. figure 1).

The global desirability D of a column set is calculated as the product of the individual desirabilities:

𝐷 = = 𝑑 2 2
The desirability approach allows to get an accurate overview of results in a straightforward manner, as in [START_REF] Vial | Pharmaceutical quality of docetaxel generics versus originator drug product : a comparative analysis[END_REF][START_REF] Varala | New biosorbent based on deoiled karanja seed cake in biosorption studies of Zr(IV): Optimization using Box-Behnken method in response surface methodology with desirability approach[END_REF][START_REF] Bukzem | Optimization of carboxymethyl chitosan synthesis using response surface methodology and desirability function[END_REF] where it was successfully applied for multicriteria optimization.

3) Software MATLAB (Mathworks, Natick, MA, USA) software v. 9.0.0.341360 (2016a) was used to process the data.

III) Material and methods

1) Chemicals and reagents

To optimize the analytical methods, a mixture of 80 standards was prepared in heptane (10 mg/kg each). The detailed composition of this mixture is available as Supplementary Material. Compounds likely to be found in human hand odor were selected [START_REF] Cuzuel | Origin, analytical characterization and use of human odor in forensics[END_REF]. A particular attention was given to cover a wide range of chemical properties (polarity, molecular weight, chemical family).

Individual standards of the 80 targeted compounds were purchased from Sigma-Aldrich (Saint Quentin Fallavier, France). Methanol was purchased from Carlo Erba Reagents (Fontenaysous-bois, France). Ultra-pure water was produced using a Direct-Q UV 3 system (18.2 MΩ/cm) from Millipore (Molsheim, France).

2) Comprehensive two-dimensional gas chromatography and mass spectrometry

Analyses were performed on a Thermo Trace GC×GC coupled with a Thermo ISQ MSD (Thermo, Villebon-sur-Yvette, France). This device was equipped with a Thermo Triplus autosampler.

Helium flow was set at a constant value of 1mL/min. The initial temperature was 40 °C, held for 1 min, then raised to 250 °C, and again held for 1 min, with gradients of 2, 2.5, 3 or 4 °C/min. The modulation was performed with a dual-jet CO2 thermal modulator, and modulation times of 6, 8 or 10 seconds were selected. The mass spectrometer was used with the electronic ionization source (70 eV) heated at 210°C. The acquisition was made in scan mode, with scan range 46-320 amu. A solvent delay of 9 minutes was used. Data were acquired with Xcalibur and processed with ChromCard (Thermo softwares).

3) Columns

Several first dimension columns (30 m x 0.25 mm, 0.25 µm) were tested and purchased from Agilent (Les Ulis, France) to conduct the first chromatographic separation: 100% Dimethylpolysiloxane (DB-1MS), (5%-Phenyl)-methylpolysiloxane (DB-5MS), (14%-Cyanopropyl-phenyl)-methylpolysiloxane (DB-1701), Polyethylene glycol (DB-WAX).

They were coupled with the following second dimension columns (1.5 m x 0.10 mm, 0.10 µm): DB-1, DB-5, DB-1701, DB-WAX columns purchased from Agilent (Les Ulis, France) and a nonbonded 1,12-Di(tripropylphosphonium)dodecane bis(trifluoromethanesulfonyl)imide phase (SLB-IL60, 1m x 0.10 mm, 0.08 µm) purchased from Supelco (Sigma-Aldrich, Saint Quentin Fallavier, France). 8 different column sets with a total of 27 analytical setups (gradient, modulation time) were evaluated (cf. table 1).

4) Head-Space SolidPhase MicroExtraction (HS-SPME)

Given the volatility of most of the targeted compounds, injection was performed using HS-SPME. Such a technique enabled to minimize the solvent peak compared to classical liquid injection. This study was conducted with a focus on the separation and not on the limits of detection. No quantitation was performed and the areas of the different peaks of interest were not used. The SPME fiber used was a 75 µm DiVinylBenzene-Carboxen-PolyDiMethylSiloxane (DVB-CAR-PDMS) from Supelco (Sigma-Aldrich, Saint Quentin Fallavier, France). For HS-SPME procedures, standards were prepared by introducing 10 µL of the mixture of 80 standards in the 20 mL vial. The fully automated HS-SPME procedure was as follows. First, the vial was equilibrated at 70°C during 1 min, then the fiber was placed into the head-space of the sample for the extraction, still maintained at 70°C for 20 min. At the end of the extraction, the fiber was desorbed directly in the GC injector for 20 min at 250°C in split mode (1:10).

IV) Results and discussion

In this paper, 27 different analytical setups were evaluated, an analytical setup being defined by a column set, a gradient and a modulation time (cf. table 1). The optimization is mainly about the column set, but the gradient and the modulation time had to be adapted to each column set to get an enhanced separation, hence the relevance of the multicriteria strategy implemented in this paper. A preliminary study was performed to define which ranges of gradient and modulation time were relevant for each column set. The modulation time should not exceed 10 seconds to avoid overloading the modulator jet, but it has to be long enough to minimize wrapping around. In a first step, all analytical setups were tested with a gradient of 3°C/min and a modulation time of 6 seconds. The eight column sets, the four different gradients (2, 2.5, 3 or 4 °C/min) and the three different modulation times (6, 8 or 10 s) led to a total of 96 different analytical setups, but a first selection was made as some analytical setups were more promising than others. The fact that developed methods could be used for routine analysis had also to be considered. Thus, data were processed with two different objectives. First, criteria were chosen to get the best separation in a reduced run time. The second objectives did not take the run time into account and gave more weight to the quality of the separation.

1) Calculation of individual desirabilities

Prior to data processing, the bounds B0 and B1 were set (cf. table 2). The arithmetic mean of NND provided information about peak spreading and extreme points, while the harmonic mean provided a complementary information about peak clustering. In Nowik et al., the orthogonality of several data sets evaluated with the harmonic mean ranged from 0.017 to 0.054, a value of 0.047 being considered satisfactory. Hence the bounds B1 and B0 were set to 0.05 and 0.045 respectively. For the arithmetic mean, the values were in the interval [0.040 ; 0.084], hence our choice of the bounds B1 and B0 at 0.08 and 0.05 respectively. When the run time was considered, analyses longer than 80 min (B1) were put at disadvantage while analyses which exceeded 150 were automatically discarded (B0). As for the orthogonality coefficient, the bounds B0 and B1 were set to 50 and 80% respectively. Concerning the occupied space, B0 and B1 were set to 20 and 45% with raw data and to 30 and 55% with normalized data.

Then, the individual desirabilities were calculated. Figure 2 shows the individual desirability function of each criterion (blue line); the data of each analytical setup were plotted as red crosses. Data spreading is different for each criterion showing the complementarity of the selected criteria. Though data are clustered with the arithmetic mean, the use of the bounds defined prior to the study (cf. table 2) was sufficient to discriminate between analytical setups and make this criterion useful. With the Spearman correlation coefficient, the majority of data receives complete fitness value, and only small numbers of analytical setups receive 0.5 or some degrees between 0.5 and 1. These clusters show that a single criterion is not be sufficient to rank the analytical setups properly. However, most criteria showed good spreading of data.

Finally, the calculation of the global desirability was performed with and without the time constraint, leading to two different rankings.

2) Computation of the global desirability 2.1 Optimization for routine analyses with run time

Considering the run time, the calculation of occupied space with raw data was preferred to the calculation with normalized data in order to account for possibly empty places at the borders of the chromatograms. For instance, space can be lost at the end of the chromatogram with too slow ramps of temperature. Moreover, the use of a column with non-adapted polarities can lead to a loss of space covering which is not taken into account using normalized data.

The quality of the separation was also evaluated with the number of detected compounds and the number of coelutions.

Optimization regardless of run time

In this part, the objectives were to get the best separation regardless of run time (cf. table 2). Thus, we chose to work with normalized data -as defined in Statistical Analysis -in order not to consider empty space at the borders of chromatograms. Here, the quality of the separation was favored over the run time. In this way, a satisfying peak scattering -even if on a limited area of the chromatogram -was preferred. According to these choices, the number of detected compounds and the number of coelutions were also used to evaluate the performance of the separation.

3) Ranking of analytical setups

The exclusion of the run time criterion played a role but was shown not to be essential since the same column set DB-1MS×DB-1701 -even with different analytical setups -appeared 3 times in the top 5, whatever the objective (cf. table 3). The column set DB-1MS×DB-1701 emerged as the most appropriate choice for the separation of the mixture of 80 compounds (cf. chromatograms shown figures 3 and 4).

The column setup WAX×1701:3-6 ranked as fourth when run time was ignored. DB-WAX was the most polar column at our disposal but was quickly discarded. Combined with a DB-1MS or with a DB-5MS, it was expected to provide quite good results, because of the difference in polarities. But the bleeding proved to be a true limitation for its use as a first dimension column (cf. figure 5). When used as the second dimension column, an inacceptable echo phenomenon occurred which had a strong impact on several analytical setups (1MS×WAX:3-6, 5MS×WAX:3-6) evaluated in this study. The set DB-WAX×DB5 did not provide convincing results but one of its analytical setups (WAX×5:3-6) was kept to see if it could give interesting results anyway.

Prior to the experiments, only slight differences between the DB-1MS and the DB-5MS were expected. But it is interesting to notice that most of the setups with a DB-5MS in first dimension were discarded because of coelutions that were resolved using the DB-1MS. Thus, only four setups with the DB-5MS×DB-1701 set were evaluated.

The use of an ionic liquid column (SLB-IL60) was expected to provide an enhanced separation combined with a DB-1MS -or a DB-5MS -as its polarity is higher than DB-WAX's. However, the preliminary tests showed it was not relevant for this study because of the dual-jet cryomodulator. This column may be far too polar and the echo phenomenon was important all over the chromatogram, thus invalidating the assumption of a phase modification due to the cryomodulation. Moreover, even when the length was reduced from 1.5m to 1m, lots of compounds kept on wrapping around even with the largest modulation time, leading to broaden peaks. These two phenomena lead to eliminate this set because it was impossible to identify compounds properly. With another modulation system -flow modulator or nitrogen cryomodulator -the results might have been different. Another hypothesis is that the problem is related to the analyte elution temperature from the first dimension: using a more polar column might increase the analyte elution temperature, and, thereby, speed-up the elution in the second dimension, thus avoiding wrap around. Using desirability is a true asset in this case because the individual desirabilities of most criteria are acceptable for the 5MS×IL60:2-6 analytical setup. Considering that a lot of compounds were not found, the number of peaks was low. But in this case, they were well scattered, leading to good results -according to the desirability bounds -with the arithmetic mean, the harmonic mean, the correlation coefficients and the orthogonality coefficient. Only the calculation of the surface coverage is of particular interest here because the number of bins was chosen according to the number of expected compounds. It gave interesting information about clustering, peak spreading and took the number of peaks into account.

In the end, the same column set DB-1MS×DB-1701 with a gradient of 2.5°C/min -2°C/min if the run time is not considered -and a modulation time of 8 seconds appeared to be the best analytical setup for the separation of the synthetic mixture of 80 compounds representative of human hand odor. Moreover, this method was successfully applied to real samples and provided good separations meeting our expectations, results which are the topic of a future paper focusing on the sampling methods.

V) Conclusion

The use of comprehensive two-dimensional gas chromatography coupled with mass spectrometry would be a true asset for the analysis of human odor. The evaluation of multidimensional separation is not an easy task and various tools are available to assess the orthogonality of the separation and choose a proper column set. A desirability analysis allowed to consider more than one criterion and to rank the different column sets according to predefined objectives. In this paper, 27 different analytical setups were evaluated using nine criteria including six complementary orthogonality criteria: Spearman' correlation coefficient, the harmonic and arithmetic means described by Nowik et al., the coefficient of orthogonality described by Camenzuli et al. and the coverage of the separation space using raw and normalized data. The use of desirability functions enabled to consider additional simple but meaningful criteria such as the run time, the number of coelutions and the number of detected compounds. Two desirability calculations were performed, one with an objective of routine analysis and one without. Whatever the objectives, the same column set DB-1MS×DB-1701 was found to be the most appropriate for the analysis of human hand odor. This approach for the evaluation of orthogonality can be easily implemented with other criteria (convex hull, synentropy, etc.) according to the operator's choice, i.e. considering the targeted compounds and the application. The methods developed in this study are a solid background for further experimentation and sampling methods development. They will hopefully be used routinely with a large number of real samples for the analysis of human hand odor to support the information provided by dogs in courts of justice.

Figures and tables captions, in order of appearance

Figure 1 :

 1 Figure 1: Example of a desirability plot

Figure 2 :

 2 Figure 2: Individual desirabilities plot

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Comprehensive GC×GC-MS chromatogram -mixture of 80 standards -DB-1MS×DB-1701 -modulation time: 8s -gradient: 2.5°C/min (detailed conditions in the experimental part)

Table 1 :

 1 Analytical setups

Table 2 :

 2 Bounds used for individual desirability calculations


Le groupe expérimental a bénéficié du dispositif mis en place dans l'étude, tandis que le groupe contrôle a, quant à lui, suivi les leçons d'une séquence classique d'apprentissage de la proportionnalité telles que prévues dans les programmes nationaux.

III -Résultats

Les analyses des résultats montrent qu'au pré-test les élèves des deux groupes catégorisent les problèmes uniquement selon leurs indices de surface. Ils ont, en outre, des taux de réussite en résolution de problèmes quaternaires relativement faibles (GE=29,73 % ; GC=29,67 %. Les différences entre les deux groupes ne sont pas significatives [t(137)= .021 ; p= .978)]. On observe cependant de réelles différences en fonction des types de problèmes. Les problèmes multiplicatifs, par exemple, ont un taux de réussite de 48,65 % (GE) et 50,00 % (GC) tandis que les problèmes de proportionnalité ont un taux de 3,38 % (GE) ou de 3,85 % (GC).

Au post-test, en revanche, les élèves du groupe expérimental ont plus fréquemment catégorisé les problèmes selon leurs structures profondes que les élèves du groupe contrôle qui n'ont eu recours qu'aux indices de surface. De plus, les taux de réussite en résolution de problèmes sont plus élevés dans le groupe expérimental (58,02 %) que dans le groupe contrôle (44,39 %). Cette différence est significative [t(114) = 2,505 ; p = .014)]. Cette fois encore, de nettes différences de difficultés sont observées selon la structure des problèmes : 78,36 % (GE) contre 60,20 % (GC) pour les problèmes multiplicatifs et 35,07 % (GE) contre 15,31 % (GC) pour les problèmes de proportionnalité.

IV -Discussion

  de notre étude est d'évaluer -en adoptant un paradigme pré-test/post-testl'efficacité d'un dispositif d'apprentissage fondé sur le recours à la catégorisation comme processus de stimulation de la flexibilité cognitive en résolution de problèmes mathématiques chez des élèves de Cours Moyen 2 ème année (âge moyen = 11, 4 ans ; SD = 0,45). À la suite de travaux menés sur les processus flexibles de découverte de solutions (Clément, 2021), nous avons émis l'hypothèse que stimuler la flexibilité cognitive des élèves en les incitant à envisager un même problème sous différents points de vue aura un effet bénéfique sur la compréhension et la résolution de problèmes de proportionnalité. En entraînant les élèves à comparer et à catégoriser des problèmes analogues partageant, ou non, des similitudes de surface, nous souhaitons leur permettre d'identifier les structures profondes des énoncés en ignorant les effets de contenus rapportés dans la littérature (e.g Gros, Thibaut & Sander, 2021). Afin d'évaluer la pertinence de cette méthode pédagogique nous avons constitué un groupe expérimental et un groupe contrôle d'élèves issu•e•s d'un même réseau d'éducation prioritaire renforcé et nous avons mesuré leurs progressions sur la base d'évaluations auxquelles ils ont été soumis avant et après les séances d'apprentissage. II -Matériel et méthode -Participant•e•s Cette expérimentation a été menée au mois de juin 2021 auprès d'élèves de Cours Moyen 2 ème année (âge moyen = 11, 4 ans ; SD = 0,45) scolarisé•e•s dans cinq écoles d'un réseau d'éducation prioritaire renforcé (REP+) en région parisienne. Le groupe expérimental (GE) comprenait 77 élèves, dont 40 filles, tandis que le groupe contrôle (GC) était composé de 70 élèves, dont 39 filles. Ces élèves sont issu•e•s de catégories socioprofessionnelles défavorisées. Selon l'indice de position sociale de ces écoles (77,5) la mixité sociale y est très faible. -Les épreuves de Pré et Post-tests. Afin d'évaluer l'efficacité de notre dispositif éducatif, les élèves des deux groupes ont été soumis à des évaluations réalisées avant (pré-test) et après (post-test) la mise en place du dispositif. Nous avons ainsi élaboré un corpus de 16 problèmes construits selon les quatre structures de problèmes quaternaires (Vergnaud, 1983) : la multiplication, la division-partition, la division-quotition et la proportionnalité. Chacun de ces types de problèmes était présenté dans quatre contextes différents décrivant soit des paquets de bonbons, soit des bouquets de fleurs, soit des albums-photos, soit des cahiers. Il était demandé aux élèves de catégoriser les problèmes puis de résoudre 8 de ces problèmes. En outre, lors du post-test, nous avons demandé aux élèves d'inventer un problème correspondant à chacune des quatre structures étudiées. -Le dispositif éducatif Chacune des 8 leçons de la séquence pédagogique présentait les caractéristiques de problèmes ayant des structures différentes et décrivant des contextes identiques ou des problèmes partageant la même structure, mais des contextes différents. Nous engagions ensuite les élèves dans l'analyse des énoncés afin de les comparer et d'identifier les similitudes superficielles ou structurelles ainsi que leurs différences. Enfin, chaque problème devait être résolu par les élèves. Un temps de correction collective suivait le temps de travail individuel. Un moment créatif d'invention d'énoncés de problèmes quaternaires était réservé à la fin de chaque séance. -Procédure Notre expérimentation s'est déroulée en quatre phases, une évaluation en pré-test suivie de quatre séances d'apprentissage, une évaluation intermédiaire de performances en résolution de problèmes, quatre autres séances d'apprentissage et une évaluation en post-test. Chaque séance avait une durée de 55 minutes.

  Évaluer l'efficacité d'une intervention éducative impose, non pas uniquement de mesurer les progrès des élèves, mais également d'estimer dans quelle mesure les apprenant•e•s exposé•e•s à cette démarche progressent par rapport à celles et ceux ayant suivi un enseignement plus classique. C'est ce que permet le recours au paradigme expérimental pré/post test. Le fait que l'on n'observe pas, lors du pré-test, de différences significatives entre le groupe expérimental et le groupe contrôle nous permet d'évaluer, à la fois, les progrès des élèves, mais également le rôle de l'intervention pédagogique mise en place dans les classes. Ainsi, nos résultats montrent que l'élaboration d'une séquence de résolution de problèmes de proportionnalité fondée sur les processus de comparaison et de catégorisation est plus efficace pour permettre aux élèves, en développant leur flexibilité cognitive, d'être capables de reconnaître la structure profonde d'un problème et, en conséquence, de le résoudre plus efficacement. Nos résultats suggèrent, en effet, qu'entraîner des élèves scolarisé•e•s en REP+ à catégoriser des problèmes en traitant leur structure profonde induit de meilleures performances de résolution. Cela conforte l'idée qu'un enseignement fondé sur la prise en compte des représentations initiales ainsi que sur le changement de point de vue et l'identification des structures profondes est une piste prometteuse pour aider les élèves à conceptualiser des notions mathématiques qui peuvent, parfois, être difficiles à appréhender.

Mots clés

Évaluation. ; Résolution de problèmes ;