Dmitrij Titov

Jean-Pierre Bibring

Alejandro Cardesin

Thomas Duxbery

Francois Forget

Marco Giuranna

Mats Holmström Francisco Gonz Lez-Galindo

Ralf Jaumann

Anni Määttänen

Mars Express: 15 years of hard work and discoveries

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

Mars Express: 15 years of hard work and discoveries

Dmitrij Titov (1), Jean-Pierre Bibring (2), Alejandro Cardesin (3), Thomas Duxbery (4), Francois Forget (5), Marco Giuranna (6), Francisco González-Galindo (7), Mats Holmström (8), Ralf Jaumann (9), Anni Määttänen (10), Patrick Martin (3), Franck Montmessin (10), Roberto Orosei (11), Martin Pätzold (12), Jeffrey Plaut (13), and MEX SGS Team (3)

(1) ESA-ESTEC, Noordwijk, The Netherlands (dmitri.titov@esa.int), (2) IAS-CNRS, Orsay, France, (3) ESA-ESAC, Madrid, Spain, (4) George Mason University, Fairfax, VA, USA, (5) LMD, Paris, France, (6) IAPS-INAF, Rome, Italy, (7) IAA, Granada, Spain, (8) IRF, Kiruna, Sweden, (9) IPF-DLR, Berlin, Germany, (10) LATMOS/ IPSL, CNRS, Guyancourt, France, (11) IRA-INAF, Bologna, Italy, (12) RIU-Uni Cologne, Cologne, Germany, (13) JPL, Pasadena, CA, USA After almost 15 years in orbit Mars Express remains one of ESA's most scientifically productive Solar System missions whose publication record now exceeds 1120 papers. Geological maps of Tharsis volcanoes have just been released. Characterization of the geological processes on a local-to-regional scale by HRSC, OMEGA and partner experiments on NASA spacecraft has allowed constraining landforming processes in space and time. Recent results suggest episodic geological activity as well as the presence of large bodies of liquid water in several provinces (e.g. Eridania Planum, Terra Chimeria) in the early and middle Amazonian epoch and formation of vast sedimentary plains north of the Hellas basin. New analysis of the subsurface dielectric properties by MARSIS radar sounder revealed that the deposits in Meridiani Planum, previously interpreted as ice-rich, may contain little or no ice at all. Mars Express observations and experimental teams provided essential contribution to the selection of the Mars-2020 landing sites.

More than a decade-long record of the atmospheric parameters such as temperature, dust loading, water vapor and ozone abundance, water ice and CO2 clouds distribution, collected by SPICAM, PFS and OMEGA spectrometers as well as subsequent modeling have provided key contributions to our understanding of the martian climate. Recent spectroscopic monitoring of the dust storms revealed dust properties and their spatial and temporal variations during the dust storm events. More than 10,000 crossings of the bow shock by Mars Express allowed ASPERA-3 to characterize complex behavior of the magnetic boundary topology as function of the solar EUV flux. ASPERA-3 observations of the ion escape during complete solar cycle revealed important dependencies of the atmospheric erosion rate on parameters of the solar wind and EUV flux and established global energy balance between the solar wind and escaping ion flow. This led to important conclusion that the ion escape at Mars is production rather than energy limited. Comparison to the similar observation record by Venus Express would allow quantification of the role of planet gravity in the escape processes.

The structure of the ionosphere sounded by the MARSIS radar and the MaRS radio science experiment was found to be significantly affected by the solar activity, the crustal magnetic field, as well as by the influx of meteorite and cometary dust. MARSIS and ASPERA-3 observations suggest that the sunlit ionosphere over the regions with strong crustal fields is denser and extends to higher altitudes as compared to the regions with no crustal anomalies. Reconnection of solar magnetic field lines carried by the solar wind with field lines of crustal origin opens channels through which the ionospheric plasma escapes to space, producing strong and narrow cavities in the density. The situation is very different on the night side where the ionosphere has patchy structure. Such patchy ionizations are observed in the regions where field lines have a dominant vertical component. Through these patches the ionospheric plasma from the dayside penetrates and supplies the nightside ionosphere. Several models of the upper atmosphere and plasma environment are being developed based on and in support of the collected experimental data. The models aim at creating userfriendly data base of plasma parameters similar to the Mars Climate Database that would be of great service to the planetary community.

A significant recent achievement was the flawless transition to the "gyroless" attitude control and operations mode on the spacecraft, that would allow mitigating the onboard gyros aging and extending the mission lifetime. The mission extension till the end of 2020 is approved pending technical evaluation in 2018. The extension plan includes both augmenting the coverage and extending long-time series, as well as new elements and potentially new opportunities for discoveries. It will be boosted by collaboration and synergies with NASA's MAVEN, ESA-Roscosmos Trace Gas Orbiter (TGO) and other missions. In 2018 the major challenge would be to pass the bi-annual extension review and elaborate a convincing science case for new extension till the end of 2022. The talk will give the Mars Express status, review the recent science highlights, and outline future plans focusing on synergistic science with TGO.


Jun Dang
email: jun.dang@vub.ac.be

Philippe Lasaygues

Dachun Zhang

Stephan Tavernier

Nicolas Felix

Benjamin Frish

Serge Mensah

Mingxi Wan

Stefaan Tavernier

Nicoas Felix

Stefaan Tavernier

Development of anthropomorphic phantoms for combined PET-ultrasound breast imaging

I. INTRODUCTION

reast cancer is the 2nd most common type of cancer [START_REF]World Cancer Report[END_REF] and the 5th most common cause of cancer death [START_REF]Fact sheet No. 297: Cancer[END_REF]. Breast cancer detection by X-ray mammography and by B-mode ultrasound scanning are in routine clinical use. X-ray mammography has a high sensitivity of about 90% for the detection of breast cancer [START_REF] Shapiro | Ten-to fourteen-year effect of screening on breast cancer mortality[END_REF], but its specificity for distinguishing malignant from benign lesions is only about 20~50% (i.e., there are many false positives) [START_REF] Feig | Analysis of clinically occult and mammographyically occult breast tumors[END_REF]. Thus, a large fraction (> 50%) of the suspicious structures identified in mammograms are non-cancerous [START_REF] Thurfjell | Screening mammography: sensitivity and specificity in relation to hormone replacement therapy[END_REF], and further diagnosis is necessary before deciding to treat the patient for cancer. The sensitivity and specificity of PET for detecting breast cancer are reported to be 92%~97% and 78%~90% [START_REF] Siegel | Positron emission tomography: instrumentation, clinical applications, and case presentation[END_REF], which is ideal for malignant tumor detection. On the other hand, ultrasound (US) has a diagnostic sensitivity 86%~100% and specificity 80%~100% [START_REF] Chan | The sensitivity of ultrasound and serum alpha-fetoprotein in populationbased antenatal screening for neural tube defects[END_REF], and has excellent spatial and contrast resolution. It provides anatomical and even functional information (with US contrast agents). In addition US real time imaging is a non-radiating and low-cost technique. This motivates the Crystal Clear Collaboration to undertake the development of a combined breast PET/US scanner (CLEARPEM Sonic System). US elastography imaging, in addition to morphological information provided by US Bmode imaging, provides the tissues' elasticity distribution. Application of this technology to breast imaging has been reported to with promising results [START_REF] Tanter | Quantitative assessment of breast lesion viscoelasticity: initial clinical results using SuperSonic shear imaging[END_REF]. This motivates us to also consider the combination of PET with US elastography. In this way it will not only be possible to obtain the functional and morphological information, but also the tissue elasticity distribution of the breast.

An anthropomorphic breast gelatin-agar phantom has been developed as part of this project. Acoustical and elasticity characteristics of the phantom will be reported, as well as the results obtained with this phantom on commercial scanners.

II. MATERIALS AND METHOD FOR PHANTOM PREPARATION

We choose (GELITA EUROPE, Ballistic 2, photographic grade) and "high gel Agar"(SIGMA-ALDRICH Co.) to prepare the phantom. We measured the propagation speed of the acoustic waves, the attenuation coefficient and the elasticity (Young's modulus) for several series of samples.

We evaluated samples with gelatin mass percentages from 1% to 15% and with agar mass percentages from 0.5% to 5%. This allowed us to find ingredients ratios that give samples with acoustic and elastic signature close to fat tissue, normal glandular tissue, fibrous/hard tissue and carcinoma in the breast.

The acoustical propagation velocity and the attenuation coefficient of several series of samples were measured using transmission and time of flight techniques using 2 parallel single US transducers immersed in a water tank. We found that the gelatin% fraction in the sample will mainly affects acoustical velocity of the sample and that the agar% fraction has a bigger effect on the attenuation coefficient of the sample. From Table I top 5 rows, we find that as gelatin% increase, the average sound velocity also increases and they can be used to mimic breast fat, glandular tissue, fibrous tissue and B carcinoma. While for the 6 th to 9 th rows it shows the average acoustic attenuation coefficient increases with Agar%. The phantom can mimic breast tissues both on velocity and side characteristics.

Once, acoustical characterization was performed the samples elasticity was measured using NEED REFERENCE OF THE TECHNIQUE USED TO MEASURE ELASTICITY). We also tried to find a elasticity correspondence between the phantom and real breast tissue. (Need a reference of proven elasticity of breast tissues) Based on these studies we were able to produce a realistic phantom simulating a human breast with a tumor.

The steps in the production of the phantom are listed below: First, measure 1% gelatin and 0.5% agar and mix them with the correct amount of water, heat the beaker in a microwave oven until the water will just boil. This is for thepreparation of the fat medium type background. Put the beaker at room temperature and measure its temperature until it become 40⁰C. Add F 18-FDG into the liquid gel and control the FDG activity 4MBq/ml.

Second, prepare inclusions with tumor type medium. Measure 12% gelatin and 4% agar, and mix them with correct amount of water. Heat them in microwave oven just before it boils. Stir the medium averagely and let it cool down to 45⁰C. Add F 18-FDG in with activity 4 times higher than fat medium, which is 16 MBq/ml. Fill the inclusions into several figure latex hats (don't understand what is a figure latex hat?) and sealed the hats tightly. Put the "inclusion balls" into fridge until they became solid.

Third, remove the inclusion latex skin and emerge them softly into the liquid fat medium when the temperature of fat medium is 40⁰C.

Fourth, wait for the whole phantom to became totally solid and make a PET scan of it.

Without any care, the phantom will degrade within days after its production. We found that adding Germall-plus [START_REF] Huber | Multi-Modality phantom development[END_REF] as a conservation agent, greatly improved the conservation of the sample. In this way will keep several weeks in a normal fridge.

After the PEM scan, a US scan can be performed immediately or several days later.

III. IMAGING RESULTS

The PET images of the phantoms were obtained with a PHILIPS GEMINI TF 64. For US imaging, as there are several different ways to generate US elastography, we have studied both static elastography by using DP-9900 (Mindray,P.R.China) and Shear Wave Elastography using SuperSonic Imagine Aixplorer system with both 1D and 3D probes (SuperSonic Imagine, Aix en Provence, France) .

A cubic shaped phantom was first prepared to study static elastography and a breast shaped phantom had been prepared later for shear wave elastography.

For static elastography, a linear US probe with center frequency of 7.5 MHz and sampling frequency of 25MHz is used. A rectangular compressor with a polyethylene plate(size 88mm*66mm) is used to enlarge the US probe contact surface between the phantom surface. US probe was fitted inside the compressor by a slot cut. US radio-frequency (RF) data were acquired by a 32 bit data I/O card. Both pre-and postcompression US RF data were acquired. A US probe compression amount of 0.5mm was applied. The strain imaging was based on the RF data from pre-compression and the post-compression. Local tissue displacements were estimated with 1D cross-correlation techniques by comparing the gated pre-and post-compression radiofrequency (RF) signals [START_REF] Ophir | Elastography : a quantitative method for imaging the elasticity of biological tissues[END_REF]. For all the elastograms, the length of the correlation window was fixed at 2.5mm, and a overlap window of 80% was used. To remove the signal deformation due to mechanical compression, temporal stretching [START_REF] Varghese | Enhancement of echo-signal correlation in elastography using temporal stretching[END_REF] is conducted on post-compression signals with a stretching factor of the mean strain. After the displacement data were obtained, the axial strain was then computed from the displacement estimates using a third-order optimum low-pass differentiator [START_REF] Luo | Axial strain calculation using a low-pass digital differentiator in ultrasound elastography[END_REF]. The dynamic range (DR) of the static elastogram was set to 0~ 1.5%. SuperSonic Imagine uses a patented concept to introduce the shear wave into the body : ultrasound beams are successively focalized and sent into different depths of tissue creating a shear wave in the medium. Using ultrafast imaging, the speed of the propagation of the shear wave at every point in the image is measured, and quantitative elasticity map can be deducted in kilo Pascal (kPa).The end result is a real-time, quantifiable, user-independent and reproducible ShearWave™ Elastography information. The technique is totally safe for the patient and respects all acoustic standards defined by the FDA and CE mark. First, we show results from PET and US static elastography. Phantom cross section images from PET and static elastography are listed below. B-mode image tells the anatomical structure, static elastography shows the Strain Compared with US B-mode imaging, US elastography shows a better ability to identify tissue differences. For inclusions whose acoustic characteristics is similar with the normal tissue around it (iso-echoic), US B-mode imaging can't show a clear interface between the inclusion and normal background part but static elastography still can distinguish them effectively from their elasticity differences(Fig. 3). Second we display phantom imaging results from SuperSonic shearwave elastography. As SuperSonic shear wave elastography provides truly quantitative elastography that allows the physician independence from compressional movements inherent in conventional elastography. Fig. 4. shows the image fusion between whole body PET image and shear wave elastography.

IV. IMAGE FUSION BY POSITIONING SYSTEM

As the final purpose for the combination of ClearPEM and SuperSonic shear wave elastography system is to superimpose PET volumes together with US volumes, a 6D magnetic positioning system has been applied to assess the 3D volume registration. The spatial accuracy of the positioning system is 1 mm and 0.1 degree ..We first make a breast shaped cone for phantom preparation and added 3 fiducial markers on the cone wall. This is used for registering PET volume. If we scan the phantom by a whole body PET/CT, the fiducial markers can be scanned by the CT scanner, so we can use the image correspondence to choose the needed PET image by CT fiducial marked images. We used the positioning system to record the positions of the fiducial markers together with the 3D US probe, including the special angle of the US probe. With these positions data we can know compared with PET image, where should the US image be superimposed. Note that there is a window on the breast phantom cone for US probe scanning. Although the phantom will have a small deformation by US probe compression, only linear transformation is performed. With SuperSonic system both Bmode volume and elastography volume are simultaneously. Fig. 5. shows the fiducial markers and positioning sensors. And image fusion results are shown in Fig. 6. The result shows an acceptable matching that demonstrates that the 6D positioning system is working properly for PET/US volume matching and registration.

V. DISCUSSION AND CONCLUSION

This gelatin-agar PET/US phantom is developed for combination research of a breast PET(ClearPEM) and commercialized US-elastography Aixplorer system (SuperSonic Imagine). The phantom has already been tested by whole body PET and it works well. With the excellent special resolution of ClearPEM( ≈ 1.5mm ), it is ideal to test the performance of ClearPEM with this phantom, e.g. we can prepare tumor type inclusions with different sizes(1mm, 2mm, 3mm, etc.) to mimic the very realistic cases to demonstrate the performances of the ClearPEM system. With this phantom it is also possible to make a benchmarking evaluation between the performance of a commercially available PET/CT and the ClearPEM system. Once volumes are obtained from ClearPEM, the phantom can still be scanned by any US imaging system to get US images for PET/US image fusion study. We already found tumor like inclusion with sizes around 1mm can be detected by SuperSonic shear wave elastography system. The in-vitro performance test of ClearPEM system should be performed in the future with this phantom. The PET/US dual modality system should be integrated together so that simultaneous volumetric imaging could be performed, and the phantom is an ideal training object for volume registration and system performance evaluation.

For the ClearPEM scanning, the patient is laying with a prone position on the bed and the breast will be scanned between two parallel detector heads(Fig. 7). A mechanical arm will be used to hold the US probe between the two detector heads for simultaneous scanning (Fig. 8). A breast shaped cone will be installed on the bed to fit the breast nature shape and different breast sizes (Fig. 9). There will also be a open US scan window on the breast cone wall to fit the US probe (Fig. 10). For conclusion, we have developed a gelatin-agar dual modality phantom for PET/US breast imaging research. This phantom could mimic different breast tissues types such as fat tissue, glandular tissue, fibrous tissue and carcinoma. It works well for whole body PET/CT and for any US imaging system. Phantom should be used to test ClearPEM performance also. A 6D magnetic positioning system has been evaluated for PET/US volume registration and fusion. , Further test with ClearPEM system will be performed to optimize accuracy of this positioning system.

Fig. 1 .

 1 Fig. 1. Cross section phantom images show the US B-mode image, static elastography and whole body PET image. The white box in the whole body image tells the US probe scan region.Below is displayed the superimposed PET/B-mode/Static Elastography results.

Fig. 2 .

 2 Fig. 2. Comparison of PET/US B-mode/US static elastography results. 1st top left image shows PET/ US B-mode superimposed result. 2nd top right image shows PET/US Static Elastogram superimposed result, 3rd bottom left image shows PET/ US B-mode /US static elastography synthesis superimposed results and the 4th shows US B-mode/US static elastogram superimposed result. In the 4th image, red color stands for strain distribution, it tells the fat tissue mimicking part.

Fig. 3 .

 3 Fig. 3. When inclusion acoustical characteristics is similar between inclusion and background, US B-mode imaging can't distinguish inclusion clearly but static elastography can still show the inclusion from background. White arrows labeled the same inclusion in B-mode image, elastography and PET image.

Fig. 4 .

 4 Fig. 4. Image fusion result of US B-mode / shear wave elastography/whole body PET images. This showed triple information of : tissue anatomical information, tissue elasticity information and tissue functional information

Fig. 5 .Fig. 6 .

 56 Fig. 5. Fiducial markers and positioning system for image matching

  Fig. 7

Fig. 8

 8 Fig. 8

Fig. 9 Fig. 10

 910 Fig. 9 Fig. 10

TABLE I .

 I PHANTOM ACOUSTIC CHARACTERISTICS

TABLE II .

 II PHANTOM ELASTICITY CHARACTERISTICSTable II shows the elasticity measurement results of the samples, which could mimic real breast tissues from acoustic point of view. The 2 nd column shows what kind of breast tissue they can mimic by acoustical requirements. The last column shows what kind of breast tissue they can mimic by elasticity requirements. The phantom can mimic breast fat medium and carcinoma both by acoustical and elasticity requirements.

Scientific Research (CNRS) -LMA,13402 Marseille Cedex 20 -France. (email: lasaygues@lma.cnrs-mrs.fr). Dachun Zhang, Mingxi Wan are with Xi'an Jiaotong University,West Xian Ning Rd. 28, 710049 P.R.China. Nicolas Felix is with SuperSonic Imagine, Les jardins de la Duranne, Aix en Provence, France. Benjamin Frish is with CERN, CH-1211 Geneve 23, Switzerland

ACKNOWLEDGMENT

We thank cooperation from LIP (Laboratory for Instrumentation and Experimental Particle Physics, Lisbon) and IPO (Instituto Português de Oncologia, Porto) for this work.


Macarena Flores González
email: macarena.flores-gonzalez@cyu.fr

Elann Lesnes

Pre-service teachers' difficulties with the semiotic representation of the decimal number line

Keywords: Decimal numbers, number line, teacher education

As a semiotic representation, the numerical line is often used in school as a tool. However, it is also considered an epistemological obstacle, which can be problematic if one is not trained to use it. In this paper, we investigate the difficulties encountered by primary and secondary pre-service teachers in representing the number line of decimal numbers when numbers are given. Using a methodology with qualitative analyses, we analysed the written productions of 103 pre-service teachers when constructing these number lines. This allowed us to identify five categories of difficulties.

Introduction

The number line is frequently used from the first grade to represent natural numbers in mathematics instruction. A few years later, rational numbers are the first new types of numbers that students meet. Different representations are used and mostly a so-called "regions" model [START_REF] Bright | Identifying fractions on number lines[END_REF] which may take the form of colored regions, fraction strips, fraction circles, etc. However, the numerical line is also reinvested to represent this type of numbers. Decimal numbers are usually introduced in the same year from decimal fractions and placed on a numerical line. However, numerical lines are mostly seen only as tools (Lemonidis & Gkolfos, 2020), there seems to be no specific teaching on this object.

In turn, pre-service teachers will have to teach number lines as representations of numbers. This is problematic considering the result that they make errors when placing unfamiliar decimal numbers on a previously drawn number line [START_REF] Thipkong | Preservice elementary teachers' misconceptions in interpreting and applying decimals[END_REF]. In France, it seems that there is no specific teaching of number lines in the initial training of primary school teachers either, even if they are used as tools of representation. Therefore, our question is: how do pre-service teachers perceive the number line?

Didactic point of view on the numerical line

There is a lot of research on problems that students from grade 4 to 12 have with decimal numbers. These difficulties can be found when students read or construct number lines. However, the numerical line brings its own difficulties and can be an epistemological obstacle (Lemonidis & Gkolfos, 2020). Indeed, numbers are abstract objects to which we have access only through their representations. One of these semiotic representations is the numerical line [START_REF] Adjiage | Un registre géométrique unidimensionnel pour l'expression des rationnels [A one-dimensional geometric register for expressing rational numbers[END_REF]. On this line, a number is considered as the abscissa of a point. But [START_REF] Bright | Identifying fractions on number lines[END_REF] explain that unlike other visual representations such as the "regions" models which can be understood without the need to use any symbols, the number line requires the use of symbols, at least to define a unit. Using a numerical line therefore requires combining two types of information: visual and symbolic (Bright et al., 1988, p. 215). On a numerical line, the difference between two numbers is represented by the length of the segment whose extremities are the abscissae of the two corresponding points. Although the numerical line can be useful for relating decimal numbers and fractions, for developing the relative magnitude of those numbers, and for representation for the density property [START_REF] Widjaja | Locating negative decimals on the number line: insights into the thinking of pre-service primary teachers[END_REF]. Nevertheless, it can also be seen as an inadequate representation for differentiating decimal numbers from rationals [START_REF] Durand-Guerrier | Conceptualization of the continuum, an educational challenge for undergraduate students[END_REF], which could be a problem for the students and create didactic problems for teachers. This is especially true when trying to visualise the density property in a static line. Moreover, as [START_REF] Castela | La droite des réels en seconde : Point d'appui disponible ou enjeu clandestin ?[END_REF] shows, for students the number line for reals seems to be confused with the decimal line, and not all students make the link between the geometric field (in reference to the points of the line) and the numerical field (in reference to the numbers). Pre-service primary teachers have more experience with mathematics, and in particular with decimal numbers, than school students (because they continued to use them at school and in their daily lives). Thus, studies show that we find some similar difficulties in the conception of decimal numbers with this audience. For example, many of the pre-service teachers consider that whole numbers are not decimal or confuse the set of decimal numbers with the decimal writing of numbers [START_REF] Flores González | Quelles conceptions sur les nombres décimaux en première année de master MEEF 1er degré après formation en INSPÉ ? [What conceptions of decimal numbers after training in French teacher training institutes (INSPÉ) for the first year of a Master's degree in primary education?[END_REF]. The density property seems particularly hard to conceive [START_REF] Alatorre | Mexican primary school teachers' misconceptions on decimal numbers[END_REF]. [START_REF] Widjaja | Misconceptions about density of decimals: insights from pre-service teachers' work[END_REF] show that it is not recognised by about half of the pre-service teachers (in Indonesia) who incorrectly apply the "discretisation" of natural numbers to the set of decimal numbers. In the context of teacher training [START_REF] Putra | Praxeological change and the density of rational numbers: the case of pre-service teachers in Denmark and Indonesia[END_REF] shows that the mathematical and didactic use of the number line depends on the mathematical knowledge of the trainee teachers. He shows that Danish pre-service teachers are able to convert between fractional and decimal notation, they understand the density property of rationals and they can use the number line to investigate these properties with (fictitious) students. In contrast, Indonesian pre-service teachers treat fractional and decimal notation separately. When they try to use the number line, they construct two lines (one for fractions, one for decimals) and do not use the density property. [START_REF] Girit | Pre-Service middle school mathematics teachers' understanding of students' knowledge: location of decimal numbers on a number line[END_REF] compare the strategies of students and pre-service teachers in Turkey when estimating the value of a decimal number on a segment of the graduated number line (with a given unit). Their results show that the two groups approach this problem in different ways. However, both tended to think of decimals in succession (i.e. ignore the density property) and to give precise answers rather than specifying a range of possible values. In addition, they investigated pre-service teachers' anticipation of students' misconceptions and difficulties with the same task. The result was that pre-service teachers could only partially anticipate them.

Research question

It seems that the numerical line is not always specifically studied in a systematic way in pre-service teacher education. However, according to the previous section, problems with the number line have been identified among pre-service teachers. Thus, we ask: what difficulties do future teachers have in spontaneously representing the decimal number line?

Methods

This research is part of a wider research that seeks to find out the conceptions of pre-service primary teachers about decimal numbers [START_REF] Flores González | Quelles conceptions sur les nombres décimaux en première année de master MEEF 1er degré après formation en INSPÉ ? [What conceptions of decimal numbers after training in French teacher training institutes (INSPÉ) for the first year of a Master's degree in primary education?[END_REF]. To do this, our methodology is close to the qualitative methodology described in (Fraenkel et al., 2012, pp. 429-431). The analysis of the data is based on description, this does not preclude some quantification when it comes to studying in more detail a phenomenon that is being observed.

We tested the knowledge of pre-service teachers from four different National Higher Institutes of Professorship and Education (called INSPÉ) in France. We have developed a ten-question test on decimal numbers, but in order to answer the research question we are asking in this paper, we will focus on the one question that dealt with the number line: "Place the following numbers on a graduated line: ½ ; 1.51 ; 0.155 ; 1.5 ; 1.05."1 . The graduated scale was not allowed. The data collected are the written responses of the pre-service teachers, which gave us direct access to the semiotic representations used. They were able to answer the question without time constraint.

The test was completed only once in each institute (the four institutes were coded as ANT, MSA, SGL and REN), but it was performed at two different times (for the first time, the number 0.155 was not present in the test). These are detailed below:

• the first one (without the value 0.155) took place during the first quarter of the year 2022, about two months after the lesson on decimal and rational numbers with primary school preservice teachers from two INSPÉ (34 from ANT and 30 from MSA) ; • the second one took place in the last quarter of the year 2022 before the lesson on decimal and rational numbers. The students of the second run are primary school pre-service teachers (13 from SGL) and middle and high schools pre-service teachers (26 from REN).

All the pre-service teachers mentioned in this paper are in the first year of the master's degree for teaching.

Analysis of the question

We want to know the semiotic elements used spontaneously by pre-service teachers on the register of semiotic representation of the numerical line as defined by Duval (2017) [START_REF] Adjiage | Un registre géométrique unidimensionnel pour l'expression des rationnels [A one-dimensional geometric register for expressing rational numbers[END_REF][START_REF] Teppo | Visual representations as objects of analysis: the number line as an example[END_REF]. According to Duval's theory, we can say that the number line results from the articulation of two different registers of representation, more precisely the geometric register (Euclidean line) and the arithmetic register (numbers) (Lemonidis & Gkolfos, 2020, p. 37). Therefore, we choose decimal numbers up to the thousandth with essentially 3 different digits (0, 1 and 5), and we vary the positioning of these digits which changes the value of the number. Only 0.5 is given in fractional writing (½) because it's a number that pre-service teachers should know, and one that research shows pupils have problems with. The numbers were chosen so that they could be placed on the same graduated line but with some spatial constraints (see Figure 1). Moreover, this is not a traditional task for pre-service teachers because they are used to placing numbers on an already drawn graded line but not to having to place the graduations themselves. Indeed, in a static line, the differences between tenths, hundredths and thousandths become difficult to represent and observe: how do pre-service teachers cope with this difficulty? We will identify the semiotic elements of the line considered by the pre-service teachers to represent the given numbers.

Method for analysing the semiotic representations

In order to analyse the difficulties in the written productions, in this article we set aside the answers that present expected semiotic representations (about 16% of answers collected) and we are focusing on responses that show signs of a struggle against the obstacle of the numerical line. Then, we establish two big categories of errors in relation to the difficulties already identified in the first section. First, there are difficulties relating to decimal numbers in general which may result here in a wrong order of numbers on the numerical line, or by an incorrect translation of ½ into decimal writing (if applicable). Second, there also are difficulties relating to the representation of the difference between two numbers by the length of a segment, especially with numbers that do not have the same orders of magnitude (for example 0.155 and 1.5). The density property, which is a major difficulty for students and pre-service teachers, is not really visible here. This research is exploratory, so we expect other categories to emerge in the future.

Results of the first experiments

A summary of the errors identified is presented in table 1. In the following, we present the different categories we identified by grouping together errors that appeared several times in the students' answers and make some hypotheses explaining them. A single answer may include several errors. Error in the decimal writing of ½

We find the confusion between ½ and 1.5 in 9 primary pre-service teachers' answers (even after the lesson on decimal and rational numbers) and 1 secondary pre-service teacher (see figure 2). 2 other primary school pre-service teachers (1 from ANT, 1 from SGL) place ½ on the scale corresponding to 1 (see figure 3). A part of these errors (4/12) appear to be related to the transition from fractional to decimal writing. However, we can explain most of these errors (8/12) by a confusion between the part-whole and the number aspects of fractions as we will see below. The number ½ can be a difficulty as we saw in the description of the first category. If the unit chosen is the whole traced segment, the confusion with the part-whole aspect of fractions leads some students to place ½ in the middle of the segment and not in the middle of the interval [0 ; 1] (see figure 4). In this category we find 6/64 primary pre-service teachers after the lesson and 2/39 pre-service teachers before the lesson.

Just ordering the numbers

In this category, we find representations of the number line that do not take into account either the magnitude of the number or a unit of length that makes it possible to assign a position to the number on the line. Thus, pre-service teachers seem to redefine the given task as "arranging numbers in ascending (or descending) order on a line" as we can see on the figure 5. There are 3 answers in this category before the lesson (1 from a secondary pre-service teacher) and 5 answers after the lesson.

No clearly defined unit

We observe that some number lines do not have a reference unit for placing the numbers requested.

Although in some cases the 0 or 1 is placed, this causes the difficulty of locating ½, as we saw with the "½ in the middle of the segment" category, but also the difficulty of locating the numbers on a line with an adequate order of magnitude. This use of the number line occurs before (11/64 pre-service teachers) or after (12/39 pre-service teachers) the lesson (see figure 6). The answers from the previous category are included in this category (see figure 5). It is possible that, according to the student, the unit chosen is clear, in particular when 0 or 1 is placed (see figure 7). However, we often find other errors on the number line, which leads us not to favour this hypothesis, which also does not correspond to a classical use of the regularly graduated line as it may be presented at INSPÉ or, in primary or secondary school.

Difficulties to perceive and/or construct the difference between one hundredth and five hundredths

This category includes responses that reveal a certain difficulty in semiotically representing differences in terms of length between 1/100 and 5/100. Thus, the distance between the points corresponding to the numbers 1.5 and 1.51 (one hundredth) is equal to or greater than that between the points corresponding to the numbers 1 and 1.05 (five hundredths) as we can see on figure 8. We assume that the digit 0 of the number 1.05 is associated with a small number and therefore "just next to" 1, regardless of the number of hundredths. In addition, 12 pre-service teachers did not place the numbers 1.05, 1.5 or 1.51 (mostly in REN and ANT), which makes it impossible to know whether they perceive this difference on the number line.

If we take into account the students who do not clearly define a unit and those who misplace or do not place 1.05, 1.5 or 1.51, we find that 75% of the constructed number lines do not correctly place these points in such a way as to highlight the relative differences between these numbers.

The position of the dot corresponding to the number 0.155 may also be an indication of a misperception of the relationships between tenths, hundredths and thousandths. Thus, 14 pre-service teachers (out of 39 because during the run before the lesson, 0.155 was not included) positioned the dot incorrectly. Either because of confusion with another number like 1.155 or 1.55 (5/14), or by placing it too close to 0 (6/14) as we can see on figure 9. On figure 8 (student n° 42), there is another error identified with the position of 0.155. However, the vast majority of students do not scale the number line finely enough to accurately locate the position of the dot. It might have been interesting to have them place another number close by, such as 0.15.

Conclusion

The aim of this research was to study the difficulties of future teachers when constructing the semiotic representation of the numerical decimal line. Thanks to the methodology used, we were able to describe and identify categories for these difficulties. First, confusion between ½ and 1.5 is common among students beginning to learn rational and decimal numbers, but it was also present in 13% of answers from pre-service teachers after the lesson. Some answers use only ½ to divide a segment into two equal parts without taking into account the location of ½ in the number line. This may be a consequence of the current teaching progression (fractions then decimals), or a problem with the status of the number ½. As well, it is not insignificant that pre-service teachers answer the question by giving a list of numbers arranged in ascending or descending order. For them, there seems to be a lack of understanding of the number line and the graduations in order to be able to represent these numbers in terms of length measures. Furthermore, the construction of the number line with decimals does not seem to be based on a unit of reference according to several pre-service teachers: without a reference, how do they place the requested numbers on the number line? Moreover, one of the striking results of this research is the large number of student teachers (almost half) who have difficulty representing the difference between one hundredth and five hundredths in terms of measuring lengths (before and after the lesson). What transposition of the number line will they make to their future students?

We agree with Teppo & Van den Heuvel-Panhuizen (2014) on the complexity of the number line as a semiotic representation of numbers, and that therefore we argue that specific teaching of the number line is needed in teacher education to understand the issues of this representation. Finally, a more detailed analysis in search of links between the categories found in this paper, as well as interviews with teachers in training, are two prospects for our short-term research in order to better understand the origins of the difficulties identified in this work.

Figure

  Figure 1: a possible number line with the numbers ½; 1.51; 0.155; 1.5 and 1.05

Figure

  Figure 2: "½ = 1.5" in two answers, students n°98 (ANT) and n°14 (SGL)

Figure 4 :

 4 Figure 4: ½ in the middle of the segment, n°7 (SGL) and n°109 (ANT)

Figure 5 :

 5 Figure 5: numbers in ascending order, student n°5 (SGL) or in descending order, student n°53 (MSA)

Figure 6 :

 6 Figure 6: no unit clearly defined, students n°28 (REN) and n°60 (MSA)

Figure 7 :

 7 Figure 7: 0 placed but no unit clearly defined, students n°10 (SGL) and n°81 (ANT)

Figure 8 :

 8 Figure 8: 1/100 equal or greater than 5/100, n°42 (REN) and n°90 (ANT)

Figure

  Figure 9: 0.155 too close to 0, student n°12 (SGL)

Table 1 : Summary of the errors

 1

	Pre-service teachers	Error in the decimal writing of ½	½ in the middle of the segment	Just ordering the numbers	No clearly defined unit	Difficulties with 1/100 and 5/100
		After the lesson on decimal and rational numbers
	ANT (34)	7	5	3	7	15
	MSA (30)	2	1	2	4	14
		Before the lesson on decimal and rational numbers
	SGL (13)	2	1	2	5	3
	REN (26 secondary pre-service teachers)	1	1	1	7	12
	Total (103)	12	8	8	23	44

Traduction from French, the decimal commas used in French have been replaced by decimal points. French version: "Placer sur une droite graduée les nombres suivants : ½ ; 1,51 ; 0,155 ; 1,5 ; 1,05".


Danxin Cui
email: danxin.cui@sorbonne-nouvelle.fr

Dominique Mariko
email: dmariko@yseop.com

Hugues De Mazancourt
email: hdemazancourt@yseop.com

A sequence to sequence transformer data logic experiment

In this paper we present experiments to evaluate how a T5 model behaves with regard to input data fidelity. The rationale behind these experiments is to evaluate if a sequence to sequence transformer can be constrained into generating the specifics of a financial report, and more generally whether it can trustfully reproduce a semantic logic, and to what extent.

Introduction

T5 by [START_REF] Raffel | Exploring the limits of transfer learning with a unified text-to-text transformer[END_REF] recently demonstrated strong constrained and data to text generation capabilities. Experiments have been lead on AQG tasks (Grover et al. (2021)) and on the WebNLG dataset as to explore the data to text capabilities of a T5 model. In particular, [START_REF] Kale | Text-to-text pretraining for data-to-text tasks[END_REF] demonstrates T5 model shows interesting capacities in generalization to new domains and relations, and [START_REF] Kasner | Data-to-text generation with iterative text editing[END_REF] proposes significant text generation experiment even without any in-domain examples.

Text generation in Finance can be very demanding as to the level of constraint a neural model should comply with. The objective of our experiment is to evaluate which data formalism we should build to achieve similar results as the results achieved by T5 models on WebNLG tasks.

In order to produce this evaluation, we chose to create a data set focusing on semantic intentions. Intentions are objects describing Natural Language Generation (NLG) pipelines based on Abstract Categorical Grammars semantic and syntactic items, as defined by [START_REF] Salmon | Natural language generation using abstract categorial grammars[END_REF], that can be specialized and combined together. We also implemented a set of metrics for NLG evaluation based on BLEU and BERT-SCORE.

Corpus

The initial corpus for this experiment is a set of 4159 public online US and UK Market Reports. We limit the experiments to the financial domain as to prove more accurate results.

Corpus generation

Raw text is extracted with a home made pdf extractor based on PDFMiner, deleting all tables and titles. A corpus analysis is performed on these raw extractions, leading to the definition of handcrafted grammars describing each intention. These grammars allow to tag each sentence belonging to one of the intentions of interest, and to extract for each sentence a set of relevant chunks which are then transformed into triples. These intentions are currently defined and used by Yseop's generation core engine. For the sake of the experiment, we choose to retain only simple intentions and sentences which can also be produced by this NLG engine.

Corpus transformation

Data logic

To ensure the precision and accuracy of the generated sentences, we have chosen a data-to-text representation method, which particularizes key elements of sentences in our financial corpus and extract triples, using an automated method close to [START_REF] Li | Triple-to-text generation with an anchorto-prototype framework[END_REF].

This method was applied as to define a corpus of intentions. An intention is a sentence corresponding to a specific expression of a financial indicator's value. Yseop has shown that a handful of such intentions are sufficient to describe a data-driven narrative in a speciality domain, such as Finance. For instance, an intention DescribeValue is a sentence stating the value of a financial indicator at a precise time and an intention DescribeVariation is a sentence describing the variation in time of a financial indicator's value. We use the prefix Merge to define a sentence composed of two or more intentions. In order to identify and extract these intentions in our corpus, a Ruta grammar [START_REF] Kluegl | Uima ruta: Rapid development of rulebased information extraction applications[END_REF] was created to automate the triples extraction, imitating [START_REF] Gardent | Deep learning based question generation using t5 transformer[END_REF] data modeling. This grammar first uses dictionaries as well as POS-tag patterns to identify financial key elements related to these intentions and characterize them into one of the following categories:

• financial indicator • reference (time and geographical element)

• measure

• predicate

Indicators, measures and dimensions are generic elements that can be found in all intentions. Predicates, on the other hand, vary according to the intention. To create a dictionary that take into account this specificity and can later be used for intention detection, we conducted a manual analysis of the market reports that allowed us to classify the predicates specific to each intention. Synonyms and antonyms have also been included to complete and enrich the dictionary (see Table 2 for some examples)

In a second stage, the grammar looks for syntactic combinations of these key elements. For example, a sentence containing exclusively a financial indicator, a state predicate, one measure and an optional time and/or geographic dimension will be extracted as an intention DescribeValue.

Sentences selected from financial corpus are then transformed into a set of triples (hereinafter referred to as complete triples), organized into subject-predicate-object structure, && serving as a connector. See Table 11 in Appendix for a detailed overview.

Data construction

There is no theoretical limit to the maximum sequence a T5 can encode, the only constraint being the memory requirements. We did not work on this specific aspect as this is not the purpose of the experiment. We choose to work with a maximum input sequence of 400, trying to keep the experiments into a small memory consumption interval. Owing to the limited capability of our T5 model, triples that are too long cannot be fully processed by the model and therefore the generated sentences will be incomplete. In order to work around this problem, we trimmed the triples by replacing the elements with simpler ones and reducing the length of predicates, then creating simplified triples. In these simplified triples, financial indicators are replaced with their semantic class (predefined in our grammar). For example, abuse tax and absolute organic operating costs both belong to the class expenses, so they were replaced with the generic short form expenses in the simplified triple. Measures are replaced a simpler number ($ + two digits), all time dimensions are substituted by a preposition (if there is one in the initial dimension time) plus a year (in 2019 , for example) and the expression in America replaces any term in the geography dimensions. We refer to this trimmed triples as simple triples.

We trained a model with simple triples to evaluate if our formalism was rich and accurate enough so the model could infer the data logic, and used the complete triples to train a model for inference.

According to the type and number of key components, target sentences are sorted into different intentions. Our grammar for now is able to recognize and annotate 8 intentions, DescribeValue and DescribeVariation being the core intentions, based on which we developed 6 others (see Table 1) Two sets of experiments have been built for each of the data sets created from simple and complete triples. Each experiment is detailed in Section 5.

Data sets

Applying the triples generator on a 4159 raw corpus files, we have collected 20615 sentences annotated for both complete and simple triples. The frequency for each intention in each set is presented in Table 1.

We randomly sampled three different training and testing partitions in order to leverage the scarcity of our data. All measures provided below are aggregated means of these three partitions.

Models

We used the T5 sequence to sequence transformer from the transformers library by [START_REF] Wolf | Transformers: State-of-the-art natural language processing[END_REF] to run the experiments, using an Nvidia GeForce RTX 2070 GPU with 8 Go RAM. We trained two models, one from complete triples, and another one from simple triples, for each of our 3 data partitions. A simple 1 and a complete 2 trained models are available for reproducibility on Hugging Face hub.

Training parameters

Our objective is to evaluate our data formalism and an associated sequence to sequence model capabilities, so we did not experiment much on fine-tuning. We used a standard set of training parameters for all models and trained for one epoch and batches of 6, using the Hugging Face transformers library and the AdaFactor optimization method, keeping all default parameters except for the following:

• learning rate lr=1e-3 • regularization constants eps=(1e-30, 1e-3) • decay_rate=0.7

Generation parameters

At inference, we tried to limit hallucinations and omissions while maintaining a good level of richness on the structure and vocabulary of the generated sentences.

We use a mix of top_k _ and top_p sampling for generating. Top_k is a sampling scheme, in which the K most probable next tokens are filtered and the 1 https://huggingface.co/yseop/FNP_T5_ D2T_simple 2 https://huggingface.co/yseop/FNP_T5_ D2T_complete probability mass is redistributed among only those K next tokens. Top_p is also a sampling scheme, managing creativity of the model. It chooses from the smallest possible set of words whose cumulative probability exceeds the probability p. This way, the size of the set of words (a.k.a the number of words in the set) can dynamically increase and decrease according to the next word's probability distribution.

We chose the following process for selecting the most suitable top_p and top_k for our generator:

• select one representative sentence and its corresponding triples for every intention.

• prepare 10 top_p (from 0.12 to 1) and 10 top_k (from 10 to 100) and combine them in a pairwise fashion to get 100 (top_p, top_k) couples.

• generate 10 sentences from a single triple.

Then measure the similarity between these 10 sentences with ROUGE3 and collect the measure under different top_p and top_k couples. We considered this average ROUGE to measure the creativity of our model. The bigger it is (less variation in the 10 generated sentence), the less creative the model is.

• compare the 10 generated sentences with the initial sentence in order to collect the ROUGE measure under different top_p and top_k couples. The bigger it is, the more accurate our model is.

The top_p and top_k selected for simple triples and complete triples are (0.72, 40) and (0.82, 90), respectively. The model gives the best performance with them, leading to results presented in section 6.

Evaluation metrics

During the experiment, we have noticed that both the length of elements in the triples and the model's familiarity with them can influence the quality of the generation. We have adopted 3 methods to assess the quality of our models.

• The generated sentences are compared with the initial sentences and the lexical similarity is measured with a BLEU score [START_REF] Papineni | Bleu: a method for automatic evaluation of machine translation[END_REF]) 4 , adapted so it considers bi-grams.

• The generated sentences are compared with the initial sentences and the semantic similarity is mesured with a BERT-SCORE [START_REF] Zhang | Bertscore: Evaluating text generation with bert[END_REF].

• The generated sentences are reintroduced into the triples generator to obtain regenerated triples. The inspiration for regenerating the triples and evaluate them against the original ones comes from Veksler et al. ( 2019) 's work on how to assess a key level of information for NLG. The degrees of similarity between the regenerated triples and the original ones offers another point of view on the quality of generated sentences and assesses the credibility of the data logic initially chosen. We used both BLEU and BERT-SCORE to evaluate these similarities.We will refer to these measures as Triple BLEU and Triple BERT.

It is important to notice that the triples comparison results is fully automated and neither human evaluation nor inter-annotator agreement statistics have been performed. Since the triples production process biases the performance measure, and is used both at training and inference, we are in fact evaluating the capability of our model to preserve the "fixed-pointedness" of T5 with respect to our representation rather than the T5 natural language generation power.

Experiments

We defined two experiments, one training and evaluating for complete triples (see subsection 5.1), the other for simple triples (see subsection 5.2), and computed the four metrics previously detailed for each experiment. The measures provided are arithmetic means of the scores evaluated for all models created from our three different data partitions.

In the following subsections, we will refer to any element issued from the original corpus sentences as original. For each table of results, we present the actual number of triples that could be regenerated in regard to the number of sentences generated at inference available for testing.

Experiment 1

In this initial experiment, we used complete triples to fine-tune a T5 model. A data sample is available in Table 3, results are provided in Table 8. The BLEU score shows important variations in between intentions, due to the fact that some intentions are more complex and contain more elements than simpler ones like DescribeValue, and because they are less represented in the training data. Having around 4000 training examples seems to be a pre requisite to obtain significant improvement on the results.

Experiment 2

Simple

In this experiment we trained another model to learn and generate from simple triples.

The objective here is to workaround the limitations of our model in low memory consumption mode. The process for training a model for simple triples is the following:

• complete triples are simplified • we simplify the original sentences by replacing the original elements by the simple ones (indexed by simple triples)

The model is trained with simple triples against these simplified sentences (see example provided in Table 4), then simplified original sentences used for training are compared with the sentences generated at inference (an evaluation sample is provided in Table 5).

Restored

To affect a metric to sentences generated at inference from simple triples models, we retain two additional features:

• in the sentences generated at inference, we restore the original elements using their index in the original sentences, and compare these restored sentences with the original ones. An example of this transformation is provided in Table 6.

• we regenerate triples from these restored sentences, and compare them with the complete triples presented in section 5.1

We will refer to the triples and sentences in this experiment as restored. The results are provided in Table 10.

Results analysis

BLEU and BERT-SCORE leads to different conclusions and the BLEU score is generally lower than BERT-SCORE. This is because the 2 metrics evaluate the sentence at different levels.

Evaluating for triples

We expect sentences in financial report to contain all key information provided in the input data. However, BLEU and BERT-SCORE are incapable of examining the completeness of generated sentences. To achieve this goal, we passed the generated sentences to the triples generator and evaluate the regenerated triples with BLEU and BERT-SCORE. The higher the score is, the more complete the generated sentence is.

We were not able to regenerate any triple for a significant amount (22%) of test set sentences generated at inference, neither for complete triples nor for simple triples. The results take into account this information loss, when this happens the Triple BLEU and BERT-SCORE are evaluated to zero. This is partly due to our triples generator, partly to the structure of the sentence generated at inference time. Our triples generator is very dependant from the lexical layout of the sentence. In some cases, generated sentences which would be qualified for triple extractions are not recognized as such and ignored. On the other hand, and for the same reasons, the triple generator will also ignore ill-formed sentences.

The attribution to each case is still a work in progress. We provide examples of ignored generated sentences from which we could not regenerate any triple in Table 7.

This leads to important discrepancy in the Triple BLEU results, between different types of intentions and between complete versus simple triples experiments. Nevertheless we can still directly link the fidelity of the results to input data with the size of the training set.

Simple triples achieve significant better Triple BLEU score than complete triples, due to the fact that sentences generated from simple triples are shorter, and usually mirror the original simple triple sinformation much better than sentences generated from complete triples (often interrupted before a human readable sentence is fully generated at inference time, thus regenerating incomplete triples or none). For complete experiment, we were able to regenerate 85 % of the indicators present in the triples at inference and 95% of the indicators for the simple triples experiment.

Evaluating for sentences

BLEU evaluates the generated sentences on lexical level. The significant difference between complete and simple results comes mainly from the number of triples we were able to regenerate in each case, the simplest intentions for which a lot of training data was available being once again favored in both cases.

We can witness an improvement on average (from 0.423 average BLEU for sentences generated from complete triples at inference to 0.656 for sentences generated from simple triples), yet the similarity between restored sentences and original sentences (0.429 average BLEU) is only slightly higher than between original sentences and sentences generated from complete triples (0.423 average BLEU), mainly due to the risk of information loss during the process of restoring the original information in simplified sentences. .

Figure 1 shows that, as the complexity of intention increases, average BLEU score for simplified sentences generated from simple triples exceeds BLEU for complete sentences generated from complete triples and also restored generated sentences.

While the sentences are short (intention is less complex), a small lexical change (change of predicate for instance) is reflected in a big drop in BLEU score. For the same intention, the simplified generated sentence is usually the shortest. Therefore, under simpler intention, (e.g. DescribeValue), simplified sentences generated from simple triples obtained the lowest score. However, as the intention becomes more complex, the length of simplified sentences increases, which offsets the influence of lexical change in BLEU score. In addition, the BLEU score of simplified generated sentences re-

Original Simplified sentence

Generated Sentence

Results was 10% compared to 11%. Results was 10% (-0.71) and remained at the same level compared with 11%) Table 5: Simplified original sentence and sentence generated from simple triples model at inference for Operating margin was 5.8% compared to 8.5%.

Restored sentence

Operating margin was 5.8% (-0.71) and remained at the same level compared with 8.5% Revenues for the nine months ended September 30, 2020 were $421.7 million, an increase of Revenue for January-September period amounted to EUR 499.6 (400.5) mains relatively steady compared to the other 2 types of generated sentences. This phenomenon tends to prove that simplification of initial sentences and initial triples does improve the performance. And another proof is that this method generates 1879 sentences with BLEU score in interval 0.98 to 1 for simple triples models, while the number of sentences generated from complete triples and restored triples models scoring within this interval is 1600 and 1783, respectively. We evaluate with BERT-SCORE on semantic level. Taking BERT as the standard, there is little difference between the aggregated measures for sentences generated at inference from complete triples, simple triples or restored triples models. It's interesting to notice that restored simple sentences for well defined intentions such as DescribeValue exhibit a BERT-SCORE close to complete triples generated sentences (0.936 and 0.944 respectively), so this technique might be a way to workaround the memory constraints of the T5.

Error analysis

We have observed notable gaps between the BLEU and BERT-SCORE measures. We identified at least 4 reasons why this might occur:

1. Different verbs of same semantic meaning are employed in generated sentences:

• Original sentence: The total gaming margin in online games during the quarter amounted to 4.7 • Generated sentence: The total gaming margin in online games during the quarter was 4.7

2. The position of dimension time or dimension geography changes (slight influence):

• Original sentence: Revenue for 2016 amounted to 245 million. • Generated sentence:

Revenue amounted to 245 million for 2016.

When the indicator in the triples starts with

a lowercase, the model adds complement to it, which may be different from the complement in the original sentence. And sometimes, different complements may lead to different conjunctions of verb:

• Original sentence: The value of deferred tax assets at 31 December 2016 was C190 million.

• Generated sentence: The total deferred tax assets at 31 December 2016 were C190 million.

4. Predicates used in the triples don't indicate the tense of verbs. Hence, the tense of generated sentence may be different from the original one:

• Original sentence: The annual savings in interest costs from this refinancing amounts to approximately US$29 million. • Generated sentence: The annual savings in interest costs from this refinancing amounted to approximately US$29 million.

The first three examples show that a lexicalbased measure as BLEU is clearly not suitable to evaluate NLG systems.We tried to leverage this issue by evaluating triples against regenerated triples, this evaluation being less sensitive to semantic variations while retaining enough syntax for comparison.

Conclusion and future work

We have evaluated how a T5 sequence to sequence transformer behaves in data to text generation, using a combination of BLEU and BERT-SCORE on triples (with simple triples achieving the best Triple BLEU score of 0.656). The result gap between simple and complete triples experiments demonstrates that transforming initial sentences into simple ones and generating sentences from simple triples contributes to increasing the completeness of the generated sentences and the data logic accuracy.

Future work will focus on leveraging the induced bias of the triple generator as to propose more accurate automation of the triple extraction, and working on the current limitations of the T5 model to extend the length of input sequences keeping memory consumption as low as possible.

  liabilities were ¥309.0 billion, an increase of ¥2.4 billion or 0.8%, from the end of the previous fiscal year Non-current liabilities | valIs | ¥309.0 billion && Non-current liabilities | incBy | ¥2.4 billion or 0.8% Non-current liabilities were The right-of-use asset and discounted lease liability related to discontinued operations are C398 million as at 1 January 2019. Discontinued operations | infBy | the right-of-use asset && lease liability | valIs | C398 million && C398 million | dTime | as at 1 january 2019 The right-of-use asset for the right-of-use asset and the right-For the nine months ended September 30, 2020, revenues were $421.7 million, up 8.9% or $34.4 million from $387.3 million in the same period in 2019. Revenues | valIs | $421.7 million && $421.7 million | dTime | for the nine months ended September 30, 2020 && revenues | incBy | 8.9% or $34.4 million

Figure 1 :

 1 Figure1: Average BLEU score on every intention for simplified generated sentence, complete generated sentences and restored generated sentences. From DescribeValue to MergeDescribeVariationWithContributor, the complexity of intention raises.

Table 1 :

 1 Complete list of Intentions

	Intention	Definition

Table 2 :

 2 Examples of predicate dictionary entries

Table 3 :

 3 Complete triples, generated sentence and regenerated complete triples example for original sentence Operating margin was 5.8% compared to 8.5%.

	Triple		Generated Sentence	Regenerated Triple
	Operating margin | valIs | 5.8% && 5.8% | comTo | 8.5%	Operating margin was 5.8% (versus 8.5%).	Operating margin | valIs | 5.8% && 5.8% | comTo | 8.5%
	Original	Generated	Regenerated
	Triple	Sentence	Triple
	Results | valIs | 10%	Results was 10% (-0.71) and remained at the	Results | valIs | 10% (-0.71)
	&& 10% | comTo | 11%	same level compared with 11%	&& 10% (-0.71) | comTo | 11%

Table 4 :

 4 Simple triples, generated sentence and regenerated simple triples example for original sentence Operating margin was 5.8% compared to 8.5%.

Table 6 :

 6 Sentence generated from simple triples model at inference restored with original elements for Operating margin was 5.8% compared to 8.5%.

Table 7 :

 7 Non regenerated triples sample

	Intention	# test	# nan triples	Triple BLEU	Triple BERT	Sentence BLEU	Sentence BERT
	DescribeValue	990	163	0.781	0.960	0.646	0.944
	DescribeVariation	1492 173	0.693	0.951	0.573	0.941
	DescribeValueWithContributor	250	146	0.304	0.864	0.364	0.896
	DescribeVariationWithContributor	61	14	0.443	0.908	0.325	0.911
	MergeDescribeValue	1149 392	0.363	0.888	0.556	0.935
	MergeDescribeValueWithContributor	15	6	0.264	0.868	0.346	0.904
	MergeDescribeVariation	146	61	0.259	0.865	0.359	0.916
	MergeDescribeVariationWithContributor 7	3	0.188	0.854	0.211	0.906
	Mean	-	-	0.412	0.895	0.423	0.919

Table 8 :

 8 BLEU and BERT-SCORE (F1) results by intention for complete triples model. nan triples stands for non regenerated triples

	Intention	# test	# nan triples	Triple BLEU	Triple BERT	Sentence BLEU	Sentence BERT
	DescribeValue	990	41	0.941	0.990	0.469	0.919
	DescribeVariation	1492 55	0.914	0.985	0.590	0.936
	DescribeValueWithContributor	250	93	0.499	0.899	0.407	0.889
	DescribeVariationWithContributor	61	14	0.564	0.919	0.453	0.915
	MergeDescribeValue	1149 110	0.819	0.964	0.604	0.931
	MergeDescribeValueWithContributor	15	5	0.397	0.891	0.471	0.899
	MergeDescribeVariation	146	27	0.656	0.934	0.491	0.919
	MergeDescribeVariationWithContributor 7	2	0.459	0.890	0.405	0.899
	Mean	-	-	0.656	0.934	0.486	0.913

Table 9 :

 9  results by intention for simple triples model. nan triples stands for non regenerated triples

	Intention	Triples BLEU	Triple BERT	Sentence BLEU	Sentence BERT
	DescribeValue	0.823	0.971	0.554	0.936
	DescribeVariation	0.684	0.944	0.537	0.930
	DescribeValueWithContributor	0.337	0.870	0.298	0.876
	DescribeVariationWithContributor	0.351	0.884	0.323	0.897
	MergeDescribeValue	0.653	0.936	0.588	0.932
	MergeDescribeValueWithContributor	0.274	0.868	0.396	0.891
	MergeDescribeVariation	0.494	0.903	0.434	0.911
	MergeDescribeVariationWithContributor 0.257	0.859	0.299	0.891
	Mean	0.484	0.904	0.429	0.908

Table 10 :

 10

results by intention for restored simple triples and sentences (# of test sentences and non regenerated triples is the same as in Table

9

)

https://github.com/pltrdy/rouge

https://www.nltk.org/_modules/nltk/ translate/bleu_score.html

Acknowledgements

We thank the FNP 2021 Committee for the opportunity to publish this research experiment.


D Titov

J.-P Bibring

A Cardesin

T Duxbury

F Forget

M Giuranna

F González-Galindo

M Holmström

R Jaumann

Anni Määttänen

Atmospheric Observations by Mars Express

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

ATMOSPHERIC OBSESRVATIONS BY MARS EXPRESS

D. Titov 1 , J.-P. Bibring 2 , A. Cardesin 3 , T. Duxbury 4 , F. Forget 5 , M. Giuranna 6 , F. González-Galindo 7 , M. Holmström 8 , R. Jaumann 9 , A. Määttänen 10 , P. Martin 3 , F.

Montmessin 10 , R. Orosei 11 , M. Pätzold 12 , J. Plaut 13 , and MEX SGS Team The structure of the ionosphere sounded by the MARSIS radar and the MaRS radio science experiment was found to be significantly affected by the solar activity, the crustal magnetic field, as well as by the influx of meteorite and cometary dust. MARSIS and ASPERA-3 observations suggest that the sunlit ionosphere over the regions with strong crustal fields is denser and extends to higher altitudes as compared to the regions with no crustal anomalies. The ionospheric plasma expands to higher altitudes where it contacts with the solar wind plasma. Reconnection of solar magnetic field lines carried by the solar wind with field lines of crustal origin opens channels through which the ionospheric plasma escapes to space, producing strong and narrow cavities in the density. The situation is very different on the night side where the ionosphere has patchy structure. Such patchy ionizations are observed in the regions where field lines have a dominant vertical component. Through these patches the ionospheric plasma from the dayside penetrates and supplies the nightside ionosphere.

  series, as well as new elements and potentially new opportunities for discoveries. It will be boosted by collaboration and synergies with NASA's MAVEN, ESA-Roscosmos Trace Gas Orbiter and other missions. The talk will give the mission status, review the recent science highlights, and outline future plans.

	Mars Express has fully accomplished its objec-
	tives set for 2015-2016. The mission provides
	unique observation capabilities amongst the
	flotilla of spacecraft investigating Mars. The
	mission extension till the end of 2020 has been
	approved by the ESA Science Program Com-
	mittee. The future observation program includes
	both augmenting the coverage and extending
	long-time


Cecilie Carlsen Bach

Rikke Maagaard Gregersen

Ingi Mathilde Kjaer Pedersen

Uffe Heinesen Højsted

Thomas Jankvist

Uffe Thomas

Ingi Heinesen Højsted

Uffe Thomas Jankvist

From design principles to implementation heuristics: A networking of theories perspective on implementation research

Keywords: Implementation research, networking to theories, design-based research, representation

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Networking of theories and design research

Networking of theories is a research practice initially described at CERME4 [START_REF] Kidron | Theoretical perspectives and approaches in mathematics education research[END_REF]. It may be defined as a meta-research practice [START_REF] Bikner-Ahsbahs | Theories of and in Mathematics Education[END_REF], often applied in combination with other methodologies, e.g., design research (DR) (see [START_REF] Alberto | Networking theories in design research: an embodied instrumentation case study in trigonometry[END_REF]. Eight networking strategies are provided and described in [START_REF] Prediger | Networking strategies and methods for connecting theoretical approaches -First steps towards a conceptual framework[END_REF], stretching from understanding own/others to integrating locally and synthesising. Two strategies, those of coordinating and combining, focus on how the use of theoretical perspectives may help to understand an empirical phenomenon. When coordinating, the theoretical perspectives must be compatible, so that the basic theoretical (and philosophical) assumptions are not be too far apart. The demand for compatibility in the theoretical basis does not appear to the same extent when combining. According to [START_REF] Niss | On the mathematical competencies framework and its potentials for combining and networking with other theories and theoretical constructs[END_REF], "A theory is a theory of something, i.e., it deals with certain sorts of objects and phenomena and includes terms for these" (p. 17). In DR, studies aim at developing both theory and practice. For example, a DR study may seek to develop and describe design principles as part of predictive theory elements. Design principles thus relate to the aim of DR as design principles are both grounded in practice and theoretical perspectives, involving the development of both [START_REF] Prediger | Theorizing in Design Research: Methodological reflections on developing and connecting theory elements for language-responsive mathematics classrooms[END_REF]. Principles that are not yet tested in practice and refined and consequently only supported theoretically, are considered heuristic (van den Akker, 1999) and humble [START_REF] Prediger | Theorizing in Design Research: Methodological reflections on developing and connecting theory elements for language-responsive mathematics classrooms[END_REF].

In [START_REF] Bach | Networking practices in design research: refining design principles[END_REF], we applied networking of theories as a meta-practice to a DR study. The aim was to develop humble design heuristics (HDH) into design principles (see section below). In continuity, in this paper, we aim to explore networking of theories in regard to implementation research (IR). Inspired by DR, we want to coin the idea of humble implementation heuristics (HIH) with an outset in our previous work. Such HIH would then be a precursor for a set of resulting implementation principles. The first underlying hypothesis here is that such HIH may serve as an informative 'link' between the developed design principles and an initial implementation process (small scale or large scale). Another hypothesis we have is that networking of theories as a metapractice has a role to play here as well. Hence, the research question of the paper: Based on the specific empirical case of [START_REF] Bach | Networking practices in design research: refining design principles[END_REF], what might a set of 'humble implementation heuristics' look like, and in what ways may the formulation of these benefit from a networking of theories perspective? Implicit in this question is of course a demand for an attempt to define HIH. For this purpose, we draw on IR theoretical constructs [START_REF] Koichu | Implementation-related research in mathematics education: the search for identity[END_REF]-constructs that may potentially themselves be subject to networking. We begin with a description of the empirical case.

Empirical case: From humble design heuristics to design principles

Based on our literature review concerning students' representation competency when using digital technologies, we recommended five HDH [START_REF] Pedersen | Mathematical representation competency in relation to use of digital technology and task design: a literature review[END_REF]. Here, we utilised van den Akker's (1999) definition of design principles, which includes a heuristic nature and may be explained as:

If you want to design intervention X [for the purpose/function Y in context Z], then you are best advised to give that intervention the characteristics A, B, and C […], and to do that via procedures K, L, and M […], because of arguments P, Q, and R (van den Akker, 1999, p. 9) Three out of the five HDH were applicable to the specific task in [START_REF] Bach | Networking practices in design research: refining design principles[END_REF]:

1. Including the linguistic register by formulating the task in natural language and asking for answers in natural language.

2. Breaking objects and windows of the tool into smaller units by introducing them gradually.

3. Using sliders, dragging and tracing for explorations of different representation and representation forms' reciprocal relations. (Bach et al., 2022, p. 2) We then developed the three HDH into design principles based on empirical testing involving two students using a dynamic geometry template. Within our study, we relied on the Danish competency framework, KOM, which is of a cognitive nature and concerns what it means to master mathematics [START_REF] Niss | Mathematical competencies revisited[END_REF]. KOM defines mathematical representation competency as "the ability to interpret as well as translate and move between a wide range of representations [...] as well as of the ability to reflectively choose and make use of one or several such representations [...]" (p. 17) KOM's description of the representation competency was not enough to understand the situation of students' use of the dynamic geometry environment and their interactions with the digital representations. Finer grained theoretical perspectives were needed. We thus applied a networking of theories practise to coordinate two other finer grained theoretical perspectives to KOM [START_REF] Bach | Networking practices in design research: refining design principles[END_REF]: (1) a semiotic register approach, which was used to analyse the representations involved [START_REF] Duval | Understanding the Mathematical Way of Thinking -The Registers of Semiotic Representation[END_REF]; and (2) the instrumental approach to mathematics education, which was applied for analysing students' use of the digital tool (e.g., [START_REF] Trouche | An instrumental approach to mathematics learning in symbolic calculators environments[END_REF].

The case presented in [START_REF] Bach | Networking practices in design research: refining design principles[END_REF] involves two students who use a pre-set GeoGebra template. While investigating the template, the students communicate about the relationship between a point P and a rectangle ABCD. See Figure 1. Based on a parallel analysis of the two students' work, the revised design principles were formulated in the following way:

If you want to design teaching with a digital tool for exercising and developing the representation competency in schools, then you are best advised to gradually relate all four registers (linguistic, symbolic, figurative and graphic) [from Duval (2007)] in a task sequence when using digital tools.

Collaboration between students is essential for all three characteristics and procedures. This may be realised by:

1. including the linguistic register (both written and oral) throughout the task (i.e., when formulating the task, exploring the environment and representations, and writing answers), and being aware of how natural language appearing in the environment relates to the intentions of the task, as the linguistic register is often neglected in a digital tool, although it is an important register expressing the relations between representations, formulating interpretations of different representations and for developing an artefact into an instrument.

2. breaking objects and tool windows into smaller units by introducing different representational registers and components of the artefact gradually (e.g., hiding the algebra view), because gradually introducing representations makes it possible to identify the specific characteristics of a representation and to help the students' processes of instrumental genesis by limiting their explorations to specific parts of the tool. The networking practice was applied to inform the choice of well-fitting theoretical constructs (see [START_REF] Prediger | Networking strategies and methods for connecting theoretical approaches -First steps towards a conceptual framework[END_REF] around KOM to be able to coordinate these. The two other theoretical perspectives shared the cognitive background of KOM, yet they concerned different objects, i.e., mathematical representations and the use of digital tools, respectively. Within our study, we found that the strategy of coordinating and DR were well suited for one another, since both aim to develop theory and understand practice. In addition, the networking practice appeared to strengthen the theory development in DR [START_REF] Bach | Networking practices in design research: refining design principles[END_REF].

Implementation in mathematics education [START_REF] Rogers | Diffusion of innovations[END_REF] defines an innovation as "an idea, practice, or object that is perceived as new by an individual or other unit of adoption" (p. 12). [START_REF] Koichu | Implementation-related research in mathematics education: the search for identity[END_REF] perceive an implementation in mathematics education as "an ecological disruption to a particular mathematics education system, through the gradual endorsement of innovation in conjunction with an action plan aimed at resolving what is perceived as a problem by (at least some of) the stakeholders involved" (p. 986). They describe the implementation process of the innovation in terms of the innovation proponents and the innovation adapters. Initially, the proponents have the ultimate agency of both the innovation and the plan of action. Over the course of the implementation, the adapters experience some or all of four sub-processes: "(1) constructing agency over the innovation, (2) gradually changing withincommunity communication or across-community communication, (3) gradually changing practice so that it accommodates the innovation, (4) adapting the innovation to their needs and aspirations" (p. 986). These four sub-processes reflect back on the innovation proponents, also in terms of the evolution of the innovation, the plan of action and not least the theories underlying the development of these. [START_REF] Koichu | Implementation-related research in mathematics education: the search for identity[END_REF] remark that the implementation process is iterative and that it comes to an end once it stops being perceived by stakeholders as an ecological disruption.

As argued on several occasions (e.g., [START_REF] Jankvist | Illustrating the need for a 'Theory of Change' in implementation processes[END_REF], an articulated 'Theory of Change' (ToC) is a crucial aspect in any implementation process for measurement of the ecological disruption. A ToC is a collection of theoretical constructs-from general IR or from 'local' mathematics education-that may be used to guide design, the phases of implementation and the evaluation of an innovation. A ToC may also encompass a description of a set of assumptions. [START_REF] Jankvist | Illustrating the need for a 'Theory of Change' in implementation processes[END_REF] state that theory-driven evaluations have two key components: a conceptual one explicating a program theory or model; and an empirical one that is to "investigate how the programs (or innovations) cause intended or observed changes and outcomes" (p. 1048).

Discussion 1: Developing humble implementation heuristics

Referring back to van den Akker (1999), we may rephrase his advice in terms of IR to define the HIH for our implementation study based on [START_REF] Bach | Networking practices in design research: refining design principles[END_REF], relying on [START_REF] Koichu | Implementation-related research in mathematics education: the search for identity[END_REF]:

If you want to implement design principles (X) seeking to exercise students' representation competency, when using digital tools in mathematics classrooms for the purpose (Y) of scaling the DP to a larger population of mathematics teachers (Z) in lower secondary schools in Denmark (e.g., in a municipality), then you are best advised to anticipate possible implementation issues regarding mathematical representations competency and the use of digital tools across schools and classrooms. This may be realised by:

• (A) clarifying the purpose of the innovation, the context of using digital tools and the mathematical content in play to the stakeholders (e.g., schools and teachers) by (K) designing possible tasks and adapter guidelines to be implemented by the adapters because (P) having tasks and teacher guidelines may help adapters to access and take agency of the innovation. The tasks built on the design principles, and teacher guidelines help the adapters with different knowledge and experiences about mathematical representation and use of digital tools.

• (B) supporting the development of communities of practice focused on this theme by (L) gradually introducing discussions groups of stakeholders (e.g., adapters at schools and across schools), as well as access to researchers/experts, mathematics counsellors or "super teacher users" with respect to digital tools and mathematical representations (both physical and online communities are established) because (Q) sharing experiences between adapters and collaboration may ease the implementation of the innovation and allow adapters to gain from other adapters' experiences. • (C) helping adapters to assess their own practice using didactical theoretical perspectives on mathematical representations and the use of digital tools (as part of the ToC) by (M) gradually presenting ToC-based objectives and rationales of tasks to be implemented and their aligned intended learning outcomes because (R) the adapters must evaluate according to the didactical theoretical perspectives (i.e., the ToC). • (D) contextualising the tasks and the design principles, presenting possible learning trajectories available, by (N) adapting these to the one's own classrooms because (S) adapters may experience difficulties when implementing the design principles depending on the involved classrooms, and on previous teaching experiences related to both the content and the use of digital tools.

The ToC for this implementation study would build on the semiotic register approach [START_REF] Duval | Understanding the Mathematical Way of Thinking -The Registers of Semiotic Representation[END_REF] and the instrumental approach [START_REF] Trouche | An instrumental approach to mathematics learning in symbolic calculators environments[END_REF] as part of the design principles from [START_REF] Bach | Networking practices in design research: refining design principles[END_REF]. Hence, from an IR point of view these theoretical perspectives make up our ToC. When assessing and evaluating the implementation, the ToC becomes an important constituent.

Our HIH are to be tested in a large-scale implementation study. In DR, these heuristics may traditionally include both design principles and learning trajectories (see [START_REF] Prediger | Theorizing in Design Research: Methodological reflections on developing and connecting theory elements for language-responsive mathematics classrooms[END_REF]). Yet, so far we have not touched upon the learning trajectories, which are closely related to the mathematical content of the study (in our case, functions). Learning trajectories may thus be more difficult to present in a generalised manner for all lower secondary classrooms, as students in 7-th grade are likely to have other learning trajectories than e.g., students in 9-th grade.

If to describe HIH to any IR study, we may again draw on van den Akker (1999):

If you (the proponent) want to implement innovation X [for the purpose/function Y in context Z, for adapters], then you are best advised to take into account the ToC in relation to the characteristics A, B, C, and D [being aware of different kinds of stakeholders], and to do that via procedures K, L, M, and N, because of arguments P, Q, R, and S.

In such cases, (A) concerns a layered communication (i.e., communication regarding the purpose of the innovation, the content of the innovation, and ways that stakeholders can use the innovation according to the intentions of the proponents) by (K) "constructing adapters agency over the innovation" (Koichu et al., 2021, p. 986) because (P) easy access and agency of the innovation to mathematics teachers, i.e., adapters, with different scope of engagement at different times is essential. (B) emphasises the necessity of communities of practice (i.e., different levels of communities with access to experts or "super adapters") by (L) "gradually changing within-community communication or across-community communication" (p. 986) across local adapter groups and, likely, having access to online communities because (Q) communities foster imperative communication, which supports adapters in the implementation process according to their needs. (C) entails that adapters have the possibility of assessing their own practice according to the ToC by (M) "gradually changing practice so that it accommodates the innovation" (p. 986) because (R) the adapters require a mechanism fostering awareness/focus on how to implement the innovation successfully into their own practices.

(D) focuses on the flexibility of the innovation by (N) "adapting the innovation" to one's own "needs and aspirations" (p. 986) because (S) implementation of the innovation depends on the contexts in which the innovations are implemented.

To sum up, DR and IR differ in various ways, e.g., DR is likely to be more local than IR. van Discussion 2: The potential contribution of a networking of theories practice

When using the networking strategies of coordinating and integrating locally, the results are of course local. In IR in general, however, results may be more global, have more context, etc. Hence, it is in this spectrum that the potential of relating networking of theories (strategies) and IR may arise.

In our DR study, we used the semiotic register approach and the instrumental approach to develop and adjust our design principles [START_REF] Bach | Networking practices in design research: refining design principles[END_REF]. From an IR point of view, the semiotic register approach and the instrumental approach act as our ToC. Hence, these theoretical perspectives act as key components when designing and assessing the innovation. Networking of theories brings practices that create consistency between the innovation to be implemented (i.e., the design principles) and how we wish to assess that. Meanwhile, IR then allows for less local testing of the theoretical work. Although both networking of theories and IR address much more fundamental issues, they both also share a focus on communication. Networking of theories includes an aim of intercommunication to the field of mathematics education research, i.e., the basic assumptions of the involved theoretical perspectives must be described and discussed [START_REF] Prediger | Networking strategies and methods for connecting theoretical approaches -First steps towards a conceptual framework[END_REF]. In IR, a successful implementation requires that basic assumptions of innovations are described clearly, and that the adapters are aware of what we wish to assess and how. Hence, providing particular practices from networking of theories focusing on presenting basic assumptions may also support clearly described innovations and well-established and coherent ToCs. Both IR and DR have theory elements of different kinds. In Prediger's (2019) perspectives on DR, e.g., categorial, normative, predictive, descriptive and explanatory theory elements, these theory elements are acting together throughout a project. As described in [START_REF] Bach | Networking practices in design research: refining design principles[END_REF], the networking of theories then provides practices to choose reflectively the theoretical perspectives as well as to relate different theoretical perspectives. The use of networking of theories in IR may help to achieve consistency in the involved conceptual component. IR may also strengthen networking of theories, since networking theories often are considered to be local in organisation. Nevertheless, it appears relevant to stress again that the theoretical perspectives connected/networked in IR are not global, unified theories. Overall, IR may be fruitful for testing the connectivity between theoretical perspectives, also in large-scale studies.

Final remarks on the idea of humble implementation heuristics

Let us return to our two hypotheses, i.e., (1) HD between the developed design principles and an initial implementation process (small scale or large scale), and (2) that networking of theories as a meta-practice has a role to play in relation to such HIH and perhaps IR in general. Besides formulating specific HIH based on the study of [START_REF] Bach | Networking practices in design research: refining design principles[END_REF], we have illustrated that HIH may be formulated using a similar structure as van den Akker's (1999) definition of design principles. Of course, such HIH may cover both learning trajectories and design principles [START_REF] Prediger | Theorizing in Design Research: Methodological reflections on developing and connecting theory elements for language-responsive mathematics classrooms[END_REF]. We suggest that formulating HIH may assist proponents in being more specific about the aim of their innovation as well as its implementation. Although HIH are indeed humble in their outset, they aim to be refined as the project progresses. The meta-practices of networking of theories are often applied to other research practices. Applying networking to IR has promising potential with respect to consistency and alignment. In our DR study, we networked the semiotic register approach with the instrumental approach to develop and adjust design principles. This theoretical work was our ToC in the IR study.

If studies focus on networking local theoretical perspectives as part of the ToC, its implementation may be more likely to be consistent with respect to the basic assumptions. Being specific about the ToC from the beginning of an IR study may result in a larger degree of alignment between the different phases and sub-processes of the implementation-and HIH may be a construct to assist the achievement of this. We prophesy that it is indeed worth looking closer at the unexplored potential in an interplay between the IR and the networking of theories practice.

  3. using sliders, dragging and tracing for explorations about different representations and the treatments and conversions caused by the dynamic features since these features make it possible to quickly move and translate between representations, and for developing conjectures about the representations' reciprocal relations. (Bach et al., 2022, pp. 6-7) Investigate the construction on your computers by dragging point A in the figure. Describe the relationship between point P and the shown figure. For point P, describe what characterises the xcoordinate and the y-coordinate.Explain why the relationship exists.

Figure 1 :

 1 Figure 1: Illustration of the task and the students' pre-set GeoGebra template (Bach et al., 2022, p. 4). The tasks and GeoGebra template were based on tasks presented in Johnson and McClintock (2018)

  den Akker(1999) remarks that developmental research (e.g., DR) does not aim for implementation, but rather at developing prototypical interventions. IR, however, aims for implementation of these interventions. Still, the two research approaches are relatable, since IR may naturally follow DR.

Acknowledgment

This study is part of project 8018-00062B under Independent Research Fund Denmark.