Lena S Jaeger
email: lena.jaeger@uni-bielefeld.de

Miriam M Lüken

The Connection between Numerical Competencies

Keywords: Probabilistic thinking, early childhood education, numerical competencies

Introduction

Probabilistic thinking is an important competence for social participation in a world dominated by uncertainties. The ability to engage meaningfully with probability-related information, and to develop probability literacy is crucial to manage real world challenges (Gal 2012, p. 4). As recent research indicates, probabilistic thinking is initiated at an early stage and long before young children have ever received formal probabilistic education (Nikiforidou et al., 2013, p. 349). Children from the age of three start to develop first signs of probabilistic thinking based on everyday experiences and have rather sophisticated understandings of probability concepts (e.g. [START_REF] Nikiforidou | Probabilistic Thinking and Young Children: Theory and Pedagogy[END_REF][START_REF] Kingston | Young Children's Use of Subjective Thinking in Response to Probabilistic Tasks[END_REF]. In this poster proposal, we aim to briefly summarise the state of research concerning probabilistic thinking in the context of early childhood education, identify existing research gaps, and outline our research project, which we will graphically present on our poster in more detail. The term young children refers to children aged three to eight years.

Probabilistic Thinking in Young Children and the Potential Connection to Numerical Competencies

Over the last decades there has been an increasing interest in gaining insights into young children's probabilistic thinkingstarting with the fundamental research of [START_REF] Piaget | The origin of the idea of chance in children[END_REF][START_REF] Piaget | The origin of the idea of chance in children[END_REF][START_REF] Fischbein | The intuitive sources of probabilistic thinking in children[END_REF], to the framework for probabilistic thinking from [START_REF] Jones | A framework for assessing and nurturing young children's thinking in probability[END_REF], to more contemporary studies such as [START_REF] Kingston | Young Children's Use of Subjective Thinking in Response to Probabilistic Tasks[END_REF][START_REF] Kingston | Young Children's Use of Subjective Thinking in Response to Probabilistic Tasks[END_REF]Nikiforidou (2019). Reviewing literature on probabilistic thinking, it ishowevernoticeable that the term is often used without a precise definition. One tends to find broad descriptions of the term, such as in the study of [START_REF] Kingston | Young Children's Use of Subjective Thinking in Response to Probabilistic Tasks[END_REF], who use probabilistic thinking "to describe young children's thinking in response to any probabilistic task." (Kingston & Twohill, 2022, p.1). Alongside these broad definitions, and as [START_REF] Borovcnik | Probabilistic thinking and probability literacy in the context of risk[END_REF]Borovcnik ( , p. 1496) ) states "little is published on an explicit description of probabilistic thinking and the mathematical competencies that are related to it". For the purpose of this poster proposal, we endorse the definition of [START_REF] Kingston | Young Children's Use of Subjective Thinking in Response to Probabilistic Tasks[END_REF], but also recognise the need to refine and clarify the term for our future research. As recent research has shown, young children already possess an intuitive understanding of the likelihood of events by defining the sample space and identifying the most/least probable outcome [START_REF] Nikiforidou | Intuitive and informal knowledge in preschoolers' development of probabilistic thinking[END_REF]. But how exactly do they perceive the sample space and draw conclusions from it? In this process, other mathematical competencies such as counting, subitizing, and a sense for structures and patterns might come into play. Numerical competencies also develop at a young age and are known to be predictors of later mathematical performance (e.g. Krajewski & Schneider, 2019). The perception of quantities is important in a probabilistic context, for example to identify all possible events (the sample space), and in a next step to assign each event a probability. However, little is known about how and which numerical competencies young children use in different probabilistic situations and with different random generators. Findings on this would provide deeper insights into the development of young children's probabilistic thinking and would imply didactical conclusions for teaching probability.

Conclusion and Outlook on the Poster Presentation

Early childhood education is of essential importance for the development of mathematical competencies, including probabilistic thinking and numerical competencies. Although there is international research on probabilistic thinking in the early childhood classroom, this fieldas far as knownhas not yet been addressed in Germany. Furthermore, the connection between the development of numerical competencies and probabilistic thinking is largely unknown. We aim to investigate and graphically present on our poster how and which numerical competencies are possibly linked to young children's probabilistic thinking in different probabilistic tasks. As our empirical research design is still in the process of development, we would like to present and discuss our theoretical considerations on this topic at the conference.


Majd Mayyasi
email: majdm@bu.edu

John Clarke

) Dolon Bhattacharyya

Jean-Yves Chaufray

Jean-Loup Bertaux

Nicholas Schneider

Improving the Detectability of Faint Emissions with MAVEN IUVS Echelle: Applications to Deuterium Lyman- and the D/H Ratio

The Mars Atmosphere and Volatile Evolution (MAVEN) mission's Imaging Ultraviolet Spectrograph (IUVS) high resolution echelle channel can resolve observations of hydrogen and deuterium Lyman-ɑ as well as O triplet FUV emissions. D Lyman-ɑ is too faint to be measured above the detector noise level when Mars is far from perihelion. Lessons learned from 5 years of IUVS echelle observations have been applied to produce a new data reduction pipeline in which the faint D emission can be better resolved.

Introduction

The MAVEN IUVS echelle is designed to optimize observations of resonantly scattered D and H Lyman-, O triplet and forbidden doublet emission lines [START_REF] Mcclintock | The imaging ultraviolet spectrograph (IUVS) for the MAVEN mission[END_REF]. In the ~5 years that MAVEN has been orbiting Mars, the background noise level of the echelle detector was found to limit the detectability of faint emissions to times when they are bright. For deuterium Lymanɑ, this brightening has been observed to be around perihelion [START_REF] Clarke | Variability of D and H in the martian upper atmosphere observed with the MAVEN IUVS echelle channel[END_REF][START_REF] Mayyasi | Seasonal variability of deuterium in the upper atmosphere of Mars[END_REF]. At remaining seasons, the data reduction pipeline developed for signal retrieval showed D brightness levels to drop to the background noise level [START_REF] Mayyasi | IUVS echelle-mode observations of the interplanetary hydrogen: Standard for calibration and reference for cavity variations between Earth and Mars during MAVEN cruise[END_REF].

A new data reduction pipeline has been developed to optimize the signal retrieval for faint emissions in the MAVEN IUVS echelle observations. The new pipeline includes a new calibration using simultaneous IUVS and Hubble Space Telescope observations. The results of the new data reduction pipeline are compared to the previous one to show higher accuracy retrieval of faint emissions [Fig. 1].

H Lyman-ɑ is an optically thick emission that is relatively bright at Mars for most MAVEN observational conditions. Improving the accuracy of MAVEN's D Lyman-ɑ emission retrievals along with the H Lyman-ɑ measurements can provide improved constraints for radiative transfer models to use in order to generate the abundances of D and H in the upper atmosphere of Mars. With D and H densities, the D/H ratio can be determined to provide estimates of water escape flux from Marsa primary objective of the MAVEN mission [START_REF] Jakosky | MAVEN explores the martian upper atmosphere[END_REF].

Figures

Figure 1 :

 1 Figure 1: A comparison of old (red), new (black) and the difference between old and new (purple) data reduction pipelines for the MAVEN IUVS Echelle channel. The H Lyman- brightness (top) varies minimally while the D Lyman- brightness (bottom) shows more pronounced variations. The new pipeline more clearly portrays the variability as well as reliability of faint emissions.


Stine Gerster Johansen

Uffe Thomas Jankvist

Making sense of distribution: A case of dépaysement

Keywords: Bildung, statistics, distribution, dépaysement

The Danish curriculum addresses the double aim of both future education and broader personal development, as captured in the notion of Bildung. In this paper, the non-trivial task of targeting both these aims through mathematics education is discussed using an empirical example from statistics teaching in a Danish middle-grade classroom. The idea of dépaysement (alienation) as a principle of Bildung is then illustrated by students making sense of the idea of an empirical distribution.

Introduction to the problématique

The notion of Allgemeinbildung, or simply Bildung, is relevant in a Danish educational context because it addresses the 'why' question and has historically served as a foundation for the syllabus; indeed, it has been discussed in the educational context in Denmark since the 19th century, drawing on German descriptions from both Hegel and Humboldt (e.g., von Oettingen, 2016a). It is part of the dual aim of the Danish curriculum: on the one hand, further education and future vocation, and on the other, Bildung, expressed through the school's purpose of 'creating a setting for experiences, immersion and enterprise' under the influence of 'freedom of spirit, equality, and democracy' (Danish Ministry of Education, 2019, our translation). Even if one succeeds in developing students' mathematical skills and competencies, it is not self-evident that this also contributes to their Bildung. In a teacher interview about statistics education conducted by the first author, an experienced lowersecondary school mathematics teacher expressed her frustration. She explained that she could teach her students to make frequency tables, draw boxplots, or calculate means but found it immensely difficult to make them experience this as anything to do with them or as relevant to their communication and understanding of phenomena-that is, to contribute to their Bildung.

In this paper, which we have structured as a discussion centred around an illustrative, empirical case, we investigate the hypothesis that the idea of being on foreign groundcalled in the German literature entfremdung and in the French literature dépaysementmay serve as 'the missing link' or 'a bridge' between the development of students' learning of mathematics and their development of Bildung. The case itself is of two students making sense of the idea of empirical distribution.

Bildung and other theoretical constructs

The Danish educational researcher von Oettingen (2016a) referred to Hegel when emphasising 'distancing' as an important task of teaching. He described how subject knowledge and objectivity as part of teaching create distance from the students' subjective points of view and emotions. In this regard, school subjects serve the role of cultural fields, each with their own characteristics to offer to the process (von Oettingen, 2016a). The individual must thus realise that the self is part of something bigger than the self, and for Hegel (1809[START_REF] Hegel | Nürnberger Gymnasialkurse und Gymnasialreden (1808-1816[END_REF], this alterity is the very core of the process of Bildung. Although he acknowledged the discomfort of that alienation, he also described it as a common driving force and instinct in all of us.

The alienation, which is a condition for theoretical Bildung, does not require this moral pain, not this suffering of the heart, but the lighter pain and endeavour of the imaginationto engage with something that is not immediate, something alien, with something belonging to the memory, thoughtfulness, and thinking. But this demand for separation is necessary in the sense that it expresses itself in us as a general and well-known instinct. The strange, the distant carries with it the attractive interest that lures us to employment and effort, and the desirability stands in inverse relation to the nearness, as it stands and is a common part of us. (Hegel, 1809(Hegel, /2006, p. 461, our translation) , p. 461, our translation) This movement away from the immediate vicinity of the student can also be framed as dépaysement, as suggested by [START_REF] Barbin | Histoire et enseignement des mathématiques: Pourquoi? Comment? [History and teaching of mathematics: Why? How?[END_REF] in relation to using primary sources from the history of mathematics as a means to place students on unfamiliar ground. [START_REF] Jahnke | The use of original sources in the mathematics classroom[END_REF] described this as a process that challenges a student's perceptions by making the familiar (e.g., mathematical content, notation, argumentation) unfamiliar, eventually causing a reorientation of views and leading to a deeper mathematical understanding. [START_REF] Johansen | Litteratur og dannelse: At lade sig berige af noget andet end sig selv[END_REF] added that self-transgression is essential, stating that the object processed by the student should be something alien, strange, and unfamiliar. In Johansen's case, the object was fiction; for Barbin, Jahnke, and colleagues, the objects were primary historical sources. [START_REF] Michelsen | Den matematiske og naturvidenskabelige dannelse [Mathematical and scientific Bildung[END_REF] contended that natural science and mathematics can contribute to Bildung, especially through modelling, which, according to him, is genuine and significant because it helps individuals both comprehend and learn in everyday scenarios and scientific endeavours. These models thus provide a particular perspective on the world, and by actively participating in such practices, students may discover that the world can be understood and revealed in distinctive ways by looking at it through the lens of these models.

In the context of Bildung, it is imperative to contemplate the nature of the 'unfamiliar ground' that each school subject can offer to students. As initially posited, each school subject serves as a 'cultural field' that offers distinctive experiencesstudying language or literature, for example, presents different opportunities to mathematics or statistics. This paper centres on exploring the potential experiences of being on unfamiliar ground that statistics teaching can provide to students in lowersecondary school. Specifically, it investigates how the concept of dépaysement can serve as a mechanism for students to derive meaning from the idea of statistical distribution.

On the meta-concept of distribution

In the statistics education research literature, crucial concepts have been identified and described in terms of fundamental ideas (e.g., [START_REF] Burrill | Fundamental statistical ideas in the school curriculum and in training teachers[END_REF], which can help us in identifying the potential experiences they have to offer statistics learners. The idea of distribution has been identified as one such substantial and fundamental idea. It is a meta-concept with several embedded statistical concepts, including variability, shape, and centres (e.g., [START_REF] Burrill | Fundamental statistical ideas in the school curriculum and in training teachers[END_REF]. According to [START_REF] Wild | The concept of distribution[END_REF], individual data values are not inherently interesting; instead, it is the identification of patterns unobscured by extraneous details that is compelling. [START_REF] Wild | The concept of distribution[END_REF] regarded distribution as the 'lens' through which we examine variability in data, and researchers have emphasised that students should develop an understanding of distribution as a whole entity rather than just learning to perform routine calculations like finding averages [START_REF] Bakker | Learning to reason about distribution[END_REF][START_REF] Mokros | Children's concepts of averages and representativeness[END_REF]. It is, however, difficult for middle-grade students to expand their view of data from the value of a measurement of an object to seeing it in terms of distribution. In a study with grade 7 students, [START_REF] Bakker | Learning to reason about distribution[END_REF] investigated the students' learning of several aspects of distribution, finding that they developed informal and qualitative notions of distribution. The researchers described a model with three levels, from data to distribution: at the bottom are data as a collection of individual values; in the middle are concepts of centre, spread, density, and skewness; and at the top is distribution as a conceptual entity. According to [START_REF] Garfield | Learning to reason about distribution[END_REF], the tasks in the textbooks that students use do not help them to understand the idea of a distribution; such tasks merely ask them to 'look at a histogram or stem plot and describe the shape, center, and spread' (p. 186). As a consequence, many students do not understand that distribution is an entity that can offer important insights into the variations in data and thus give them a particular view of the world. [START_REF] Konold | Data analysis as the search for signals in noisy processes[END_REF] highlighted a problem with students' understanding of averages: that they often do not see them as representatives of whole distributions. To address this issue, they proposed viewing statistics as the study of a 'noisy' process in which signals can be detectedcomputed statistics, like the mean or median, can thus represent a 'signal' in the 'noise' of individual data points. However, they also observed that students may know how to calculate these while still struggling to apply or interpret them. Mokros and Russell (2002) also voiced concerns about the 'unappreciated complexity' of the mathematical mean: it is trivial to calculate but complex for students to find meaning in. In their study, they focused on averages as representatives of whole distributions, distinguishing between approaches in which averages are representatives of distributions and approaches in which they are notthe latter tending to focus on modal and algorithmic approaches to averages. The representative approach is subdivided into approaches in which mathematics is intertwined with real-life reasonableness to approaches in which the meaning of a mathematical balance point is developed. Mokros and Russell (2002) also stressed that the idea of averages does not make sense unless a data set is seen as a unit instead of a series of values. They warned about the premature introduction of the algorithm for finding the mean because it can interfere with using it meaningfully for describing data and solving problems. Both [START_REF] Mokros | Children's concepts of averages and representativeness[END_REF] and Konold and Polatsek (2002) have identified a potential conceptual lever in the act of comparing distributions; when students undertake the task of describing similarities and differences between distributions, they use various methods to do so and, in the process, foster an understanding of the concept of distribution as a distinct entity.

To understand how to describe a distribution in specific ways, one must be able to envision how it might vary. To identify and explain a distinctive feature of a distribution, students need a point of reference, and comparing distributions provides such a reference. To identify features of a distribution and establish connections to variations in the data, students must be able to imagine alternative possibilities. For example, a statistician with greater experience may describe a distribution as having a 'fat tail', which can only be understood in relation to different tail shapes and their associated contexts. Novice learners may encounter difficulties in this regard because they have only encountered a limited number of distributions, restricting their ability to discern the features of a distribution and to identify formal descriptors that would allow them to view the distribution as a distinct entity.

Research question

The objective of this study is to introduce the concept of dépaysement in the teaching of distribution to students in a Danish lower-secondary mathematics class. As discussed, the capacity to envision alternative possibilities may be a fundamental requirement for comprehending distributions, and based on the descriptions above, we can formulate a research question: Based on dépaysement as a principle of Bildung, what are the indicators of students' creation of meaning within formal descriptors, such as mean, median, and range, when they attempt to describe a statistical distribution?

The setting and illustrative empirical example

The following example is part of a critical case (Flyvbjerg, 2006) that follows a Danish grade 7 class during their statistics learning (a total of nine lessons of 45 minutes each). Two challenges arose along the way; the first was that the students knew how to calculate the mean, range, and median when asked by the teacher, but it was difficult for them to see meaning in them when describing the data. The second challenge was that the students tended to focus on the mean alone as a way to describe the distribution and felt that they had completed the task of describing the distribution once they had found the mean, even if pressed to describe in their own words what they noticed. The first challenge is consistent with the research literature outlined above, but the second related to seeing the distribution as an entity with certain characteristics.

The following task was designed to bring the students to a point where a strategy of describing the distribution solely using the mean would not work. In the 'mathematics test' task, the students were asked to describe and compare three different fictitious data sets. They were told that the data were from a mathematics test taken once a year at a Danish free boarding school. Every year, the school received 100 new students, and the mathematics teacher had developed a test for those students that was scored between 0 and 10 points. The data sets were said to be from 2020, 2021, and 2022 and were constructed so that all three had the same means (6.4), but their shapes (Fig. 1) were clearly different and described three very different situations. Thus, when asked to compare the test results from the three years, the students were compelled to supplement the mean with other measures of central tendency as well as natural language. The empirical example is presented in the form of transcripts of semi-structured interviews (Kvale & Brinkmann, 2015) conducted with two students, Frida and Iris, immediately after the teaching unit, during which the students still had their computers with the work they had done for the tasks. The purpose of the interviews was to investigate the students' methods of making sense of the data. The interviews were video recorded, transcribed, and translated from Danish into English for this paper.

Frida and Iris on measures of central tendency: 'The median is strange'

In the following, Frida and Iris are discussing the usefulness of the descriptors they calculated during the task: the mean, median, mode, and range. The students were asked which descriptors they liked best, meaning those that they found most meaningful.

Frida:

I think the median is a bit strange because I don't quite see what it's supposed to be used for. So, I think the mean is best, because it kind of takes everyone into account more, including those who are really low and those who are really high. Iris:

Yes, I think so too. So, if there was someone who got 100,000, then the mean probably wouldn't make much sense, but otherwise I also like the mean best.

In their responses, Frida and Iris refer to the impossible value of 100,000, but Iris is exaggerating to explain that the mean may be sensitive to outliers. Frida prefers the mean, referring to it being the division of the sum of all values in a data set by the number of values. Iris agrees but is aware that outliers can distort the information that one extracts from the mean. Even though the students do not initially consider the median to be useful, Iris acknowledges that, in some cases, the data can be distributed in a way that affects the mean.

Interviewer: Okay, so there will be some cases where you think the median is better? Frida: Yes, if there is one that stands out. Iris:

Yes, if it stands out a lot from the others, then the median is better. But the mean is still the best. Interviewer: But would it be nice just to know them both? Frida:

Yes. Or maybe instead of knowing the median, know the range and the mean.

In this excerpt, they refer to a possible outlier. Frida is expanding upon Iris' idea with the thought that the median could be a 'better' measure if there is an outlier in the data; Iris agrees but still thinks the mean is the best, yet she does not explain further what she means by that. Frida introduces the range as a supplement to the mean, but the range is also the same in all three data sets and thus is not useful for explaining the differences between them. Even if Frida does not accept the idea of the median being useful for describing data, she does seem to accept the ideas of flexibility and situation dependence in that context.

In the following, Frida tries to explain the situation in which the range could be a relevant supplement by imagining a different distribution of test results. She refers to the Danish scale for grades: -3, 0, 2, 4, 7, 10, and 12, where -3 and 0 are failing grades.

Frida:

Because it may well be that everyone in a class got a 7. Interviewer: Yes, then the range must be small, if they all got a 7. Frida:

Yes, exactly. But it may well be that there are some who got -3, and some who got 12, so the level is a little more spread out in the class. Interviewer: Yeah, okay. So, could you say that you understand more if you know more than one thing about your data? Frida:

Yes.

Here, Frida refers to an imagined example of everyone in the same class getting the same grade. This would mean that the range is small (or without variation); it would not be likely, but rather it is a hypothetical intended to explore the meaning of the range. Even if not expressed directly, creating an imagined distribution from a descriptor verges on viewing the distribution as an entity.

The interviewer points at the 2021 data set on Frida's screen (see fig. 1), represented by a dot plot in the digital tool that they used (Tinkerplots). She asks Frida to suggest a way to describe the data.

Interviewer: I find that interesting. Can you look at the data there? … What kind of descriptors might be interesting to look for? [Frida looks at the shape and, after a while, proceeds.] Frida:

Eh … The range again because it's very large. It's as big as it gets. And then the mean.

She found the range relevant because the spread is large, even if it might not be useful because it is the same in all three data sets. She also found the mean relevant, but she does not offer a reason. Nevertheless, she still does not find the median relevant, but there is a desire to use more than a single descriptor, and she considers the usefulness of two ways of describing the data after considering its shape.

Discussion and conclusion

Frida and Iris were given a task that involved comparing distributions in which the means were not useful for differentiating them. The intent behind the task was to challenge the students' belief that the mean is the only relevant measure of central tendency, and in this, the task had mixed success because both Frida and Iris expressed reservations about the usefulness of the median. However, Frida engaged in a discussion about alternative ways to handle outliers, showing that she grasped the idea of using multiple methods to describe data. It is important for students to be able to judge what is relevant rather than simply calculating and reporting statistics; if the students thus sense the value of their personal judgement, this point may also relate to freedom of spirit as a guiding principle of the Danish curriculum. This task highlights the delicate balance between including formal descriptors, like the median, and allowing for personal sense-making. As cautioned by [START_REF] Mokros | Children's concepts of averages and representativeness[END_REF], including formal descriptors too early can create problems with understanding.

Although neither the mean nor the range could help the students in contrasting the three distributions, Frida's preference for the mean indicates a preference for a representative viewpoint (as defined by [START_REF] Mokros | Children's concepts of averages and representativeness[END_REF]. Frida believes that the mean is preferable because it 'takes everyone into account', which pertains to the algorithm for its calculation, in contrast to the median.

In this study, Iris and Frida employed alternative examples to elucidate their understanding of the mean, median, and range in different ways. Iris demonstrated how an outlying value in relation to the grading scale could potentially obscure the mean, while Frida illustrated the use of the official Danish grading scale instead of the hypothetical scale provided in the task, emphasising the significance of the range, which can extend from the minimum to the maximum or have no variation at all. By devising alternative scenarios, both Iris and Frida were able to articulate how they comprehended the distribution. This ability to envision different scenarios and assess their impacts on the distribution was critical in creating personal meaning, which supports the idea of providing students with opportunities to explore unfamiliar territory and investigate different contexts and distributions and thus learn about the different views of a phenomenon available through the lens of distribution.

There were two important initial considerations; the first was the teacher's intention to provide the students with an opportunity to recognise that statistics entails more than simply generating tables and performing calculations. The second was to challenge the students' preconceived notion that the mean is the only way to depict the central tendency in a data distribution. The notion of dépaysement was crucial in creating the task: by placing the students in a situation in which their existing linguistic and conceptual knowledge was insufficient to complete the task, they would be encouraged to expand their perspectives. Although Frida and Iris were not entirely convinced of the value of the median as an adjunct to the mean, they still analysed the characteristics of the mean to compare it to the median.

Describing a distribution is not a preprogramed process. Flexible movement between the levels, from data as individual values to a distribution as an entity, demands a great deal of imagination and judgement, and the process of placing the students on unfamiliar ground demands a sense of what defines their 'comfort zone' and what might push them away from it. There might also be a discussion about how unfamiliar that force should be. In the two dépaysement situations briefly mentioned earlierprimary historical sources in mathematics education and self-transgression in literature teachingthe alien is particularly unfamiliar to the students. Johansen (2019), especially, is both extreme and provocative in pleading for the introduction of literature that is hard to understand, while in the case of primary historical sources, the force could be something as alien as Babylonian tablets.

For Frida and Iris, it was evident that they knew other ways to describe a distribution even though they still preferred the mean; the context of the data was also easy for them to relate to. Still, the key is to consider what might distance the students from their immediate point of view by creating the possibility of experiences that transgress the students' horizons and eventually lead to the 'versatile individual development' that Bildung is.

Figure 1 :

 1 Figure 1: The students were introduced to three different sets of data (purportedly from 2020, 2021, and 2022) with the same means and ranges but which clearly represented different situations