F Signol

C Barras

J.-S Lienard

Evaluation of the Pitch Estimation Algorithms in the monopitch and multipitch cases

Reliably tracking the fundamental frequency F 0 of the components is an important step in the separation of superimposed speech signals. Several Pitch Estimation Algorithms (PEAs) are potentially usable and a rigorous evaluation method is needed. However, even in the monopitch case, many variations between them render such a comparison difficult. The F 0 min-F 0 max interval extent, the use of a priori information on the whole sequence or database and above all the arbitrary voicing threshold setting lead to large differences in the results. These biases can be removed by setting the F 0 bounds to fixed values acceptable for many voices, by proceeding with the evaluation on a strictly frame-to-frame basis, and by fixing the voicing threshold in order to get an equal error rate for overvoiced and undervoiced frames. In the multipitch case any frame may exhibit 0, 1 or 2 valid voicing according to the coincidence between the voiced and unvoiced parts of both signals. This problem is treated by defining a metric linking the PEAS's hypotheses to the pitch values of the isolated signals. The proposed methodology is applied to several PEAs on several databases in the monopitch case.

Introduction

In the last five decades, numerous PEAs have been developed so that a proper evaluation is mandatory. Most PEAs process the input speech signal as successive shorttime frames. A F 0 estimate is produced for each voiced frame. For unvoiced frames the F 0 estimate is replaced by a zero value. This involves two distinct processes: Voiced/Unvoiced (VuV) frame decision and F 0 estimation. In some cases those processes are performed sequentially.

In [START_REF] De Cheveigné | YIN, a fundamental frequency estimator for speech and music[END_REF][START_REF] Camacho | SWIPE: a Sawtooth Waveform Inspired Pitch Estimator for Speech and Music[END_REF] the first step is the computation of a voicing criterion (typically between 0 and 1). Then this voicing criterion is compared to a threshold and the Voiced/Unvoiced (VuV) decision is taken. This is a typical sequential frame-to-frame approach. In some other cases the two processes are not distinct and sometimes they are mutually dependent. For instance in [START_REF] Boersma | Accurate Short-term Analysis of the Fundamental Frequency and the Harmonics-to-noise Ratio of a Sampled Sound[END_REF][START_REF] Secrest | An integrated pitch traking algorithm for speech systems[END_REF] several F 0 candidates are selected for each frame and a dynamic programming algorithm is used to select the globally optimal sequence of candidates. In this case, the F 0 estimation influences the VuV decision and vice-versa.

Eventually evaluating a PEA means simultaneously evaluating the two above processes: VuV decision and F 0 estimation.

PEAs evaluation requires a database, VuV references, F 0 references and quality measures. Thus several variability sources should be addressed, which means numerous features to set and control. First of all, one has to control the database specificities such as the speech recording conditions (noise level, anechoic room, telephone, car...) or the type of speech (read, broadcast news, conference, spontaneous, expressive...). The reference voicing and F 0 labeling has to be perfectly reliable. It may differ according to the protocol used (manual, fully automatic or manually corrected); it may come from the analysis of the speech signal itself, or from an electroglottographic waveform (EGG) recorded simultaneously. The VuV decision may have been taken on the basis of the experience of trained phoneticians, or from the examination of a waveform on physical criteria, which may be quite different. Finally each PEA's specificity should be taken into account. It may or may not offer the operator the capability of disabling some pre-or post-processing, or of modifying the main parameters (frame duration, frame hop, voicing threshold, lower and upper bounds of F0 estimation, number of F0 candidates per frame). For all the above reasons, it is extremely difficult to perform a fair comparison between algorithms.

In this paper we propose an evaluation methodology that could help containing some of the main sources of variability. Section 2 presents what has been done in PEA's evaluation, highlights how the VuV decision is problematic to evaluate and proposes a way to deal with it. The usual PEA use is the monopitch case where it has to deal with a one-speaker speech signal. Section 3 proposes a new evaluation methodology and illustrates it on a database in the monopitch case. Section 4 extends our monopitch evaluation approach to the multipitch case, in which several voices may be mixed in a single signal.

The role of the VuV decision in the evaluation of F 0 estimation

The first quality measure for which a consensus in literature exists is the quality of the pitch estimation. What is generally reported is a gross error rate (GER). A gross error is raised when the pitch estimate lies more than a certain distance away from the reference (typically 20%). This rather high threshold has to be put in perspective with the typical pitch errors which are mainly octave or sub-octave errors and most seldom third, triple or two thirds errors [START_REF] Liénard | Using sets of combs to control pitch estimation errors[END_REF].

The gross error rate does indeed capture the principal harmonic or sub-harmonic errors.

F 0 estimation is performed on the frames which are considered as voiced. In that sense, the GER calculation is subjected to the VuV decision. However, the voicing state may be ill-defined for many frames, often located at the bounds of the voiced speech segments. In those regions F 0 estimation may remain unsteady, whereas for frames where the voicing is strong, the F 0 estimation is quite reliable. For any PEA, avoiding the litigious frames may drastically reduce the F 0 error rate. This phenomenon is illustrated on Fig. 1, which shows that the GER decreases and falls to 0% when the VuV threshold is increased so that only the reliably voiced frames are taken into account. This example was performed on a short sample of speech with the YIN algorithm, but the same trend can be observed with any algorithm, on any speech excerpt.

Therefore, for a given PEA the GER calculation should not be presented alone. It should be linked with an evaluation of the VuV decision.

The results published in [START_REF] Camacho | SWIPE: a Sawtooth Waveform Inspired Pitch Estimator for Speech and Music[END_REF] are established on the frames declared as voiced by all PEAs involved in the evaluation. Thus they discard from the statistics the frames whose voicing is not reliably established. This protocol may reflect the intrinsic quality of the algorithm, in dealing with perfectly voiced sounds. But it does not give a fair idea of the PEA's behavior in the difficult situation encountered at both ends of the voiced segments. Thus, we have to normalize the algorithms behaviors in terms of VuV decision.

To formalize our voicing behavior normalization let us look at table 1 which illustrates the kinds of errors that could appear between the VuV state of a reference frame and the VuV state of an hypothesis frame. A false alarm (FA) called overvoiced error occurs when the hypothesis frame is voiced and not the reference frame. A false rejection (FR) called undervoiced error occurs in the opposite situation.

Reference Hypothesis Agreement

UV UV OK UV V FA V UV FR V V OK
Table 1. Full set of reference/hypothesis couples and their agreement result.

Table 1 refers to a typical problem of decision theory. This problem may be addressed within the Equal Error Rate (EER) statistical framework. A way to compare the different PEAs without the bias attached to their specific VuV behavior is to tune it so that the FA/FR ratio gets a constant value. Let EER be the ratio of the overvoiced error rate to the undervoiced error rate. The value chosen for the EER depends on the voicing behavior requested by the application where we want to use a PEA. For instance, if the application is only interested in strongly voiced segments, we should set EER at a value between 0 and 1 in order to get more undervoiced than overvoiced frames. For the present evaluation purpose we shall fix it at the value 1.

Monopitch case

In this chapter we present our evaluation methodology beginning with the database presentation. Then we explain precisely how we build our reference. Then we provide a general formalization of how could be represented a set of reference annotations and F 0 hypotheses. At last, we propose some evaluating features and formalize them in the monopitch case.

Evaluation method

Database

Our mono-speaker database is composed of three speech corpora, all recorded in clean acoustic conditions. The Keele database (noted K) comprises 10 English speakers (5 males, 5 females) reading the same text one time and quite neutrally, for a total duration of 337 seconds. The Bagshaw database (noted B) comprises 2 English speakers (1 male, 1 female). Both are saying the same 50 shorts utterances, for a total of 331 seconds. The Daless database (noted D) comprises 4 French speakers (2 males, 2 females) reading various texts quite neutrally. This database contains 1359 seconds of speech. For all the databases, the authors give the EGG waveform for each speech signal.

To build a 2-speaker mixture, two normalized signals are simply added. Then, the whole 2-speaker mixtures databases are built by forming all possible signals combinations.

Reference annotations

This step needs a particular care because it determines the quality of the evaluation results. It raises the "groundtruth" problem.

Manual or manually corrected reference annotation is a very tedious task. It must be performed by a group of specialized operators to be reliable. However, the protocols and groups of operators differ from one database to the next, so that it may happen that a given algorithm gets different results when evaluated on different databases. This is why we chose to automatize the annotation process, even if we know that it may produce more errors than the manual annotation of a given database. Also, as there are few manually annotated databases, this approach permits the use of a non-annotated database. For instance our largest database is not manually annotated; an automatic annotation procedure was the only way to use it with the same annotation quality than the other two. In any case, a careful examination of each database is needed in order to adapt some parameters such as the F 0 range.

As EGG signals are available for all databases, we generate the VuV decision with them. Although it is not perfect, the EGG waveform is a direct trace of the vocal cords vibration so it should give a more reliable and steady voicing measure than the acoustic speech signal.

The EGG waveform is first shaped by a bandpass filtering between 50 and 1600 Hz to cancel undesired noises and low-frequency components. Then, a positive saturation threshold is computed which corresponds to the amplitude exceeded by 5% of the samples. The same negative saturation threshold is computed with the negative part of the waveform. If the negative saturation threshold's absolute value is bigger than the positive saturation threshold then only the absolute value of the waveform's negative part is kept (negative half-wave rectification). Else a positive half-wave rectification is done. The obtained waveform is normalized at 0.9 and then the temporal envelope is computed by linear interpolation between local maxima. A 18-point histogram is computed from the envelope. The histogram exhibits two peaks, corresponding respectively to the unvoiced and to the voiced segments. The local minimum between these two peaks is chosen as the voicing threshold. Below this threshold the sample is considered unvoiced and above the sample is voiced. A refinement is added to avoid too short voiced or unvoiced segments. All unvoiced segments shorter than a first This algorithm returns a series of voiced segments defined by a beginning time and a finishing time. A frame is voiced if at least half of its width is included in a voiced segment.

To compute the F0 reference value for a voiced frame, a basic and easily reproducible algorithm is used. The EGG signal is windowed. On each window, normalized autocorrelation is computed, its first main lobe is removed and the abscissa of the highest peak is considered as the period of the fundamental for this frame.

This algorithm returns a series of frames described by two values: the frame's time which is the middle instant of the EGG window studied, and the estimated F 0 which is strictly positive if the frame is voiced and 0 else.

Quality measures

Let us define some notations.

We note N S the number of speakers mixed in the speech signal. N R corresponds to the number of F 0 values in each reference frames. N R is equal to N S because there are as much references values as speakers. N H is the number of F 0 hypothesis candidates given by a PEA per frame. N H is superior or equal to N R . We also note N F the total frame's number.

Let us note ∩ the intersection operator, U the union operator, Ω the cardinal operator and | the writing shortcut which signifies "such as".

We define R as the references frames set and R t as the t th reference frame. R t contains the set of F 0 references values denoted R x t . R x t is the x th F 0 reference value in the t th reference frame. The same kind of notation is applied to the hypotheses. H is the set of hypotheses frames. H t is the t th hypotheses frame and H y t is the y th F 0 hypothesis value in the t th hypotheses frame. The range of t, x and y are given in Eq. [START_REF] De Cheveigné | YIN, a fundamental frequency estimator for speech and music[END_REF].

{ } { } H F N N H H H R R R ... ... 1 1 = = and { } { } t N t t t N t t H R H H H R R R ... ... 1 1 = = (1)
An unvoiced F 0 candidate is set to zero. Else, it is strictly positive. We note VC R the set of voiced F 0 references values and uVC R the set of unvoiced F 0 references values. VC H is the set of voiced F 0 hypotheses values and uVC H the set of unvoiced F 0 hypotheses values. A mathematical formulation is given in Eq.( 2).

( )

{ } ( ) { } 0 , 0 , > = > = t y H t x R H t y VC R t x VC and ( ) { } ( ) { } 0 , 0 , = = = = t y H t x R H t y uVC R t x uVC (2)
Concerning the VuV decision, we consider a frame as voiced if at least one of its F 0 values is strictly positive. Thus let us note (R t >0) as a voiced references frame and (H t >0) as a voiced hypotheses frame. We also note (R t =0) an unvoiced references frame and (H t =0) and unvoiced hypotheses frame. These definitions are illustrated in the Eq.( 3).

0 0 0 0 > ∃ ⇔ > > ∃ ⇔ > t y t t x t H y H R x R and 0 0 0 0 = ∀ ⇔ = = ∀ ⇔ = t y t t x t H y H R x R ( 3
)
VF R is defined as the set of voiced references frames and VF H is the set of voiced hypotheses frames. uVF R is the set of unvoiced references frames and uVF H the set of unvoiced hypotheses frames.

{ } { } 0 0 > = > = t H t R H t VF R t VF and { } { } 0 0 = = = = t H t R H t uVF R t uVF (4)
We define two series of quality measures: the first one deals with the VuV decision, and the second one deals with F 0 estimation.

In the monopitch case, N R and N S are equaled to 1 and the number of references frames is the same than the number of F 0 references values.

The overvoiced frame rate (OVR) corresponds to the number of FA frames over the number of unvoiced references frames as explained in Eq.( 5). The undervoiced frame rate (UVR) is the ratio between the FR frames number and the number of voiced references frames. It is given in Eq. [START_REF] Secrest | An integrated pitch traking algorithm for speech systems[END_REF].

[ ] [ ] R H R uVF VF uVF OVR Ω ∩ Ω = 100 (5) [ ] [ ] R H R VF uVF VF UVR Ω ∩ Ω = 100 (6)
The voiced decision agreement rate (V ok ) is given in Eq.( 7). It corresponds to the number of common voiced frames between the reference and the hypothesis over the number of voiced frames in the reference or the hypothesis. The unvoiced decision agreement rate (uV ok ) is described in Eq.(8).

[

] [ ] H R H R ok VF VF VF VF V ∪ Ω ∩ Ω = 100 (7) [ ] [ ] H R H R ok VF VF VF VF uV ∪ Ω ∩ Ω = 100 (8)
The F 0 estimation quality measure involves a Gross Error Threshold (GET) fixed at 20%. A F 0 reference value is said "in accordance" with a F 0 hypothesis value whether their absolute relative distances is inferior to GET. Eq.( 9) presents the mathematical formalization of this definition. We note the "in accordance" operator by the ≈ symbol. ( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ≤ - ∃ ∧ > ⇔ ≈ GET R H R y R H R t x t y t x t x t y t x 0 ( 9 )
The "in accordance" operator is extendable to frames. A reference frame is in accordance with a hypothesis frame if all the F 0 references values are in accordance as showed in Eq.(10).

t y t x t t H R y x H R ≈ ∃ ∀ ⇔ ≈ , ( 10
)
The set of references frames in accordance is noted R ok and is described in Eq.( 11).

{ } t t ok H R t R ≈ = (11)
The "in accordance" rate corresponding to a recall rate is noted RER and is given by Eq.( 12). It is the ratio between the number of "in accordance" reference frames and the number of common voiced frames between reference and hypothesis.

[

] [ ] H R ok VF VF R RER ∩ Ω Ω = ( 12
)
The precision rate (PRR) corresponds to the number of "in accordance" reference frames over the number of voiced F 0 hypotheses values needed to put in accordance the reference frame. It is an indication of the PEA efficiency to find the rights F 0 values in the first hypotheses candidates. PRR is explained in the Eq.( 13) below.

[ ] { } [ ] [ ] ∑ ∈ ∈ > Ω Ω = ok H H t N y t y ok H R PRR , , 1 0 (13)
In the case of PEAs which provide a single F 0 hypothesis value per frame, PRR always equals to 100%.

The Gross Error Rate (GER) is theF 0 estimation accuracy indicator. It corresponds to the number of frames not "in accordance" over the total number of voiced references frames. We note R ko the set of reference frames not "in accordance". The GER is given in Eq.( 14).

[

] [ ] R ko VF R GER Ω Ω = (12)
An hypothesis F 0 can be associated with several reference F 0 values. Thus if some F 0 reference values are equal in a same frame (crossing streams of F 0 ), PEAs probably return a single hypothesis for the ensemble of equal F 0 reference values. This multi-association allows the evaluation to deal with this problem.

Once a hypothesis F 0 value is associated to one or more reference F 0 values, the hypothesis F 0 value is locked and can not be associated with other reference F 0 values.

In the monopitch case, a F 0 candidate, a frame and a stream are the same. Thus, it remains only four voicing quality measures.

Monopitch evaluation results

Four PEAs are evaluated and compared: YIN, SWIPE, PRAAT and PSH. YIN [START_REF] De Cheveigné | YIN, a fundamental frequency estimator for speech and music[END_REF] provides an aperiodicity criterion which can be easily converted into a voicing feature between 0 and 1. It gives one hypothesis per frame. SWIPE [START_REF] Camacho | SWIPE: a Sawtooth Waveform Inspired Pitch Estimator for Speech and Music[END_REF] provides a voicing strength criterion between 0 and 1 and also gives a single hypothesis per frame. PSH [START_REF] Liénard | Speech Fundamental Frequency Estimation Using the Alternate Comb[END_REF] provides a voicing strength which needs to be converted in a voicing feature between 0 and 1. It gives a tunable number of hypotheses and is adapted to the multipitch case.

In the monopitch case, we fixed it at 1 like the others tested PEAs. PRAAT [START_REF] Boersma | Accurate Short-term Analysis of the Fundamental Frequency and the Harmonics-to-noise Ratio of a Sampled Sound[END_REF] provides an autocorrelation coefficient between 0 and 1 as voicing feature. In our series of algorithms, it is the only PEA with a post-processing. We tuned its parameters to remove this post-processing effect. All PEAs are tuned to provide a single F 0 hypothesis value so in Table 1 the PRR column is removed because PRR is always equal to 100%.

The particularities of each PEAs may lead to some small time misalignments between the references and the hypotheses so that a time alignment step is needed. It consists in a linear interpolation between the first before hypothesis value and the first after value. There is an undefined interpolation in the case of an unvoiced hypothesis frame and a voiced one. These frames are discarded from evaluation. As the reference F 0 range differs from the hypothesis F 0 range, all the F 0 references out of the hypothesis F 0 range are discarded from the evaluation. For a given PEA, all statistics are means obtained on the whole of the three corpora. The results given in table 2 do not aim at a classification of the algorithms tested. They show that the performance differences are not as large as they look in the original publications, and they may encourage a detailed analysis of their respective strengths and weaknesses. The PRAAT settings that we adopted to remove the post-processing are clearly not adapted to this algorithm.

Multipitch case

In the multipitch case, several F0 streams need to be tracked simultaneously. This complicated task is usually performed by top-down approaches. These approaches use continuity hypotheses from one frame to the next or other general knowledge concerning the whole sequence. As there is also a lack of consensus in literature concerning these top-down processes, it is still difficult to evaluate their contributions to voicing and pitch estimation. That is why we are interested in a frame-to-frame evaluation without knowing more than the information extracted from the studied frame. We are aware of the fact that the better are the frame-level feature estimators, the simplest will the higher-level processes be. The reference annotations in the multipitch case are directly based on the monopitch reference annotations. The annotations consist in a simple concatenation of the monopitch reference annotations.

Evaluation method

Database an reference annotations

Quality measures

All the notations and definitions posed in 3.1.3 are still available in this section. In the monopitch case, a speech signal contains only one speaker so that it was not necessary to introduce the notion of F 0 stream. A F 0 stream corresponds to the set of F 0 values belonging to one speaker in a speech signal. In the multipitch case we have to introduce it due to the mixing of F 0 streams. One defines S R the set of F 0 streams contained in the signal and S R S the particular F 0 stream of the s th speaker. There are as many F 0 streams as speakers so there are N S F 0 streams. An ideal PEAs should be able to reproduce the exact references annotations on his hypotheses output.

{ }

S N R R R S S S ... 1 = and { } F N S S S R R R S ... 1 = (13)
An F 0 hypothesis value should be associated with several reference F 0 values. Thus if some F0 reference values are equal in a same frame (crossing streams of F 0 ), PEAs probably return a single hypothesis for the ensemble of equals F 0 reference values. This multi-association allows the evaluation to deal with this problem. Symmetrically once a F 0 hypothesis value is associated to one or more F 0 reference values, the hypothesis F 0 value is locked and can not be associated anymore.

With the VuV definition adopted in 3.1.3, the VuV decision evaluation criteria remain the same in the multipitch case than in the monopitch case. One frame is voiced if one of her value is strictly positive. Mathematical definitions of V ok , uV ok , OVR an UVR remain unchanged. Nevertheless as the reference change, a new adequate VuV decision threshold has to be computed.

There is an evolution in the F 0 estimation evaluation criteria. Contrary to the monopitch case where a F 0 reference value is the same than a frame reference, in the multipitch case we can clearly distinguish between them and any simplification is no longer available.

The "in accordance" F 0 references values rate RECR given in Eq.( 15) is the number of F 0 references values in accordance over the number of voiced F 0 reference values (VC R defined in Eq.( 2)). Eq.( 14) provides the definition of RC ok which is the set of F 0 reference values in accordance. One can define the same rate for frames. We call REFR the "in accordance" reference frames rate. It corresponds to the number of reference frames in accordance over the number of voiced references frames. It is a first feature to quantify a PEA quality in reconstructing F 0 streams. Indeed higher is REFR higher is the number of frames where all F0 references values are associated with an F0 hypothesis value and better may be the F 0 streams tracking. Eq.( 16) and (17) formalize this feature.

{ }

{ }

)

t y t x ok H R t RF ≈ = (16) [ ] [ ] H R ok VF VF RF REFR ∩ Ω Ω = (17)
The precision rate on F 0 values explained by Eq.( 18) is the same than in Eq.( 13) except that here it may exists more than one F 0 reference values.

Conclusions

In this paper we provided an evaluation methodology to compare PEAs not only on their F 0 estimation efficiency but also on the VuV decision which is a PEA step at least as important as the F 0 estimation. We illustrated this view in the monopitch case, by using several PEAs on a set of 3 different databases including the EGG signals, for which an automatic annotation protocol has been worked out. The methodology has been elaborated to extend easily to the multipitch case.

This work reinforces the idea that F 0 estimation and VuV decision are deeply related, in complex ways. Voicing was considered here as a two-state, binary problem. In the future it may become necessary to treat separately two notions of voicing, one at the signal level, essentially continuous, and the other, binary, incorporating some upper-level perceptive and linguistic considerations.

Fig. 1 .

 1 Fig. 1. Rate of frames declared as voiced and Gross Error Rate as functions of the VuV decision threshold

Fig 1 .

 1 Fig 1. (a) speech signal (b) egg signal (c) filtered, saturated, half-wave rectified, normalized egg (black line) and temporal envelope (red line) (d) speech signal (black line) and voiced segment (red line) (e) envelope 18-points histogram. The voicing threshold is 0.1.

Table 2 .

 2 Evaluation results on all the corpora.

		EER V ok uV ok OVR UVR RER GER
	YIN	1.00 76.5 74.7 12.3 12.4 93.9 5.1
	PSH	1.01 85.6 85.7 6.3	6.3 95.5 4.2
	SWIPE 0.98 80.1 77.8 10.3 10.4 96.4 3.0
	PRAAT 1.00 62.2 60.6 17.1 16.7 94.1 3.7

  To build a 2-speakers mixture, two 60 dB energy normalized signals are simply added. The energy normalization method is the "Scale Intensity..." PRAAT function. Then, the 2-speakers mixtures databases are built by adding several possible signals combinations. The mixtures are cut to the shortest component. Three types of mixtures are available: the female/female mixtures, the male/male mixtures and the male/female mixtures. K contains 1411 seconds of speech mixtures, B contains 13693 seconds of speech mixtures, D contains 79869 seconds of speech mixtures for a total of 26 hours 22 minutes of speech mixtures artificially collected.


domain: math shs

text:
Daniel Frischemeier

Rolf Biehler

Stepwise development of statistical literacy and thinking in a statistics course for elementary preservice teachers

Keywords: Elementary preservice teachers, design based research, cooperative learning, statistical literacy and thinking, TinkerPlots

. We will refer to design principles of the course and show in which way a stepwise development of statistical literacy and thinking with TinkerPlots works in cooperative learning environments.

Introduction

Since the implementation of the leading idea "Data, Frequency and Chance" [START_REF] Hasemann | Daten, Häufigkeit und Wahrscheinlichkeit[END_REF] in mathematics classrooms in primary schools in Germany, statistics has become a central topic in primary school. This has set requirements not only for schools and teachers, but also for universities who have to educate preservice teachers in statistics for their upcoming school career. Requirements for teacher education in statistics can be found on German national level (e.g. AK Stochastik, 2012), and on international level (e.g. [START_REF] Batanero | Teaching statistics in school mathematics-challenges for teaching and teacher education[END_REF]. Two important aspects appear at both levels: applying a whole data analysis cycle (like PPDAC, see [START_REF] Wild | Statistical thinking in empirical enquiry[END_REF] and analyzing data with digital tools (see [START_REF] Biehler | Technology for enhancing statistical reasoning at the school level[END_REF]. We decided to use TinkerPlots [START_REF] Konold | TinkerPlots 2.0[END_REF] for our purposes, since it is easy to learn, no formulas are needed, and it enables learners to create multiple representations of data. In our sense, TinkerPlots can serve as educational software for pupils from grade 4, as software for teachers for analyzing data, and as medium for demonstration purposes in classroom. This was our motivation to design, realize and evaluate a statistics course for elementary preservice teachers with TinkerPlots on the basis of the Design Based Research paradigm [START_REF] Cobb | Design experiments in educational research[END_REF].

Our course to develop statistical literacy and thinking with TinkerPlots

The main goal of this course is to develop statistical literacy and thinking components (for a definition see Garfield & Ben-Zvi, 2008, pp. 34) and the technological knowledge of our participants. At Paderborn University in Germany elementary preservice teachers for mathematics attend an obligatory course "Elementary Statistics", which is about data analysis, combinatorics and probability theory. Due to limits of time, there is no space for going through a whole data analysis cycle like PPDAC or to do further data explorations in multivariate datasets. For that reason our course was designed taking into account the principles of the "Statistical Reasoning Learning Environment" (Garfield & Ben-Zvi, 2008), to expand preservice teachers' knowledge in data analysis and to introduce a new tool to them, which better fits to primary and lower secondary school. Fundamental ideas realized in our new designed course are to "focus on developing central statistical ideas", to "use real and motivating data sets", to "use classroom activities to support the development of students´ reasoning", the integration of "appropriate technological tools", to "promote classroom discourse that includes statistical arguments and sustained exchanges that focus on significant statistical ideas" and the "use of formative assessment" (see Garfield & Ben-Zvi, 2008, p. 48). On the paradigm of the PPDAC-Cycle [START_REF] Wild | Statistical thinking in empirical enquiry[END_REF], the course has the aim to encourage participants to define a statistical problem and to pose statistical questions (First "P" in PPDAC), to plan and to prepare a data collection (Second "P" in PPDAC), to collect data (with regard to data management and cleaning -"D" in PPDAC), to analyze data ("A" in PPDAC) and to make conclusions of the data explorations (interpretation -"C" in PPDAC). We implemented cooperative learning environments like the "Think-pair-share" method to develop the statistical literacy and thinking components of our participants (see [START_REF] Roseth | Collaboration in learning and teaching statistics[END_REF] and to support peer-learning, peer-feedback and expertfeedback. In this respect, in a "Think-pair-share" setting, students first deal with the task on their own ("think phase"), discuss about their findings in peers ("pair phase") in a second step and finally discuss their findings in class with the teacher ("share phase"). All in all, the course consists of four modules: The first module deals with the generation of statistical questions, the preparation of data collection and the collection of data. Here the participants get to know the "PPD" elements of the PPDAC cycle with a special emphasis on the generation of adequate statistical questions (see [START_REF] Biehler | Statistische Kompetenz von Schülerinnen und Schülern -Konzepte und Ergebnisse empirischer Studien am Beispiel des Vergleichens empirischer Verteilungen[END_REF]. The second module has the intention to introduce the participants into data analysis with TinkerPlots. Here the participants can learn first steps in data analysis using data cards and hands-on-activities and then use TinkerPlots for first explorations in small datasets. Because the focus is on the "AC" (Analysis & Conclusion) elements of the PPDAC cycle, the datasets are given to the students in an already prepared form. Furthermore the students learn to describe and interpret distributions of categorical and numerical variables with special emphasis to the elements and characteristics of distributions like center, variation, etc. as it is proposed in [START_REF] Rossman | Workshop statistics: Discovery with data and Fathom[END_REF] and Biehler (2007a[START_REF] Biehler | Students' strategies of comparing distributions in an exploratory data analysis context[END_REF]. Module three builds on module two and covers advanced data analysis with TinkerPlots in large multivariate datasets. Here the learners are introduced into comparison of groups [START_REF] Pfannkuch | Year 11 Students' informal inferential reasoning: A case study about the interpretation of box plots[END_REF]. A major aspect in module two and three is to enable the participants to explore datasets and make their own statistical investigations with their own statistical questions (for a typical task for a statistical investigation in this respect see Figure 1). Module four has the intention to introduce the participants into inferential statistics, especially into randomization tests with TinkerPlots [START_REF] Frischemeier | Design and exploratory evaluation of a learning trajectory leading to do randomization tests facilitated by TinkerPlots[END_REF]. Further details on the course design and the lesson plans can be read in [START_REF] Frischemeier | Statistisch denken und forschen lernen mit der Software TinkerPlots[END_REF].

Accompanying research of the course: Stepwise development of statistical literacy and thinking

The course was taught by the first and second author and consisted of 14 sessions, each session lasted 90 minutes. One major goal of the course was to develop the elementary preservice teachers' statistical literacy and thinking components with TinkerPlots. Since statistical investigations of complex datasets are new to our participants, we want to evaluate the statistical investigations in the introductory stage (module 2) and see in which way the quality will improve over time in cooperative learning environments. Two major research questions arise: How is the quality of the statistical investigations in the intermediate steps in module two? How does the quality of the statistical investigations develop in process of module two?

Participants, task and data collection

All in all 22 elementary preservice teachers participated in the course. All of them attended the course "Elementary Statistics" as described in the introduction. As a typical task a multivariate dataset with an exercise sheet consisting of four subtasks was given. As example you can see the "KinderUni"-task in Figure 1, where the dataset "KinderUni" had to be explored. The KinderUni dataset, is a (non-random sampled) dataset with 28 variables containing information about leisure time and school activities of 39 pupils in the area of Kassel, Germany. In the introductory phase of module two the idea was, that learners at first explore small multivariate datasets to get used to data explorations with TinkerPlots and then to explore larger datasets in module three. So when working on the "KinderUni" task, the participants are at first (subtask (i)) asked for a short description of the dataset to get familiar with it. The second part (ii) of the task was to generate an appropriate statistical question. This statistical question of subtask (ii) is the starting point for subtasks (iii) and (iv). In subtask (iii), the participants are asked to create suitable graphs with TinkerPlots, which allow answering the statistical question arisen in part (ii). In subtask (iv) the participants are supposed to describe and interpret the TinkerPlots graphs of (iii) and finally to answer the statistical question posed in (ii). The participants worked in pairs of two on the "KinderUni" task. So all in all, we had 11 pairs, who remained constant all over the course. When working on the task "KinderUni", the participants were asked to document the procedure of their statistical investigations in written form in Microsoft Word with the TinkerPlots graphs implemented. We collected all word documents from the "KinderUni" task. As mentioned above, one major idea of the course was to improve the quality of the statistical investigations by peer-feedback and expert-feedback. This happened with cooperative learning activities like "think-pairshare". First, in the "think" phase, all pairs worked on the task on their own and produced the preliminary version of the task (preliminary version: V1). Then two pairs came together and discussed the products of their statistical investigations (not necessarily with the same questions) in peers in the "pair" phase with the goal to find improvements for the TinkerPlots Graphs, for the descriptions of the TinkerPlots Graphs, etc.. Finally after revising the documents after the "pair" phase (version after peer feedback: V2), as a last step, the revised documents were discussed in plenum with the first and second author. After this phase the participants were again asked to revise their products for a final version (version after peer-and expertfeedback: V3). So for our data analysis we have the documentations on the statistical investigations of the participants as preliminary version (V1, n=11 documents), as version after peer feedback (V2, n=11 documents) and finally as version after expert-feedback (V3, n=10 documents).

Methodology for data analysis and coding

Our main goal was to rate the quality of the statistical investigations by points. Due to the huge amount of data, we used qualitative content analysis [START_REF] Mayring | Qualitative Inhaltsanalyse: Grundlagen und Techniken[END_REF] for rating the quality of the subtasks. We decided to weigh the subtasks (ii), (iii) and (iv) with equally two points maximum since these tasks are fundamental for the statistical investigation. In subtask (i) only one point is given, since this is an introductory task and easier than subtasks (ii), (iii) and (iv).

For subtask (i) we expected a description of the dataset (number of cases and variables, description of variables). In our course we have set the norm to begin every statistical investigation with an introduction. So two codes are given: "subtask (i) done correctly" and "subtask (i) not done correctly". If subtask (i) is done correctly, one point is given, if it is not done correctly no point is given for this subtask. Details given with examples can be found in Frischemeier (2017, p. 350).

To distinguish the quality of statistical questions posed in subtask (ii), we took into account the classification of [START_REF] Biehler | Statistische Kompetenz von Schülerinnen und Schülern -Konzepte und Ergebnisse empirischer Studien am Beispiel des Vergleichens empirischer Verteilungen[END_REF] in "one-variable-" and "two-variable-" questions from a deductive point of view. So we distinguished whether the questions take into account one variable (example: "What is the distribution of the variable height?" -variable: height) or two variables (example: "In which way do boys and girls differ in respect to the variable height?" -variables: gender and height). For questions taking into account only one variable one point is given as maximum, because the exploration coming out of questions containing one variable is easier than for questions taking into account two variables. For "two-variable-questions" a maximum of two points are given. In between we inductively identified different qualities of statistical questions: So there can be "one-variable-questions", which have just "yes" or "no" as answer (example: "Do 60% of the pupils have a mobile phone?")rated with 0.5 points, whereas "one-variable-questions" in regard to a characteristic of a distribution (example: "How many pupils have a personal computer?")rated with 1 point -are a little bit more sophisticated. Also in the set of "two-variable-questions" we find different types: There are questions leading just to a "yes"/"no" -answer (example: "Is there a difference between boys and girls in their time spending on computer use?"), whereas other types of questions lead to working out differences between the distributions (example: "In which regard does the computer use differ between boys and girls?"). Questions of the first type are rated with one point, questions of the second type are rated with two points. There is also another type of "two-variable questions", which we call "open and complex"-questions like "which differences exist between boys and girls in regard to their leisure time activities?" This type of "two-variable-question" is also rated with two points. In this course we have set the norm to try to pose statistical questions which aim at two variables. As an example for our rating in regard to subtask (ii) we take the question "How many kids have a way to school of 30 or more minutes?" of the pair Anne and Alice. We rated the question with one of two points, since it only covers one variable ("way to school") and it is aimed at one characteristic of a distribution ("how many …?"). Further details for the categorization of questions are given in Frischemeier (2017, p. 350).

For subtask (iii) an adequate TinkerPlots graph has to be created, which enables participants to answer the statistical question posed in (ii). Since we want our participants to focus on the distribution of the investigative variable and on the influence in regard to further variables, we have set the norm in our course that the icons should be stacked in TinkerPlots and further explorations (taking into account other variables) have to be made. If all three requirements (informative TinkerPlots graph, stacked dots and further explorations) are fulfilled, subtask (iii) is rated with the maximum of two points. Table 1 shows the several ratings for subtask (iii).

Informative TinkerPlots graph, stacked and further explorations Informative TinkerPlots graph, stacked and no further explorations Informative TinkerPlots graph, not stacked and further explorations Informative TinkerPlots graph, not stacked and no further explorations Non informative TinkerPlots graph/missing TinkerPlots graph 2 points 1.5 points 1.5 points 1 point 0 point

Table 1: Overview of ratings and their definitions of subtask (iii)

As an example for our rating with regard to subtask (iii) we take the pictogram (with stacked icons) of Anne and Alice in Figure 2. With this TinkerPlots graph they are able to answer their question ("How many kids have a way to school of 30 or more minutes?") posed in subtask (ii). Since icons are stacked, but no further explorations are made, this graph is rated with 1.5 of 2 points. Further details and examples on the ratings of subtask (iii) can be read in Frischemeier (2017, pp. 354). In subtask (iv) the TinkerPlots graph (see Figure 2) has to be described adequately in at least one aspect and the question arisen in subtask (ii) has to be answered correctly. A maximum of two points are given, if both conditions are fulfilled. As adequate descriptions of the TinkerPlots graph we see elements like center, variation, shape, peaks, clusters and outliers (see Rossman et al. 2001, p. 48) but also absolute and relative frequencies of bins. For adequate elements to be carved out in group comparisons, see Frischemeier (2017, p. 42). In Table 2 we see the ratings for subtask (iv).

Component of TinkerPlots graph described and question (ii) answered correctly Component of TinkerPlots graph described and question (ii) not answered correctly Component of TinkerPlots graph not described and question (ii) answered correctly Component of TinkerPlots graph not described and question (ii) not answered correctly 2 points 1 point 1 point 0 points

Table 2: Overview of ratings and their definitions of subtask (iv)

As an example for our rating in regard to subtask (iv) we have a look at the conclusion of Anne and Alice in subtask (iv): "We can see that 12+3 pupils have a way to school of 30 minutes or more." This was rated with the maximum of two points, since one component (absolute frequency of pupils in bins 30-59.9 and 60-90) of the graph is described and the question posed in (ii) is answered correctly. Further details on the ratings of subtask (iv) can be read in Frischemeier (2017, pp. 362).

Results

Let us have a look at the quality of the statistical investigations for the "KinderUni" task in module two in the different stages V1, V2, V3. For each team we rated the subtasks and calculated the success rate "points gained in all subtasks divided by the maximum points in all subtasks" for the "KinderUni" task in each stage (V1, V2, V3). In Figure 3 we see the distributions of the success rates in stages V1, V2 and V3. Regarding to our research questions we can say that the median and also the mean (see blue triangles in Figure 3) of success rates of the different teams increase in the process of the several stages: In preliminary version (V1), where the pairs where on their own, 10 of 11 statistical investigations have a rate below 0.50, the median of the rates is 0.40, the mean of the rates is 0.3636. After the peer feedback phase ("pair"), there is a big positive shift in quality from V1 to V2. Exemplarily one peer feedback component which has often occurred was the advice to stack the dots in the plot to get a better view on the distribution of the data. The quality of the statistical investigations in V2 has increased a lot (mean=0.5236; median=0.56), since in this version only 3 of 11 statistical investigations are below the 0.50 rate. After the expert feedback ("share") in stage V3 all reports are over the 0.50 rate, the median of the rates is 0.65 and the mean of the rates is 0.6520. The expert feedback concentrated most notably on prompts which suggest a better description of the TinkerPlots graph and a more adequate answer to the statistical question posed in subtask (ii). Finally we can identify a positive development of the quality from V1 to V3. We can also see that the distributions in Figure 3 are heterogeneous at the beginning (stage V1) and become more homogeneous in V2 and V3. For a more detailed look we will have a look at Table 3, which identifies the changes within the development of quality in between the four subtasks (i), (ii), (iii) and (iv) in the stages V1, V2, V3. We see that there is an improvement of quality in all subtasks, but the amount of the improvements differ on the kind of subtask. In subtask (i) there is a high quality (0.73) even in the beginning at the preliminary version (V1). This quality improves over time in the different stages V2 (0.91) and V3 (1.00). In V3, the rate is 1.00, which means that every pair began their report at this stage with an introduction of the dataset. In subtask (ii) we see the smallest development of quality: for the questions in the preliminary stage (V1) the rate is 0.47 on average, there is no improvement in V2 (0.47) and only a small improvement in V3 (0.50). Even at the stage V3, in subtask (ii) all questions were only rated with one point, since none of the questions went beyond single characteristics of a distribution or beyond "yes"/"no" answers. One reason might be that the peer and also the expert feedback concentrated too much on the improvement of subtasks (iii) and (iv) but not enough on the development of the quality of the statistical questions. In subtask (iii) the tasks were rated 0.39 on average at the preliminary stage and improved over time (0.51 at V2 and 0.68 at V3). In subtask (iv) the performance was very poor at the beginning (0.19), but improved in progress: In V2 the rate was 0.55 on average and in V3 the rate was 0.74 on average. So in summary we can say that peer feedback and expert feedback in a think-pair-share environment enhances a stepwise development of statistical literacy and thinking components with TinkerPlots. Especially with regard to subtask (iii) und (iv) the creation of TinkerPlots graphs and their description seem to improve after peer and expert feedback. Only in subtask (ii) problems with the generation of statistical questions occur and there was no "big" improvement of quality.

Discussion and implications

The quality of statistical investigations depends on the statistical question rised for the investigation. As we could see, some questions only lead to a short exploration because the answer to that question is just "yes" or "no", wheares there can be also other questions which are aiming at carving out many differences between two or more variables. The analysis of the reports on the "KinderUni" task shows, that especially the creation of informative TinkerPlots graphs and also their description and interpretation with regard to the statistical question succeeds and the peer-and expert-feedback can improve the TinkerPlots graphs (subtask (iii)) and the descriptions and interpretations of the TinkerPlots graphs (subtask (iv)). The key point is the generation of adequate statistical questions aiming at more than only one variable. Although this was taught in our course, too many statistical questions lacked quality. For the re-design and the upcoming cycle of the course it would be important that there will be feedback on the statistical questions to improve their quality as well. Here it could be helpful to discuss adequate and non-adequate statistical questions in class to help learners to differentiate between adequate and non-adequate statistical questions.

Figure 1 :

 1 Figure 1: Task "KinderUni" as typical statistical investigation task in module two

Figure 2 :

 2 Figure 2: TinkerPlots graph for "KinderUni" task of Anne and Alice

Figure 3 :

 3 Figure 3: Success rates of the statistical investigations in stages V1, V2 and V3

Table 3 : Development of quality (average of rates) between the subtasks

 3

	V1	V2	V3

Proceedings of CERME10


text:
Christian Büscher
email: christian.buescher@math.tu-dortmund.de

Common patterns of thought and statistics: Accessing variability through the typical

Keywords: Informal inferential reasoning, design research, concept development

Allowing students to construct meanings of statistical concepts like variability requires building on their individual experiences. The notion of patterns of thought is utilized to conceptualize the difference between formal statistics and learners' initial thinking and to describe a pathway to bridge this gap between individual and mathematical thinking. Students' patterns of thought and the processes of their conventionalization are reconstructed in a qualitative study with n = 10 students of grade 7. An outlook is given on the possibility of connecting students' patterns of thought with formal statistics.

Introduction

One of the aims of statistics education is to foster informal inferential reasoning (IIR), the ability and disposition to use data in order to reason about some wider universe [START_REF] Makar | The reasoning behind informal statistical inference[END_REF]. In IIR, statistical concepts are combined with contextual knowledge under certain statistica l norms and habits, such as a "critical stance towards data" (see [START_REF] Makar | The reasoning behind informal statistical inference[END_REF] for a more thorough overview). While the framework of IIR can explain the role of statistical concepts in producing informal statistical inferences (ISI, see [START_REF] Makar | A framework for thinking about informal statistical inference[END_REF], it does not account for the development of statistical concepts. Accordingly, there exists a lack of research concerning statistical concept development of students with little experience in statistics. Tracking of the development of concepts, [START_REF] Bakker | Lessons from inferentialism for statistics education[END_REF] draw on the background theory of inferentialism, linking students' emerging concepts in inferential practices with IIR. Looking at the micro level of students' reasoning processes, Bakker and Derry argue that students can show complex ideas regarding statistical concepts such as center, variation, distribution, and sampling. These ideas are not formally articulated, but can rather be seen as "precursor notions" of regular statistical concepts (Bakker & Derry, 2011, p. 20).

These precursor notions could provide a promising basis for the development of more formal statistical reasoning. The challenge remains however to design tasks that draw on these precursor notions in order to foster students with little prior experience in formal statistics. These tasks need to draw on students' individual ways of reasoning as a resource in order to develop statistical concepts meaningful to the students. This requires paying careful attention to students' learning processes, and to uncover the links between formal statistical concepts and students' everyday thinking. This study aims at reconstructing students' individual ways of thinking on a micro level in order to find ways to connect students' everyday thinking to regular statistical concepts.

Patterns of thought in everyday thinking

The mathematician and philosopher [START_REF] Wille | Allgemeine Mathematik" als Bildungskonzept für die Schule[END_REF] argues that in order for mathematics to become learnable and possibly meaningful to non-experts (i.e. the general public), the discipline of mathematics itself has to be restructured in a program he coined Generalistic Mathematics ("Allgemeine Mathematik"). Wille calls for mathematicians to reveal the reasons, the aims, the common patterns of thought, and the boundaries and dangers of mathematical concepts and of whole mathematical theories. This however must not be done in the language of mathematical theory. Opening mathematics to the general public necessitates the use of common language, rather than specialized vocabulary, to describe those reasons, aims, patterns, and boundaries. This approach to make mathematics open and meaningful resonates well with [START_REF] Lengnink | Das Verhältnis von Alltagsdenken und mathematischem Denken als Inhalt mathematischer Bildung[END_REF], who see the challenge and goal of mathematics education in explicating the connection between everyday thinking and mathematical thinking. For them, mathematical thinking manifests as conventionalized everyday thinking. In this way, confidence intervals can be seen as a conventionalized form of the generalizations taking place in daily life: being reasonably sure that some observation of daily life can be taken as true, with a certain intuitive degree of variation.

This places a firm emphasis on the importance of common thought patterns and practices, so that a task of mathematics education becomes the identification of these patterns of thought. While [START_REF] Wille | Allgemeine Mathematik" als Bildungskonzept für die Schule[END_REF] sees this task in the hands of (the philosophy of) mathematics, [START_REF] Prediger | Do you want me to do it with probability or with my normal thinking? Horizontal and vertical views on the formation of stochastic conceptions[END_REF] points out that empirical insights into mathematical learning processes form a valuable and even necessary basis for identifying connections between everyday thinking and mathematical thinking. This also means that thought patterns cannot be approached in a general way, but rather are tied to the specific mathematical content of the learning process under investigation. This study adopts the approach outlined by [START_REF] Prediger | Do you want me to do it with probability or with my normal thinking? Horizontal and vertical views on the formation of stochastic conceptions[END_REF], in order to reveal patterns of thought and their connection to the statistical concepts of center and spread when comparing distributions.

The typical as a common pattern of thought

One way students' thinking differs from formal statistics is in the use of measures. In statistics, measures such as median or standard deviation function as highly specialized tools for talking about statistical concepts like center or variability. In their intuitive approaches to statistics, students use strategies such utilizing "modal clumps" instead [START_REF] Konold | Students' use of modal clumps to summarize data[END_REF] to point out 'the majority' of the data. Since to the students the location and the width of the clump both are important, these modal clumps can simultaneously address the center of a distribution as well as a form of spread. Thus, in their everyday language, learners integrate many different statistical concepts that would formally be strictly distinguished through use of different specialized measures [START_REF] Makar | Variation talk: Articulating meaning in statistics[END_REF]. This raises the question on what would be promising patterns of thought to build on. One such candidate would be the practice of identifying 'typical' values or ranges of values within datasets. Some research indicates that 'typical' might be a good way for students to think about the average (Makar, 2014), while other research finds 'typical' or 'normal' to be a term for talking about ideas combining center and spread (Büscher, 2016a;[START_REF] Büscher | Students' informal measures between objects and tools[END_REF]. Thus, the pattern of thought of identifying the 'typical' of a distribution seems a promising candidate as a resource for concept development, although it remains unclear for which concepts exactly.

Patterns of thought and concept development

To address the question of how patterns of thought can support concept development, this study follows a conceptual change approach [START_REF] Duit | Conceptual change: A powerful framework for improving science teaching and learning[END_REF]. Learning is understood as a restructuring of prior conceptions, occurring when these conceptions no longer satisfactorily exp lain phenomena. The goal of statistics teaching is then to initiate the development of conceptions into statistical concepts. Since statistical concepts show a high degree of connection to each other, learning trajectories in statistics however should not address concepts in an isolated way, but rather in a holistic way [START_REF] Bakker | Lessons from inferentialism for statistics education[END_REF]. This calls for organizing structures that (a) connect to learners' prior conceptions, (b) holistically address statistical concepts, and (c) lead to regular statistical concepts. Patterns of thought can provide just such a structure, as they encompass different prior conceptions and thus enable the connection of everyday thinking to mathematical thinking.

Research questions

The theoretical background of this study suggests that designing learning trajectories towards statistical concepts should start from fruitful patterns of thought. It is thus an issue for empirica l investigations to find those patterns of thought in students' thinking which can indeed serve this function as starting points in learning trajectories. Looking at what is 'typical' was identified as one potential pattern of thought that could result in such concept development. This study therefore aims to answer the following research question: What concepts do students address and develop when conventionalizing the vague pattern of thought of 'identifying the typical'?

Research design

This study adopts the methodological framework of topic-specific didactical design research [START_REF] Prediger | Topic-specific design research with a focus on learning processes: The case of understanding algebraic equivalence in Grade 8[END_REF] with a focus on learning processes [START_REF] Prediger | Design research with a focus on learning processes: An overview on achievements and challenges[END_REF]. This approach aims at providing empirically grounded local theories on topicspecific learning processes as well as design principles and concrete teaching-learning arrangements for the topic. While the framework utilizes iterative cycles of design experiments, this study reports on the third cycle of design experiments.

Data gathering

Design experiments were conducted with five pairs of grade 7 students (aged 12 -14) who had only little experience with statistics within the mathematics classroom. Experiments consisted of two sessions of 45 minutes each, with each session having its own arrangements of data and tasks. They were fully videotaped and partially transcribed. The process-focused analysis of the video data from the first session of the design experiments allows to reconstruct the development of students' patterns of thought, in their relation to task design.

Task Design

For designing the teaching-learning arrangements, various design principles have been impleme nted throughout the different cycles of design research, two of which play an important part in this study.

Drawing on 'typical' as pattern of thought. As outlined above, connecting to the pattern of thought of characterizing what is 'typical' of a certain distribution can potentially provide a starting point for processes of conventionalization that lead to meaningful use of formal statistics. Therefore, 'typical' has to be explicitly addressed in the task design, and the setting of the task has to provide a context in which this pattern of thought can naturally occur.

Criticizing conventionalizations. As [START_REF] Lengnink | Das Verhältnis von Alltagsdenken und mathematischem Denken als Inhalt mathematischer Bildung[END_REF] point out, mathematical learning has to explicitly address the relation between everyday thinking and mathematical think ing. Following this, it is not enough for a task to just utilize students' patterns of thought and to encourage conventionalization of those patterns of thought. It is rather these conventionalizations that have to become the object of investigation. Under a Generalistic Mathematics perspective this could mean addressing reasons, aims, and boundaries of these conventionalizations. These design principles were realized in the design of the Antarctic weather task. The initia ted activity puts the students into the role of consultants to researchers at the Norwegian Antarctic research station Troll forskingsstasjon. In the first phase of the task, the students investigate dot plots of daily temperatures for the month of July 2004 and predict the weather for next July by giving a distribution of ten days. In Phase 2, additional data for July 2002 and 2003 were included (Figure 1). Predicting the weather was chosen as activity because it is rooted in everyday thinking, combining experiences of short-term variability (one can never be too sure about the weather…) with longterm signals (… but there are typical temperatures after all). In order to support conventionalization of patterns of thought, the third phase introduces a design element called report sheets (cf. Figure 2 and3). The report sheets are introduced to serve as a brief summary of the Antarctic weather in July. They combine graphical representations and the use of measures with a brief inference. First, the students are asked to fill out their own report sheet. After that, in the fourth phase, the students receive three different filled-in report sheets by fictitio us students (Figure 2). These filled-in report sheets differ in their interpretation of typical. This serves as a basis for discussing and criticizing the different conventionalizations of typical, as the students are asked to evaluate the correct use of typical.

Data Analysis

The students' patterns of thought were reconstructed in an interpretative analysis using concepts-inaction and theorems-in-action from [START_REF] Vergnaud | The theory of conceptual fields[END_REF] theory of conceptual fields. Concepts-inaction are "categories (objects, properties, relationships, transformations, processes etc.) that enable the subject to cut the real world into distinct elements and aspects, and pick up the most adequate selection of information according to the situation and scheme involved" (Vergnaud, 1996, p. 225). Theorems-in-action are statements held to be true by the learner.

Which concepts-in-action and theorems-in-action are activated depends on the pattern of thought utilized by the learners. Thus, patterns of thought are conceptualized as groups of concepts-in-actio n concurrently occurring in the learners' activity. Concepts-in-action and theorems-in-action are reconstructed from the students' point of view, and do not necessarily correspond to regular statistica l concepts. In the analysis, the reconstructed concepts-in-action are symbolized by ||…||, while theorems-in-action are denoted by <…>.

Empirical Insights: The case of Maria and Natalie

The first snapshot starts with Phase 2 of the Antarctic weather task. After getting the additional data of the years 2002 and 2003, Maria and Natalie, Grade 7, try to explicate their view on the data. This excerpt serves as an illustration of the starting point in the students' reasoning. The students are trying to characterize the differences observed in the distributions. At first, the students formula te differences between 2003 and 2002/2004 in terms of ||spread||: in 2003, the temperatures "came apart" (#2). Another difference they notice is the difference of the location of the ||center|| between 2002 and 2004, indicated through modal clumps ("the majority", #9).

It is important to note that at this stage, the students face difficulties in trying to express their findings. The distributions of 2004 and 2002 are found to be "somewhat similar" (#8) though "different" (#8) to 2003, with further explanation supplied by Natalie through use of gestures and improvised vocabulary ("like, like the most dots", #9). Few minutes later, the students find a way to deal with the complexity.

21 Maria: Well, we first should look at how many degrees it has risen or falle n.

Generally. In two years. […] 27 Natalie

You mean average, like… 28 Maria

The average, and then we look at how the average changed in two years.

By introducing the notion of reducing the distributions to a ||general value|| ("Generally", #21), the students are able to handle the complexity of the differences between the distributions. For this general value, they appear to already know an adequate measure: the ||average||. To the students, <the average represents the general value of a distribution>. The average acts as a summary to be used in further procedures, as <the differences between distributions can be described by differences in general values>.

In the following exchange, after having estimated the averages to be -12 (2002) and -14 (2004), Maria and Natalie try to use their result for a linear extrapolation of the weather in 2015 to be predicted.

41 Natalie: Wait. If it gets colder by 2 °C in two years, then it gets colder by 1 °C each year, so we have to… 42 Maria:

Nah, eh, yeah okay 43 Natalie:

13 °C colder average temperature. Right? 44 Maria:

Yes. 45 Natalie: But that's too much, isn't it?

When reflecting on their result however, the students realize that a decline of the average temperature by 13 °C is not a realistic proposition (#45). While their knowledge of the real-world context helps them to identify this contradiction, they are not able to find another solution. In the minute following (not shown here), the students stay insistent in using the average and a linear extrapolation of the trend. It is important to note that, at this stage, their ideas concerning ||spread|| as expressed in the first excerpt seem to have disappeared, replaced by the stronger notion of ||general value|| expressed through the more conventionalized form of ||average||.

The design experiment progresses through the third phase, in which the students create their own report sheet (Figure 3). The analysis picks up at beginning of the fourth phase, with the students comparing the different interpretations of 'typical' in the filled-in report sheets. Well, no. Typical is more like where the most… no… 67 Maria:

The average temperature isn't the typical after all. Because it's only the general, the whole. The typical would be for example for this distribution, expressed through the ||Typical|| ("the most common", #68)although at this point it is not yet clear if Typical consists of a number or an interval. With the ||most common|| corresponding to the notion of ||center|| expressed earlier ("the most dots", #9), Typical seems to help the students to express ideas that got swept aside by the more conventionalized average. Both, average and typical, start to act as conventionalized tools for talking about specific aspects of distributions.

Some minutes later, Natalie summarizes her view on the relation between 'typical' and average.

81 Natalie: And average is pretty imprecise, because it doesn't say anything about a single day. And with typical, I'd say, that it's a span between two numbers, because that way you can better overlook how it is most of the time.

In the end, typical and average provide two different applications. Whereas the average acts as a summary, 'typical' gives an overview into a distribution. The average can be used in comparing distribution in an efficient way, while 'typical' gives an insight into a range of 'normal' or 'expected' temperatures, to which any single day can be compared. In this way, 'typical' combines aspects of ||center|| and ||spread||.

Conclusion

The aim of this study was to examine the interaction of concept development and students' patterns of thought. The students showed two different patterns of thought: Characterizing data through a general value and through a range of typical values. These patterns differed in their degree of conventionalization. While the general-pattern was addressed through use of the average, the students lacked a conventionalization corresponding to the typical-pattern. This resulted in the typical-patter n to be suppressed in favor of the general-pattern, as the ideas addressing spread disappeared. Only when the students were supplied with different conventionalizations of Typical, they were able to reconnect to their typical-pattern. This then allowed them to express ideas combining center and spread.

This identification of thought patterns provides a promising starting point. Work still remains however utilizing these thought patterns to develop regular statistical concepts. This could be achieved by reconceptualizing formal statistics in terms of a typical-pattern. One possible connection could be interpreting the 'box' of a box plot as the typical area of a distribution. Additionally, more insight into processes of conventionalization is needed in order to be able to successfully guide students on their way to meaningful statistics (for one example see [START_REF] Büscher | Entwicklung von informellen statistischen Maßen zwischen Werkzeugen und Objekten[END_REF].

Figure 1 :

 1 Figure 1: Distributions of the Antarctic weather task (translated from German)

Figure 2 :

 2 Figure 2: Fictitious students' filled-in report sheets (translated from German)

Figure 3 :

 3 Figure 3: Maria and Natalie's report sheet Comparing the different interpretations of 'typical', Maria and Natalie are intrigued by the possibility to use an interval to characterize 'typical'. This leads them to reflect on their use of the average. 61 Natalie: But the average temperature isn't really typical, is it? 62 Maria: What, typical? Of course the average temperature is the typical. […] 66 Maria:Well, no. Typical is more like where the most… no… 67 Maria:The average temperature isn't the typical after all. Because it's only the general, the whole. The typical would be for example for this[2004] here [points to -14 on the 2004 dot plot]. 68 Natalie:Typical I think simply is what is the most or the most common.

  [2004] here [points to -14 on the 2004 dot plot]. 68 Natalie:Typical I think simply is what is the most or the most common.The students utilize Typical to differentiate between two different ideas: The ||general value|| that is expressed through the ||average|| ("the general, the whole", #67), and the ||most common|| part of the graph was drawn first, labeled a mistake, and immediately replaced by the red graph. Typical was first assigned as -15, then after the fourth phase changed to -19 to -15.)

Proceedings of CERME10


A Yousfi

S Fréour

F Jacquemin

Eshelby-Kröner viscoelastic self-consistent model: Multi-scale behavior of polymer composites under creep loading

Keywords: Multi-scale, polymer composites, creep, viscoelastic self-consistent model

. In order to achieve the computations, the visco-elastic Eshelby Kröner self-consistent model was used.

Introduction

The thermal aging of polymers is a slow process occurring mainly on structural elements designed to experience mechanical loading, while being subjected to high temperatures for long periods. In many cases, matrix creep can occur. As the role of the matrices is essentially to deform and support stresses primarily in shear, it becomes then important to consider this particularity within a phenomenological modeling framework [START_REF] Nedjar | A time dependent model for unidirectional fibre-reinforced composites with viscoelastic matrices[END_REF]. This context corresponds to the case of graphitepolyimide composites used in aerospace applications [START_REF] Rupnowski | The response of a woven graphite fiber/polyimide composite to aging in nitrogen[END_REF][START_REF] Wilson | Recent advances in polyimide composites[END_REF]. According to, for instance, references [START_REF] Soden | Experimental failure stresses and deformations for a range of composite laminates subjected to uniaxial and biaxial loads[END_REF][START_REF] Kaddour | Behaviour of ±45° glass/epoxy filament wound composite tubes under quasi-static equal biaxial tension-compression loading: experimental results[END_REF], below the glass transition temperature of its polymer matrix, composite materials exhibit a linear or at least a weakly nonlinear mechanical behavior until the damage threshold is attained. However, it is not the case anymore at high temperatures (relative to the glass transition temperature of the matrix). Indeed, recent experimental investigations performed at high temperatures on polyimide matrix composites, have shown very strong discrepancies from the linear elastic behavior [START_REF] Rupnowski | The response of a woven graphite fiber/polyimide composite to aging in nitrogen[END_REF][START_REF] Rupnowski | Mechanical response of a woven graphite/polyimide composite to in-plane shear dominated loads at room and elevated temperatures[END_REF][START_REF] Benedikt | Residual stress analysis in aged graphite/PMR-15 composites using X-Ray diffraction[END_REF]. These nonlinearities can be explained by two main mechanisms coupled together: the visco-elastic behviour of the polyimide matrix subjected to long-term loading at high temperatures, on the one hand, and the effects related to thermal aging, on the other hand.

Multi-scale Modeling of the Viscoelastic Behavior

Viscoelasticity is the generalization of a dual behavior, simultaneously elastic and viscous. It is characteristic of creep, which reveals itself as the strain states evolves over time, whereas the mechanical load imposed on the structure is kept constant. The combination of a mechanical load and a relatively high temperature can yield such a visco-elastic behavior of polymer matrix reinforced composites. Indeed, a creep strain, generally increasing as a function of the time is added to the purely elastic strain. The resulting total strain ε does not only depend on the initial applied stress σ. It also becomes a function of time (t) and temperature (T):

( ) T t, , f σ σ σ σ ε ε ε ε =
. Maxwell model with three parameters (often called the standard viscoelastic model) is useful to describe the viscoelastic response of amorphous polymers or elastomers. This model allows highlighting every feature observed in practice for these materials, at least qualitatively. Accordingly, in order to correctly reproduce the experimental data in the time space, one often uses the Maxwell viscoelastic model generalized to the order n, also called Wiechert model. The viscoelastic modulus then follows a Prony series:

( ) ( ) ∑ = - ∞         - - + = N 1 i i i 1 i g τ t exp Y Y Y T t, Y . (1)

Linear Viscoelasticity

The thermodynamics of irreversible processes allows for expressing the constitutive laws for the behavior of linear viscoelastic materials. The theory developed in [START_REF] Biot | Theroy of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena[END_REF] leads to the most general forms, which are often summarized by the following convolution integrals:

( ) ( ) ( ) ∫ - = - t i i i dt dt t x d t t L t x 0 / / / / 1 , , σ σ σ σ ε ε ε ε . (2) ( ) ( ) ( ) ∫ - = t i i dt dt t x d t t L t x 0 / / / / , , ε ε ε ε σ σ σ σ . ( 3
)
Where σ is the stress tensor, ( ) t L i the viscoelastic stiffness tensor of the constituent noted by the superscript i, while

( ) t L i 1 -
is the viscoelastic compliance. Although each component constituting the ply composite satisfies a differential Maxwell equation, the constitutive law providing the effective response of a composite ply is not Maxwellian because of the coupling existing between the elastic and inelastic part of the strain rate. This coupling comes from the mechanical interactions occurring between the plies constituents, which induce interface stresses. However, without loss of generality of the model, one can assume that the overall macroscopic response of a given ply, represented by the superscript I, also satisfies a viscoelastic constitutive law whose form is that of a Stieltjes convolution:

( ) ( ) ( ) ∫ ∞ - - - - ≡         = t I mn I klmn kl I I I kl dt dt t d t t L t L / / / / 1 1 * σ σ σ σ σ σ σ σ ε ε ε ε . ( 4
)
Where I ε ε ε ε and I σ σ σ σ are the macroscopic stress and strain tensors, respectively.

( )

t L I 1 -
is the viscoelastic stiffness tensor of the composite ply.

Self-Consistent viscoelastic Model

The macroscopic visco-elastic stiffness tensor of the composite ply, appearing above in equation ( 4), is a priori unknown. In the present work, it will be calculated owing to a scale-transition homogenization model. As part of the calculation process, each constituent is considered as an inclusion immersed in the Homogeneous Equivalent Medium representing the effective behaviour of the composite ply. Eshelby-Kröner self-consistent model (see [START_REF] Jacquemin | A Hygroelastic Self-consistent Model for Fiberreinforced Composites[END_REF][START_REF] Fréour | On an analytical Self-Consistent model for internal stress prediction in fiber-reinforced composites submitted to hygro-elastic load[END_REF] for a description in pure elasticity) will be extended in this study to the case of creep loads will be applied here.

A traditional method used for modeling the viscoelastic behaviour of a material consists in transforming the integral equations associated to the problem by means of the Carson transform [START_REF] Cruz | Micromechanical modelling of the viscoelastic behaviour of an amorphous poly(ethylene)terephthalate (PET) reinforced by spherical glass beads[END_REF]. The Carson transform of a function or a tensor f (t) is defined as:

( ) ( ) ( ) ∫ ∞ - = 0 ^. . . exp dt t f t s s s f . ( 5
)
The application of the Carson transform ( 5) to the so-called "Hill's average relations" [START_REF] Hill | The elastic behaviour of a crystalline aggregate[END_REF] yields:

( ) ( ) ( ) ( ) ( ) ( ) ( ) s : s L s : s B : s L s s ^I Î ^I i î î i ^i ^I σ = σ = ε = ε - - 1 1
.

(

Where

( ) ( ) ( ) ( ) ( ) ( ) ( )           +           + = - - - - - s L : s A s L : s L : s A s L s B Î I Î Î I î i 1 1 1 1 1 , ( 6 b)
is the stress concentration tensor expressed in the Carson transform space.

( )

s A I
is the viscoelastic strain localization tensor expressed in the Carson transform space. This tensor relates the local pseudo-macroscopic strains experienced by a constituent to the corresponding macroscopic quantity: The comparison of expression [START_REF] Soden | Experimental failure stresses and deformations for a range of composite laminates subjected to uniaxial and biaxial loads[END_REF] with the generic behavior law of the composite ply leads to an implicit equation to be satisfied by the macroscopic viscoelastic tensor:

( ) ( ) ( ) ( ) s E s E I s A I I I : 1 - - = . (7) ( ) s E
( ) ( ) ( ) i i i I s B s L s L 1 ^1 : - - = . ( 8
)

Viscoelastic Behaviour of the Polymer Matrix PMR-15

The previously published contribution [START_REF] Rupnowski | The response of a woven graphite fiber/polyimide composite to aging in nitrogen[END_REF] ( )

        - +         - + = m 2 m 2 m 1 m 1 m 0 m λ t exp Y λ t exp Y Y t Y . ( 9
)
where

m 0 Y , m 1 Y , m 2 Y , m 1 λ et m 2
λ are unknown parameters.

As suggested by the previously published work in this field of research, the time-dependent behaviour of the material should be described through a "sufficiently large" number of discrete calculations. Accordingly, a number N = 500 of points uniformly distributed on a logarithmic time scale between the initial stage t = 1 s and the final time t max has been chosen. The resulting time step t step on the logarithmic scale would thus be given by the relation:

1 N t Log t max step - =
In order to achieve the identification of the numerous components of the stiffness tensor, the following series of exponentials was used:

( ) t L 1 m 11 - , ( ) t L 1 m 12 - et ( ) t L 1 m 44
of the PMR-15 matrix assumed to remain isotropic in the viscoelastic regime. This approach provides an analytical form for the Carson transform of the time-dependent evolution of each component. Thus, for example, one would easily write:

( ) ( ) ( ) ( ) ∑ ∫ = - ∞ - + = - = 12 1 p p m 11 1 m 11 0 1 m 11 λ s 1 p a s dt t L s.t exp s s L .
The procedure is even more straightforward for the carbon fiber, assuming that the stiffness tensor of this constituent is independent of time. Accordingly the following relation holds:

( ) ( ) s L L t L f f e f = =
. The properties considered for the T650-35 graphite fibers are summarized in Table 1 below: For each value of ( ) ( )

n t 1 n s =
, an initial value of the macroscopic viscoelastic stiffness tensor expressed in the Carson transform space is estimated, using the following simplified model historically established in pure elasticity [START_REF] Herakovitch | Mechanics of Fibrous Composites[END_REF]:

( ) ( ) ( ) s Y v s Y v s Y f 1 f m 1 m I ini 1, + = , ( ) ( ) ( ) 1 f 2 f m 1 m I ini 2, s Y v s Y v s Y -         + = , ( ) ( ) ( ) s ν v s ν v s ν f 12 f m m I ini 12, + = , ( ) ( ) ( ) ( ) ( ) s Y s ν v s Y s ν v s ν m 2 f 23 f f 2 m m I ini 23, + = , ( ) ( ) ( ) 1 f 12 f m m I ini 12, s G v s G v s G -           + = .
The initial macroscopic compliance tensor ( )

s L I ini 1 -
in the Carson transform space is primarily evaluated for each of the N=500 discrete s(n) values considered. Then the self-consistent approach is applied, in order to determine iteratively the effective macroscopic compliance tensor ( )

s L I 1 -
in Carson transform space. It is assumed that each component of the macroscopic stiffness tensor expressed in the Carson transform space satisfies a Dirichlet finite series of 12 terms similar to that defined above for the components of the composite ply. As a consequence, it is necessary to use a numerical optimization method in order to find the 12 constant coefficients defining the evolution, in the Carson transform space for each of the six components of the macroscopic viscoelastic tensor. Then, the inverse Laplace transform is applied to the found Dirichlet series in order to obtain the time-dependent evolution of the macroscopic stiffness tensor. Following a similar method, one can determine the time-dependent evolution of the coefficients of thermal and/or hygroscopic expansion. In the first case, for example, the following homogenization relation should be applied instead of (8) in order to determine a discrete value of the coefficient of thermal expansion macroscopic α I in the Carson transform space:

( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) T s : s L : I s L s L : s E : s L s i i I i I I I ∆ α + - = α - - 1 1
.

The time-dependent evolution of the macroscopic coefficients of thermal expansion and/or their hygroscopic counterparts is determined through a method similar to the procedure described above for the macroscopic viscoelastic stiffness tensor.

Results and Discussion

Figure 1a shows the evolutions of the visco-elastic modulus Y m of the pure PMR-15 matrix as a function of time and temperature. The elastic modulus of the studied polymer decreases during the creep process. Besides a strong evolution of Y m , as a function of the temperature, is observed. The stiffness decreases from the value corresponding to the elastic behaviour at the initial stage but eventually reaches a constant, asymptotic value in the permanent regime. In between, a transient stage occurs where the visco-elastic modulus changes continuously. The duration of this transient stage decreases as the applied temperature increases. The asymptotic creep modulus dramatically falls from its reference elastic counterpart as the temperature increases as shown on figure 1b. According to that curve, at T = 350 °C, the Young's modulus is experiences a division by as much as 12, for instance. Conversely, the longitudinal Young's modulus of a macroscopic ply composite reinforced with parallel fibers is not significantly modified by the softening of the organic matrix. The keeping of such high properties in the longitudinal direction should ensure the durability of composite parts made of PMR-15/T650-35, intended to undergo this type of loading during their service life. As for the macroscopic shear plane modulus (G 12 ), it is reduced when the material is placed under creep conditions, but the weakening observed for this module, while not negligible, is intermediate between what is observed for longitudinal and transverse Young's moduli. Since both the thermal and hygroscopic properties of the matrix may vary under creep conditions. Consequently, the corresponding macroscopic thermal and hygroscopic properties will be affected, also. The macroscopic coefficients of expansion predicted by Eshelby-Kröner self-consistent model are shown on figures 4, 5 and 6. The macroscopic coefficient of thermal expansion was calculated from three independent sets of constituents properties fort the polymer matrix: its coefficients of thermal expansion were assumed (a) to be constant (see figures 4a-5a); (b) to follow the behavior law observed in practice by Rupnowski et al. [START_REF] Rupnowski | Mechanical response of a woven graphite/polyimide composite to in-plane shear dominated loads at room and elevated temperatures[END_REF] (see Figures 4b-5b) and (c) behavior law described by Benedikt et al. [START_REF] Benedikt | X-ray diffraction experiments on aged graphite fiber/polyimide composites with embedded aluminum inclusions[END_REF] (see figures 4c, 5c)). Figures 4c and5c highlight the effect of viscoelasticity on the macroscopic coefficients of thermal expansion (α 11 and α 22 ), the latter decreasing during relaxation. From the time t = 4000 s the CTE reaches an asymptotic value corresponding to a fall of around 15 % in relative value. It is noticeable, however, that the effective macroscopic coefficient of thermal expansion in the direction parallel to the reinforcing fibers remains almost constant throughout the creep process. The magnitude of the variations calculated for α 11 stay below 5.10 - 7 /K. This result should be attributed to the small influence of the properties of the polymer matrix on the effective behavior of the bulk material in this specific direction. matrix (see [START_REF] Rupnowski | Mechanical response of a woven graphite/polyimide composite to in-plane shear dominated loads at room and elevated temperatures[END_REF][START_REF] Benedikt | X-ray diffraction experiments on aged graphite fiber/polyimide composites with embedded aluminum inclusions[END_REF]). The differences between the curves displayed on figures 5a, 5b and 5c precisely come from the variability of experimental data sets available for the considered organic matrix. The magnitude of the effective properties remains similar, however. As a result, the overall shape of the 3D evolutions of the macroscopic transverse coefficient of thermal expansion, as a function of both the time and the temperature looks rather similar, whatever the organic matrix properties accounted for.

Conclusion

This study is devoted to the durability of organic matrix composites reinforced with unidirectional long fibers. The effect of thermal aging on the effective mechanical behavior of composite plies PMR-15/T650-35 subjected to creep conditions at high temperatures (T = 275 °C -350 °C) was investigated. The macroscopic hygro-thermo-mechanical properties were evaluated numerically using the viscoelastic Eshelby-Kröner self-consistent scale transition model. In the longitudinal direction (parallel to the reinforcing fibers), the model predicts that the effective properties of the studied material are not affected by the softening of the organic matrix that constitutes the composite plies. This should ensure the reliability of the structural parts submiotted to such thermomechanical loads. In the transverse directions to the reinforcing fibers, the macroscopic properties (Young's modulus, as well as the coefficients of thermal expansion and hygroscopic expansion) undergo a significant reduction during the creep process. This decrease could modify the distribution of the multi-scale mechanical states within the components of the plies. As a result, the durability of the composite parts could be impaired. The results obtained in the present work will feed further investigations focused on the multi-scale mechanical states experienced by structures made from unidirectional composites plies PMR-15/T650-35 submitted to creep conditions.

I

  is the transform of Eshelby's tensor and can be calculated in the general case of an ellipsoidal inclusion embedded in an anisotropic medium from the classical integral forms, provided that one replaces the elastic stiffness I e L by the corresponding creep stiffness, i.e. ( )

  least-squares fitting procedure was used in order to determine the 36 unknown coefficients the time-dependent evolution featured by the N = 500 discrete values of the stiffness components

Figures ( 2 - 6 )

 26 Figures[START_REF] Rupnowski | Mechanical response of a woven graphite/polyimide composite to in-plane shear dominated loads at room and elevated temperatures[END_REF][START_REF] Rupnowski | An evaluation of elastic properties and thermal expansion coefficients of medium and high modulus graphite fibers[END_REF][START_REF] Nedjar | A time dependent model for unidirectional fibre-reinforced composites with viscoelastic matrices[END_REF][START_REF] Wilson | Recent advances in polyimide composites[END_REF][START_REF] Soden | Experimental failure stresses and deformations for a range of composite laminates subjected to uniaxial and biaxial loads[END_REF] show the effective hygro-thermo-mechanical properties for a unidirectional composite ply PMR-15/T650-35 containing 60 % fiber volume, submitted to creep conditions, calculated as a function of time using Eshelby-Kröner self-consistent model. The numerical predictions of the

Fig 2 :

 2 Fig 2: The effective longitudinal modulus (Y 11 ) (a) and transverse (Y 22 ) (b) predicted as a function of time for the composite ply PMR-15/T650-35, at different creep temperatures.

Fig 4 :

 4 Fig 4: The effective coefficients of thermal expansion (α 11 ), as a function of time, the composite ply PMR-15/T650-35 for different temperatures creep. (a) the CTE of the matrix is constant, (b) the CTE of the matrix follows the law described by Rupnowski [2], (c) the CTE of the matrix follows the law described by Benedikt [15].

Fig 3 :

 3 Fig 3: The effective shear moduli (G 12 ) (a) and (G 23 ) (b) predicted as a function of time for the composite ply PMR-15/T650-35, at different creep temperatures.

Fig 5 :

 5 Fig 5: The effective coefficients of thermal expansion (α 22 ), as a function of time, the composite ply PMR-15/T650-35 for different temperatures creep. (a) the CTE of the matrix is constant, (b) the CTE of the matrix follows the law described by Rupnowski [2], (c) the CTE of the matrix follows the law described by Benedikt [15].

Fig 6 :

 6 Fig 6: The effective coefficients of hygroscopic expansion (β 11 ) (a) and (β 22 ) (b), as a function of time, the composite ply PMR-15/T650-35 for different temperatures creep.

  reports the results of relaxation tests performed on samples made of pure PMR-15 polyimide resin. The relaxation tests were at four different temperatures: 275 °C, 300 °C, 325 °C, 350 °C, during 2 hours. The time-dependent Young's modulus Y m (t) was deduced from these series of measurements by means of an optimization technique. The authors have found that the Young's modulus of PMR-15 matrix satisfies the following Prony series:

Table 1 :

 1 Thermo-elastic properties of the fiber T650-35 from Rupnowski[START_REF] Rupnowski | Mechanical response of a woven graphite/polyimide composite to in-plane shear dominated loads at room and elevated temperatures[END_REF][START_REF] Rupnowski | An evaluation of elastic properties and thermal expansion coefficients of medium and high modulus graphite fibers[END_REF].

	E fL [GPa]	E fT [GPa] G fL [GPa]	G fT [GPa]	ν fL	α fL [10 -6 /°C]	α fT [10 -6 /°C]
	224 ± 3	15.4 ± 0.5 21.1 ± 1.1 5.8 ± 0.4 0.44 ± 0.02	1.2 ± 0.05	13.5 ± 0.8