Defne Yabaş
email: defneyabas@gmail.com

Batın Örene
email: batinorene@gmail.com

M Sencer Corlu
email: sencercorlu@gmail.com

Exploring the statistical practices in classroom: STEM teachers' experiences using dynamic data analysis technology

Keywords: Statistics education, statistical investigation, dynamic data analysis technology

In the 21st century, schools have been increasingly expected to raise students prepared for a world of quantities and uncertainty. As statistical practices become an important skill for every individual, researchers and educators work on ways to translate the methodologies of professional statisticians in the teaching and learning environments. Technological tools are helpful in translating statistical investigations into the classroom. The purpose of our research is to understand teachers' experiences about integrating statistical practices into their STEM lesson plans by using LabStar, a dynamic data collection and data analysis tool. Findings were investigated under four categories: personal development, in-class implementation, views about the mobile application and the tool, and statistical investigation environment. Future research was recommended to investigate teachers' professional development opportunities to integrate meaningful statistical practices in their lesson plans.

Introduction

In the Big Data era being competent in making data driven decisions in all aspects of daily life is inevitable. Many decisions in politics, economics and social sciences are based on thorough analysis of data. The omnipresence of data requires individuals with strong analytical and reasoning skills who can base their decisions on their inferences beyond the data in the way that professional statisticians do [START_REF] Frischemeier | Building statisticians at an early age-statistical projects exploring meaningful data in primary school[END_REF]. However, international reports point out the lack of individuals, who demonstrate necessary skills to cope with Big Data even at the tertiary level [START_REF] Manyika | Enhancing statistics instruction in elementary schools: Integrating technology in professional development[END_REF][START_REF] Puang-Ngern | Comparison of Graduates' and Academics' Perceptions of the Skills Required for Big Data Analysis: Statistics Education in the Age of Big Data[END_REF]. From this point, practice of statistics became a crucial element to integrate in primary and secondary school curricula.

In 21st century, schools have been increasingly expected to raise students prepared for a world of quantities and uncertainty. With the availability of data from variety of sources and in a variety of formats students are required to manage the complexities to reach data driven conclusions [START_REF] Makar | A framework for thinking about informal statistical inference[END_REF]. Statistics, therefore, has become an essential element of school mathematics curricula of many countries (e.g., Australian Curriculum, Assessment and Reporting Authority, 2015;Common Core State Standards Initiative, 2010;Ministry of National Education, 2018).

As statistical practices become an important skill for every individual, researchers and educators work on ways to translate the methodologies of professional statisticians in the teaching and learning environments. Statistics education is becoming an agenda across many levels of schooling. Debates about how to help students gain an understanding about statistics requires us to think about the nature of statistics. Statistics is about carrying out statistical investigations on a specific statistical question to understand a phenomenon. Statistical investigations are based on exploring the variation, by engaging data collection, data representation, data reduction, level of certainty and informal inference practices [START_REF] Watson | The practice of statistics[END_REF].

Statistical investigation is recommended to be the focus of the statistics teaching-learning activities in classroom contexts. The parts of the investigation process have been also included in mathematics curricula starting from elementary grades. As the practices of statistical investigation become a part in mathematics curriculum, the role of the teachers comes forth. Teachers are expected to design authentic contexts where students can engage in data collection, analysis, and informal inference processes. Research also underlines the importance of teacher knowledge on statistics and statistical thinking to design effective learning experiences for their students (Rodrigues & de Ponte, 2020).

Technological tools are helpful in translating statistical investigations into classroom. Dynamic technologies that allow students to experiment with data collection and analysis techniques were started to be researched in terms of the opportunities they provide for statistical practices (Meletiou-Mavrotheris et al., 2008).

COVID-19 caused schools to get shut off, which made STEM and statistics education to be conducted on online portals. This probably led to changes in the teaching method and the environment. The aim of creating statistics literate students that use statistics tools, software, and can interpret the data became the priority. In order to fulfill this aim, collaboration, communication and multidisciplinary studies that involves data science and education must be addressed.

Based on this rationale the purpose of our research is to understand teachers' experiences about integrating statistical practices into their STEM lesson plans by using LabStar, a dynamic data collection and data analysis tool. Teachers were provided with educational videos about integration of statistical investigations into STEM lesson plans and the use of LabStar tool. After watching all the videos teachers implemented the STEM lesson plans, designed with authentic statistical investigation tasks. Our aim is to explore the lived experiences of teachers about the implementation of these lesson plans. Our research questions are:

•

How STEM teachers contextualize statistics and mathematics into daily life problems using STEM integration? • How STEM teachers perceive their roles in terms of integration statistical investigation in STEM education? • How STEM teachers evaluate the educational videos in terms of designing and implementing statistical investigations in STEM lessons?

Methodology

The research used a phenomenological method among qualitative research designs. The study group is four STEM teachers who used statistical investigation practices in their classrooms during 2020-2021 Spring Semester. T1 was a middle grades mathematics teacher, T2 and T3 middle grades science teachers, and T4 was a high school physics teacher. They implemented STEM lesson plans integrated with statistical investigation in various grades and with different disciplines in the center of the lesson plan. They used LabStar as a dynamic data analysis technology. LabStar is a dynamic data collection and analysis tool. It works with the physical tool and the mobile application. The mobile application can be used also separately from the tool by analyzing open resource data. All teachers and students had the LabStar mobile application, one of them (T4) had both the physical data collection tool and the mobile application. Teachers used data from open resources to analyze in their in-class implementations. Before the implementation they were provided with training videos regarding ways of using dynamic data analysis technology in their STEM lesson plans.

For data collection we conducted face-to-face interviews with the teachers. To ensure a prolonged relationship, interviews lasted at least 60 min. Audit groups were formed within the research group and member check processes were planned to ensure trustworthiness. Researchers are also using peer-debriefing during data collection and analysis phases. STEM lesson plans and field notes of the teachers were used as data sources to ensure triangulation in data collection. For data analysis, interviews were transcribed, and thematic analysis was conducted by using constant comparative method. Because multiple coders will analyze the transcripts, the methods for ensuring intercoder reliability were used. Documents of teachers (lesson plans and notes) will be analyzed by content analysis. Data analysis will be finished in November 2021.

Findings

Analysis of the interviews and the documents revealed myriad findings about the experiences they had during integration of STEM education into their classes. Findings were investigated under four categories: personal development, in-class implementation, views about the mobile application and the tool, and statistical investigation environment.

Personal Development

This theme emerged as the participants mentioned about the training they have received and research they have done before and/or during applying a STEM activity in the class. All teachers expressed that the educational videos that they have watched before attending to the classes, helped them to learn and grasp the nature of the application before introducing the software to the students. One of the teachers, T1, commented that "With the help of the videos, especially the fourth one, we were able to lead the students in the right direction". T2 added that: "The videos were sufficient, I received STEM education last year, I knew nothing. In a year I learnt about STEM education and some of the questions were given to us that are in the videos."

Furthermore, some of the teachers had to get help from various resources because they felt that they did not have sufficient knowledge.

I am a science teacher, when I was studying on it, I got help from my husband who is a mathematician, (I asked about) upper quartile, lower quartile etc. I taught students with the information I got from him too. I also got tremendous help from the educational videos. (T3)

This view also showed that teachers also had an opportunity to discover knowledge related to other disciplines. A teacher had sufficient technological knowledge and even integrated STEM education to work with a software simultaneously.

I thought about what I could do, I set up a meeting with the students from 9 th and 10 th grades. A student said that we should test its (LabStar's) accuracy. We tested it and it was accurate. We conducted two experiments, firstly we created sound waves and measured its decibel. We have a software called Scratch; we created the sound waves then integrated it with LabStar. The data were almost equal. We used it to compare the sound waves, their wavelengths. I found it usable for advancing in physics. It is a nice software (T4).

The teachers started their lessons with introductory activities to motivate students. T1 stated "We asked the students about number of cases of covid-19. Then we led them to the relevant website.". T2 added "I asked the properties of microscopic life forms. With the feedback I have received, I mentioned about the bacteria then gave the table". Although T2 did not ask anything before the lesson, she let the students explore by themselves. This allowed the students learn by discovery.

I did not ask anything, I let them do experiments because I was experimenting too. When I asked what do you see, they told me that the wave is increasing, I asked how a wave can increase? Is it on the direction x or y? Then we got data, but now they tell numbers instead. Are they increasing or decreasing? We call them data. As the waves increase, the numbers increase, what do we call it? Wavelength. (T4) Some teachers worked as a group to put out the lesson plan.

We worked as a group. I usually build interdisciplinary relations with mathematics, but I always get help from the biology. We do gas pressure, sound waves, they are all about biology. There were machines in STEM laboratory, I used the distance sensor there. (T4)

The teachers explained that they plan to apply their knowledge into their other lessons too.

Maybe it is not possible to apply it to all topics, since physics, chemistry and biology are different. But for velocity, race, order of the velocity, density of a matter, there can be data and it can be commented on. I think that it could be integrated depending on the area. (T2) T4 also asserted that, "I will use it to explain illumination density. I have the topic movement on 10 th grade, location, distance, translocation, I will use graphs. It is fun to make students active in 9 th and 10 th grades."

In-class implementation

This theme has emerged as the teachers expressed their experiences about the application of a STEM lesson in a class. A problem occurred because of the mismatch between the current grade's curricula and the statistical terms in the tool. T1 claimed that "Some terms were troublesome since they did not fit to my class". T2 added that "In my opinion it was not suitable for 5 th graders, the terms mode, median in the graphs". T1 further explained that: "The students didn't know the terms and their meanings since the terms do not fit to fifth grade's curriculum. They did not understand it because they did not learn it, and I did not pre-teach the terms."

Some teachers had problems because of the students' technological skills being insufficient. T1 claimed "Some students found it hard to transfer data to an Excel sheet", She further said "At first they found it hard to create a table. There were students who did not know the shortcuts".

The teachers explained that they were not able to hold the attention of the whole class nor make all students participate in the activity. T2 claimed that "If you have 50 students, only 10 students give positive feedback. There were students who did not download the mobile application or participate in the lessons".

The teachers who had the mobile application but not the physical tool claimed that they could have reached to more students if they had the physical tool to collect real time data. Some teachers observed that some students were interested in the lesson:

There were many students who were interested in the lesson. There was data related to basketball, it drew the attention of male students. If the topics were different there could be different things. But I received positive reactions, some students asked if they could collect and analyze another data. (T2)

Views about the mobile application and the tool

This theme has emerged as the teachers stated their views about the mobile application and the tool that they have used to carry out their STEM lesson plans.

Another challenge was born because of the language of the application being English. This was a common problem amongst all teachers. T1 stated "The only problem was the application being in English, it could be in Turkish. If they don't know English, they need to use translation". Some schools did not receive the physical tool, but they only had the mobile application. The teachers believe that they need the tool itself.

It would be better if we had the tool. For instance, in a 7th grade, I have conducted a lesson about darker colors absorbing light thus being colder. I used heat and light sensor simultaneously to observe the graph. It was a brilliant lesson. I did not have such fun in any lesson before. So, it would be better if we had the data collection tool. (T3)

T2 claimed that. "It is vague for 5 th graders, for a fifth grader, the tool being here and the process of collecting the data is necessary. It needs to be concrete". There were some problems that originated from the application and technology use. Overall, the teachers were satisfied with the experience they had with LabStar. T2 claimed "The teachers can use LabStar easily and create STEM lesson plans.", T4 added "LabStar can analyze data, it is accurate and fun".

Statistical Investigation Environment

The teachers have different expectations about the statistical terms that the students must know and express in a statistical investigation environment.

It is crucial to be able to read the graphs, where is the peak, what is the increment this month, being able to comment, creating a cause-and-effect relationship is important for science. Mathematical interpretation, peak point, outliers. I believe that LabStar application is related to mathematics at most. (T2)

The students do not know how to read the graphs, they cannot even read tables. They have learnt how to do it; I believe that that is the significant contribution. (T3)

The teachers claimed that STEM integration made observing statistical terms easy.

In the topic waves, there are fundamental notions. Frequency, amplitude, and wavelength. You can create a sinus wave, you can observe the wavelength, then we can measure it. We talked about range and modes, we created 2-3 waves, x wave and y wave then we compared them. At first, they talked about their image, then we compared them as data. (T4) Some students were not interested in terms related to statistics. When teachers were asked if the students asked about statistical terms, T2 said "No, they did not ask. In 6 th and 7 th grades the activities would be more efficient. However, since the total number of lessons are less in 5 th grades, it limits us." T1 added that, in online environment, only the students who are passionate about mathematics asked, but we did not have a whole-class discussion. Since the number of curious students were low, they contacted us through private messages. (T1)

Discussion

The teachers reported that by seeing the graphs live, students grasp the concepts of mathematics and statistics in a fun and interactive way. This may be because in most cases formal statistics teaching leaves students in a confused state as most of the time they are not able to apply the theoretical knowledge into real life [START_REF] Bakker | An inferentialist perspective on the coordination of actions and reasons involved in making a statistical inference[END_REF], whereas the fun factor comes into play with the activities. The teachers worked together with other teachers to set up a multidisciplinary context, this is because developing mathematics, and integrating it with the other disciplines in STEM is the main aim in STEM education [START_REF] Li | Problematizing teaching and learning mathematics as "given" in STEM education[END_REF]. Even though context integration should be used to teach STEM [START_REF] Kertil | Mathematical Modeling: A Bridge to STEM Education[END_REF], the teachers most of the time failed at integration of STEM with other disciplines, they mostly worked on their own disciplines. Therefore, creating statistical investigation environments was evaluated to be essential [START_REF] Anderson | Advances in STEM Education Series Editor: Yeping Li Integrated Approaches tooSTEM Education An International Perspective[END_REF].

Teachers used real data from various resources to analyze with the mobile application and believed that real life situations are shown with excellence by using the tool and mobile application. it is probable that go prepare students to real world and its requisites, they feel that real life problems should be the main scope of the curriculum and the lessons in the schools to provide problem solving skills in which they need to provide a solution, a model to achieve the goals clearly [START_REF] English | Mathematical Modelling With 9-Year-Olds[END_REF]. The teachers preferred experimenting with the students mostly because In the 21 st century, using computers and being proficient is necessary and in STEM education, for the most of the educators experimentation is the preferred way rather than the computation [START_REF] Li | Computational Thinking Is More about Thinking than Computing[END_REF]. The teachers dealt with problems that arise from the statistical terms, but they were caused from the curriculum not including the related terms, and the teacher's lack of self-awareness to do a pre-teach session. Future research was recommended to investigate teachers' professional development opportunities to integrate meaningful statistical practices in their lesson plans.


Zsolt Lavicza

Bea Kristinsdóttir

Freyja Hreinsdóttir

Development of silent video tasks as a tool for formative assessment

Keywords: Task design, formative assessment, silent video tasks

Introduction

This poster reports on results from a design-based research project (e.g. [START_REF] Bakker | Design research in education: A practical guide for early career researchers[END_REF] focusing on the definition, design, implementation, and development of silent video tasks-tasks were students watch and add their voice-over (in the form of explanations, descriptions, narrative) to short, silent animated mathematics films. Experiences made with teachers in the first phase of the research project formed the basis of the first drafts of a definition of silent video tasks and a description of their instructional sequence that were presented in the form of comics on a poster at CERME11 in Utrecht [START_REF] Kristinsdóttir | Silent video tasks: Towards a definition[END_REF]. Based on reactions from participants in TWG15 at CERME and at ICTMT-14 in Essen, a second phase of the design-based research project was prepared and conducted in fall 2019 with the aim to develop further the process of assigning silent video tasks. The second phase of the research project-which could be viewed as a case study-was conducted in collaboration with three Icelandic upper secondary school mathematics teachers, who had some previous experiences with the use of formative assessment. This poster presents the refined instructional sequence of silent video tasks, in the form of comics, based on results from the case study.

Methodology

Prior to the case study, some potentials of silent video tasks to be used as a tool for formative assessment had been identified, using a list of technology-based formative assessment strategies by [START_REF] Wright | Designing for formative assessment: A toolkit for teachers[END_REF]. Hence, teachers with experience of formative assessment practices were purposefully selected to take part in the case study. It so happened that all three teachers, that accepted participation, taught a course for low-achieving students entering grade 11 (age 16-17) in upper secondary school. Based on teachers' suggestions and in accordance with the course curricula, I (the first author) created three one-minute-long silent videos on the topics of coordinate geometry and linear equations. Teachers planned to implement all three videos within the course of one semester. They were encouraged to suggest refinements to the tasks' instructional sequence already in the first interview prior to the silent video task implementation.

Data collection was mainly in the form of teacher interviews that were conducted before and after each implementation of a silent video task and classroom observation notes written during each implementation of a silent video task. Furthermore, during three of the interviews, the teacher who tried out all three silent video tasks was asked to do some think-aloud exercises to prepare and reflect on his task implementation. These think-aloud exercises proved to be helpful to understand the reasons behind his actions and decisions regarding the task implementation.

All interviews were transcribed verbatim in Icelandic. Analysis started immediately after the first interview, focusing on the instructional sequence design and development. After transcribing the last interview, an iterative process of open coding and detailed notes-writing was used to gain an overview of how teacher's ideas, experiences, and expectations developed over time. Thus, creating a base for forming the design principles of silent video tasks, using a format suggested by van den Akker et al. (2013, p. 67) that included characteristics, procedures, and underlying theoretical and empirical arguments for the design.

Results

In the end, only one of the participating teachers implemented all three videos, and the other two teachers both implemented one of the videos in their teaching. Based on teacher interviews, thinkaloud reflections, and classroom observations, the instructional sequence of silent video tasks was further developed. Prior to the case study, the silent video tasks' instructional sequence involved the selection of a silent video; whole class viewing of the video; students working in pairs to prepare and record their voice-over for the video in a first lesson, teacher preparation (listening to all students' responses) before the follow-up lesson, and a group discussion based on a part of students' task responses in the follow-up lesson. Teachers in the case study, however, wanted feedback to be immediate and instead of only listening to selected task responses, they wanted a whole group listening and discussion as a reaction to all students' responses to the task. This proved to be challenging for teachers. Nevertheless, the teacher who implemented all three silent video tasks developed his ways of conducting the group discussion and during the third implementation, he was observed to make connections between students' discourse (as presented in students' responses to the task) and the mathematical discourse (as had been presented in classes before the silent video task implementation). This might be connected to the process of reification (Sfard, 2008, p. 44), which involves a transition from describing processes towards talking about objects.


Kim André

Stavenaes Refvik
email: refvikki@hivolda.no

Annette Hessen Bjerke
email: annette.hessen@oslomet.no

Morten Misfeldt
email: misfeldt@ind.ku.dk

Students Undertaking an Elective Programming Course: Their Views on the Connection Between Programming and Mathematics

Keywords: mathematics education, problem solving, computational thinking, programming

In this study, we investigate how students undertaking an elective programming course experience the connection between programming and mathematics. Based on stimulated recall interviews with six Grade 8 students who have solved test items from PISA 2003, we identify various experiences among the students regarding how they draw on programming skills in mathematics and vice versa. We propose that three findings, in particular, are worthy of further discussion and investigation: the way some experience programming as a context for mathematics and mathematics as a context for programming; the challenges that follow from how some students assert that programming and mathematics complement each other while others struggle to see a connection; and, perhaps the most important, the way in which the students report that experiences from programming have equipped them with skills such as being systematic and exact, which they find they need in mathematics.

Introduction

After recognising computational thinking (CT) as a vital 21st century skill [START_REF] Bocconi | The Nordic approach to introducing computational thinking and programming in compulsory education[END_REF], several countries have recently introduced CT to their curricula, either as a separate programming subject or as part of existing subjects. In Norway, CT is introduced in mathematics across grades (Directorate of Education, 2020), adding to being an elective programming subject in lower secondary school (Grades 8-10, ages 12-16). In the new Norwegian curricula, after Grade 4 in primary school, CT is exclusively addressed and talked about as programming. In programming, CT entails the ability to develop programmes to perform different algorithms and is considered a systematic description of how to solve a problem using a specific approach [START_REF] Kaufmann | Programming in mathematics education[END_REF], p. 1030).

As several studies have asserted that programming can induce improved understanding in mathematics [START_REF] Moreno León | Code to learn: Where does it belong in the K-12 curriculum[END_REF], and even more preciseimproved problem-solving skills in mathematics [START_REF] Husain | Engendering problem solving skills and mathematical knowledge via programming[END_REF][START_REF] Sinclair | The dynamic geometrisation of computer programming[END_REF] it is easy to understand why many European countries introduce programming in mathematics [START_REF] Balanskat | Computing our future. Computer programming and coding-Priorities, school curricula and initiatives across Europe-update 2015[END_REF]. However, [START_REF] Kaufmann | Programming in mathematics education[END_REF] warned against assuming that the programming's effect on problem solving is automatic. Exploring these findings while considering [START_REF] Kaufmann | Programming in mathematics education[END_REF] warning, we seek to better understand whether those students undertaking an elective programming course in lower secondary school can communicate how mathematics and programming are interrelated. Hence, we give students a voice in the matter. It is imperative that the students themselves experience this connection. Hence, we ask the following research question: How and to what extent do students undertaking an elective programming course experience a connection between programming and mathematics?

Theoretical Framework

To investigate the interplay between programming and mathematics, we turned to [START_REF] Kilhamn | Programmering i svensk skolmatematik[END_REF] and [START_REF] Elicer | Nature of the relations between programming and computational thinking and mathematics in danish teaching resources[END_REF]. [START_REF] Kilhamn | Programmering i svensk skolmatematik[END_REF] investigated the relationship between mathematics and programming in 32 mathematics lessons and found four 'categories' of relationships: only programming (lessons only focusing on programming); mathematics as context for programming (lessons where no new mathematical concepts are used, but programming can be used to repeat or confirm mathematical knowledge); programming as a computational tool in mathematics (lessons that clearly have mathematical content, and programming is used to carry out calculations efficiently); and programming as a tool for exploring mathematical concepts (lessons where programming is used to explore mathematical concepts and relations, where programming adds new insights and develops one's mathematical competence). Similarly, [START_REF] Elicer | Nature of the relations between programming and computational thinking and mathematics in danish teaching resources[END_REF] analysed activities from the Danish 'Texforsøget' and described the role of mathematics and what they refer to as programming and computational thinking (PCT) in different activities. They found six categories, where the first two are no mathematics involved and no PCT involved, implying either no mathematics or no PCT involved in the tasks, and that the focus of the task is either purely mathematical or purely programming. The two following categories, mathematics as a context and PCT as a context, are categories where mathematical concepts are explored using PCT operations or where PCT concepts are explored using mathematical operations. The next category is conceptual integration, which are tasks that are "not solved with mathematical or PCT actions but involve concepts in both math and PCT" (Elicer & Tamborg, 2021, p. 5). The last category is operational integration, where "mathematical and PCT competencies are interdependent" (Elicer & Tamborg, 2021, p. 6).

While exploring what these two studies revealed on the relationship between programming and mathematics, both at the level of a lesson and at the level of the tasks given, we sought to see how a discussion around some problem-solving tasks can work as a starting point for investigating how students see the relationship between programming and mathematics. Problem-solving tasks were a natural point of departure, as research seems to unanimously assert that problem solving is the most obvious overlap between mathematics and programming. For instance, in pursuing the definition of CT for mathematics and science classrooms, [START_REF] Weintrop | Defining computational thinking for mathematics and science classrooms[END_REF] proposed a taxonomy comprising four practices, one of which focuses on computational problem-solving practices. In this practice, [START_REF] Weintrop | Defining computational thinking for mathematics and science classrooms[END_REF] identified problem solving as a central practice for developing CT in mathematics (and science), which sometimes takes the form of conventional programming (p. 138). This made us take a set of mathematics problem-solving tasks as our point of departure when interviewing students on how and if they see a connection between programming and mathematics.

Methods

In Norway, students choose one elective subject in each of their three years in lower secondary school. This study used data collected from a larger project, where all participants (N = 247) completed a test comprising five different problem-solving tasks from the PISA 2003 test (these were the tasks on the library system, course design, transit system, irrigation and energy needs; see Faculty of Educational Sciences UiO [2003] for details). Due to how the library system task was mentioned and discussed in most interviews, we give a brief account of it here to inform the reader:

The task starts by giving a flowchart that shows a library lending system: If you are a teacher (yes), you are allowed to lend books for 28 days; if you are not (no), you are allowed to lend books for 7 days. Based on this system, the students are asked to build more complicated flowcharts.

Participants selected for this subproject are six Grade 8 students (ages 13-14) who chose programming as their elective subject (hereafter referred to as the elective programming course). The participants (five girls, one boy) came from four different schools (a mix of city-centre and rural schools) in four different counties in the western and southern parts of Norway and were chosen for interviews based on a) their choice of elective, b) their answers on the initial test, and c) how their methods on the initial test stand out from the rest of the students. Also, they were selected to ensure distribution across schools. The names presented later in the Results section are fictional.

Each of the six participants was interviewed by the first author of this paper on the same day as they had completed the initial PISA-2003-based test. The semi-structured interviews lasted for about 15 minutes (between 11-28 minutes) and were recorded. The overall intention of the semi-structured interviews was to explore the students' accounts of their solutions and to see if there were any crossreferences between their experiences in mathematics and the elective programming course. The first author used stimulated recall when asking questions about what knowledge they drew on when solving the five selected problems. Additionally, the participants were asked if they had examples of when they had drawn on knowledge learned in their elective programming course in their mathematics class and vice versa.

The interviews were transcribed in full in their original language (Norwegian) and analysed in two steps.

Step one involved a process of consistent coding [START_REF] Mason | Qualitative researching[END_REF] that entailed reading and rereading the transcript to identify if and how the interviewees related programming and mathematics and their accounts of how they drew on their programming knowledge in mathematics and vice versa.

Step two was to identify the central experiences and understanding of the students, which led to two main categories in the results: programming in mathematics and mathematics in programming. These two categories also build on the categories where mathematics is the context for programming and programming is the context for mathematics from [START_REF] Kilhamn | Programmering i svensk skolmatematik[END_REF] and [START_REF] Elicer | Nature of the relations between programming and computational thinking and mathematics in danish teaching resources[END_REF].

Results

This results section comprises two parts. In the first part, we will consider how the 8th grade students saw the possibility of using methods, knowledge and skills from their elective programming course when solving problems in mathematics, while in the second part, we will focus on how students used methods, knowledge and skills from mathematics in their elective programming course.

Drawing on Elective Programming Skills in Mathematics

When talking about the library system task, three of the students explained how they recognised the need to use a flowchart, which they had been introduced to in their programming elective course.

Peter immediately recognised the flowchart and explained how he had used it as part of his programming process: I recognised this one immediately [points at the flowchart] because we have had many flowcharts in the elective programming course when making programmes in different programming languages. The first thing we have to do is make pseudocodes and flowcharts.

We see how Peter drew on knowledge from his programming elective course when solving the Library system task. This also applies to two other students, Andy and Nick, who recognised the flowcharts as a method from the elective programming course.

While the section above provides examples of how students draw on methods from their programming elective course when solving problems in mathematics, we also found that the programming elective course provided the students with a new vocabulary that came in handy in their mathematical problem solving. During interviews, the students revealed how they tended to use programming concepts to solve and explain how they solved the library system task. Nick and Michael used TRUE/FALSE statements or IF-ELSE statements to explain how they solved the library system task and related this problem to earlier experiences:

(…) is it a magazine? If yes, seven days. If no, go to the next step. If you do not return books or magazines, then you are not eligible to borrow (…) (Nick) Peter, however, described how he solved other problems during regular mathematics classes using the programming concept variable and loops: One time, I developed a programme in Trinket or Python. Then I used a counting variable. I got a task where I could use the counting variable to solve it. This is a formula I can use in the programme because I have learnt to calculate this way using variables and 'x in range' [a command for loops in Python].

The third finding is connected to how the students identified whether or how the elective programming course changed their mathematics problem-solving approach. Due to how they had learnt to work in the elective programming course, Robbie revealed that he made more use of diverse strategies and focused more on finding multiple solutions to mathematical problems, and both Nick and Peter said they saw themselves as working more systematically when solving mathematical problems. Adding to this, Nick talked about how he felt more accurate in solving mathematical problems since this is vital in the elective programming course:

My mathematical view has become slightly more systematic. (…) In programming, one must be more exact to make a programme work.

Overall, these findings indicate that students acquire general skills through the elective programming course that they see as positive for their mathematical problem solving. Andy even highlighted that he sees a clear connection between programming and mathematicshe sees that both subjects are about solving problems:

Programming is much about solving different problems in a code, and some of the tasks in mathematics are about the same: solving a problem. However, some students did not see a connection between the elective programming course and mathematics. Nora, like Michael, claimed that there is no connection between programming and mathematics, adding that there is much she has not understood in the elective programming course. As she finds the elective programming course challenging, kind of softenings her statement on the disconnect. When asked if the elective programming course had affected him in mathematics, Michael put it like this:

That is a good question; I do not think so since programming does not involve, or it involves numbers, of course, but I am not sure if it is under [has to do with] mathematics. (…) It is not addition or subtraction (…)

Adding to this and despite his vague idea that the elective programming course might have changed his mathematical competency, Robbie revealed that he had never experienced any programming in mathematics and therefore saw no apparent connection. However, he said that one must know mathematics to programme, which we will return to in the second part of the results, which we turn to now.

Drawing on Mathematics Skills in the Elective Programming Course

Our data revealed three ways in which the students reported how they drew on mathematics skills in their elective programming course. The first entails how they report on the use of mathematical concepts in programming tasks, such as figure numbers and geometry. Five of the students had experience with mathematics in programming, one of which is Nora. She explained that she had experienced using rotation and angles when programming how to make different figures move. Likewise, Peter connected experiences from algebra to the concept of variables in the elective programming course:

There is much mathematics in programming because there is much algebra in programming. One uses many variables to make calculations, and that is almost the same as in algebra.

The second entails that students report on how they find their experiences of using different digital tools in mathematics to better understand what is going on in the elective programming course. Andy described an experience with GeoGebra, where he talked about making and using different functions to solve problems in mathematics. Similarly, Nick drew on an experience of using Excel and claimed this to be programming when he was asked if he used programming to solve mathematical problems: "Excel is technically programming". Both Andy's and Nick's experiences come from well-known computational tools in mathematics, and they reported that their experiences with those tools can help them in the elective programming course.

Third, our analysis shows that all the students, in one way or another, agreed that one needs to understand mathematics to programme. For example, Robbie, who mentioned the four arithmetic operations earlier, used arithmetic to argue that one needs to know them to develop a programme. Nick adds to this, saying "… one has to know much mathematics to programme properly". From this, we can see that Nick sees how mathematics plays a role in the elective programming course.

Discussion

In answering the research question -How and to what extent do students undertaking an elective programming course experience a connection between programming and mathematics?we decided to investigate how students tend to draw on programming in mathematics, and vice versa. Due to how, for instance, [START_REF] Weintrop | Defining computational thinking for mathematics and science classrooms[END_REF] identified problem solving as a central practice for developing CT (and conventional programming) in mathematics, we decided to use the students' solutions on a set of five PISA 2003 problem-solving tasks for knowing their thoughts on the connection between programming and mathematics.

Our analysis resulted in three main findings, where the first two confirm, and considerably elaborate on, the findings of [START_REF] Kilhamn | Programmering i svensk skolmatematik[END_REF] and [START_REF] Elicer | Nature of the relations between programming and computational thinking and mathematics in danish teaching resources[END_REF]. First, we saw that our students reported programming as a context for mathematics and mathematics as a context for programming. This became apparent in how some students argued that one needs to know mathematics to programme, and conversely, in how they revealed that they had experienced programming in mathematics. Andy, for instance, saw problem solving as a potential connection between programming and mathematics. We believe that this result argues for a potentially fruitful integration between mathematics and programming.

Second, we found differences in how the students talked about and experienced the relationship between mathematics and programming. If we lend the vocabulary of [START_REF] Kilhamn | Programmering i svensk skolmatematik[END_REF], we can say that some students tended to talk about how programming can work as a computational tool in mathematics. Peter, for instance, explained how he uses concepts from programming to solve problems in mathematics. Moreover, he asserted that programming tools made him approach and view mathematics differently. Robie added to this picture when revealing how he saw programming helping him find multiple solutions to a given problem. This adds insight into Kilhamn et al.'s (2021) category of "programming as a tool for exploring mathematical concepts", where they propose that programming can be used to explore mathematical concepts and add new insights. However, there are students, Nora and Michael in particular, who experienced the elective programming course and mathematics as two different subjects. We see how this adds insight into the categories of [START_REF] Elicer | Nature of the relations between programming and computational thinking and mathematics in danish teaching resources[END_REF] in the way Nora and Michael's experiences seem to reflect an understanding of "no PCT in mathematics" and "no mathematics in PCT". Hence, we see how some students enhance their understanding of mathematics when engaging in programming activities, while others struggle to see how they can leverage programming in mathematics and vice versa.

Our third finding adds to the "categories of interplay" between programming and mathematics set forward by [START_REF] Kilhamn | Programmering i svensk skolmatematik[END_REF] and [START_REF] Elicer | Nature of the relations between programming and computational thinking and mathematics in danish teaching resources[END_REF]. Both Nick and Peter stated that the elective programming course provided them with new skills. They emphasised that their experiences with programming had made them see the added value of being systematic and accurate in mathematics. We propose that this is a new category, and more research is needed to elaborate on it.

Concluding Remarks

This paper contributes to the ongoing discussion on how (and if?) programming and mathematics should be integrated in an educational context. We report on the experiences and views of the students, which, in sum, speak for an integration of the two. In addition, we assert, that by comparing our results with those of [START_REF] Kilhamn | Programmering i svensk skolmatematik[END_REF] and [START_REF] Elicer | Nature of the relations between programming and computational thinking and mathematics in danish teaching resources[END_REF], we strengthen their findings regarding how we see that the students' utterances confirm most of their categories of interplay.


Janka Medová
email: jmedova@ukf.sk

Algorithms in mathematics education

Keywords:

This paper summarizes my contribution to the plenary panel discussion Big Questions in Mathematics Education and focuses on the role of algorithms in mathematics education. Mathematics and computer science are interrelated from their substance. They share several common concepts including the algorithms, but the way they work with them differ among the two disciplines. Algorithms provide an additional dimension to mathematical knowledge, the deep procedural knowledge. Involvement of coding and work with algorithms in mathematics instruction seems to be a promising activity bridging the two disciplinary approaches. The questions: (i) how the work with algorithms contributes to students' learning and (ii) how to prepare teachers for this kind of activities should be investigated.

Mathematics and computer science are interrelated since the inchoative stages. Various fields of mathematics, including Boolean algebra, algebraic structures, discrete mathematics and probability provided the theoretical base for the emerging field of computer science. Some years later, in 1976, computer science started to pay off its debt by solving the problem of four colors using the thousands of hours of computing time [START_REF] Appel | The Four-Color Problem[END_REF]. The nature of mathematics and natural sciences as they are practiced in the professional world is developing towards computational thinking [START_REF] Weintrop | Defining computational thinking for mathematics and science classrooms[END_REF]. Nowadays mathematicians perceive computing as an inherent part of doing mathematics [START_REF] Lockwood | Computing as a mathematical disciplinary practice[END_REF]. The use of computers allows processing the calculations in mathematical statistics, even the work with big data.

Algorithms lie on the border between mathematics and computer science. Each discipline looks at algorithms from a different point of view. Computer scientists (e.g., [START_REF] Wirth | Algorithms and Data Structures[END_REF] define algorithms as finite, general, deterministic, resultative and elementary which has only limited use in mathematical proofs. Mathematicians investigate the existence and correctness of algorithms, whereas computer scientists look at the algorithms in a more utilitarian way, looking for the realtime instantiations, preferring obtaining the result to its preciseness. Therefore, heuristics were developed for solving problems with high complexity.

Algorithms occur in curricula of both mathematics and computer science. Algorithms for basic mathematical operations were the traditional part of mathematics education. In recent years the algorithms have started to disappear from mathematics curricula as conceptual understanding was in main focus of mathematics educators and procedures were often perceived as the 'rote learning'. On the other hand, [START_REF] Star | Foregrounding Procedural Knowledge[END_REF] sees the particular value in the procedural knowledge itself. Star foregrounds the deep procedural knowledge where the procedure is "known deeply, flexibly, and with critical judgment" (p. 133). Deep knowledge of algorithms, its principles and correctness might be seen as an amalgam between conceptual understanding and procedural fluency, so-called precept [START_REF] Gray | Duality, Ambiguity, and Flexibility: A "Proceptual" View of Simple Arithmetic[END_REF]. Some studies [START_REF] Lockwood | Reinforcing key combinatorial ideas in a computational setting: A case of encoding outcomes in computer programming[END_REF] demonstrated the added value involving coding in students' mathematical problem solving. Furthermore, using coding for solving mathematical problems may strengthen the computational thinking of the learners. The questions about how knowledge about algorithms contributes to students' learning of particular mathematical concepts and procedures and how can students profit from generalization of the mathematical procedures in an algorithmic way remain open.

If we want to approach algorithms from both sides, mathematics and computer science, we should decide who should teach about algorithms. There are countries, e.g. Slovakia, where algorithms and programming have had a strong position in curricula since the 1980s, as a part of computer science education. In other countries, e.g., France, algorithms are part of mathematical curricula. Both possibilities have their strengths and weaknesses. Teachers, specialists in mathematics or computer science focus on different characteristics of algorithms, in accordance with the disciplinary practices. Kortenkamp (this proceedings) describe several types of activities usual for work with algorithms in computer science: (i) design of algorithms, (ii) describing the algorithm in a (formal) language; (iii) carrying out algorithms, (iv) proving the correctness of algorithms, and (v) comparing algorithms with respect to complexity, elegance, or simplicity. These activities can be seen also in mathematics lessons and may serve as a bridge between the two disciplinary practices.

The teachers are the agents of any educational change [START_REF] Kieran | Linking Research to Practice: Teachers as Key Stakeholders in Mathematics Education Research[END_REF] and their work is shaped by their resources, orientations and goals [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision making and its educational applications[END_REF]. How the mathematics teachers' predispositions influence their practice regarding the algorithms, what is the knowledge needed to teach algorithms as deep procedural knowledge and whether mathematics teachers see any value in algorithmic approaches to mathematical objects should be examined in further studies.


Ingi Heinesen Højsted
email: ingih@setur.fo

Big Questions for Research on Digital Technology in Mathematics Education

Keywords: Digital technology, implementation, DGE, the teacher dimension

This paper outlines central points from my presentation at the plenary panel "Big Questions in Mathematics Education" of CERME12. I begin by briefly recounting the historical struggles with implementation of digital tools in mathematics education, and refer to the importance of how digital tools are utilized. To provide an example of the nuances and complexity of the research involved on digital technology in mathematics education, I focus on the specific digital tool of dynamic geometry environments and describe pertinent dimensions of research foci in relation to this software: student learning, task design, the teacher. I consider some of the major challenges with regards to the implementation of dynamic geometry environments as well as other digital resources in mathematics education. Finally, I reiterate the three broad questions posed for the plenary panel debate.

Looking back

Since the introduction of digital technology in mathematics education, extensive research has been conducted in relation to the implementation of different types of digital tools. In fact, already in the mid-eighties, the very first ICMI study [START_REF] Churchhouse | The Influence of Computers and Informatics on Mathematics and its Teaching[END_REF] considered the consequences computers and informatics might have on mathematics and mathematics education, and at that time, researchers were optimistic. However, two decades later, in the 17th ICMI study, Michelle Artigue looked back and reported that "The situation is not so brilliant and no one would claim that the expectations expressed at the time of the first study have been fulfilled" (Artigue, 2009, p. 464). During his CERME11 plenary speech, Paul Drijvers reflected on the low effect of integration of digital technology in mathematics education, acknowledging that "the mathematics education community is still struggling with the integration of digital technology in teaching and learning." (Drijvers, 2019, p. 8), while suggesting that the quality and exploitation of the digital tools are essential ingredients for effective integration. The manner in which digital technologies are utilized seems to be essential [START_REF] Drijvers | Embodied instrumentation: combining different views on using digital technology in mathematics education[END_REF][START_REF] Højsted | Signs emerging from students' work on a designed dependency task in dynamic geometry[END_REF][START_REF] Jankvist | CAS-Induced Difficulties in Learning Mathematics? For the Learning of Mathematics[END_REF]. In fact, [START_REF] Niss | Mathematics standards and curricula under the influence of digital affordances: Different notions, meanings and roles in different parts of the world[END_REF] proposes that "the very same piece of digital technology can give rise to 'marvels' as well as to 'disasters' in mathematics education. This means that no ICT system, hard or soft, is, in and of itself, good or bad for mathematics education." (p. 247).

Of course, there are many different types of software, and many different software of the same type. If we take the paradigmatic example of dynamic geometry environments (DGE), which has received a lot of research attention, more than 40 DGE had been developed by 2012 [START_REF] Hollebrands | Preparing to teach mathematics with technology: An integrated approach to geometry[END_REF]. Initially, much of the DGE research focused on students learning with DGEs, for example, Arzarello and colleagues' (2002) seminal work that categorized seven spontaneous ways in which students drag objects in DGE. Another example is [START_REF] Mariotti | Transforming Images in a DGS: The semiotic potential of the dragging tool for introducing the notion of conditional statement[END_REF] elaboration of the semiotic potential of the dragging tool to introduce conditional statements.

More recently, research attention has shifted towards task design, with authors describing DGE task design principles for specific mathematical aims, for example non-constructability tasks and dependency tasks in relation to conjecture generation and reasoning [START_REF] Baccaglini-Frank | Designing nonconstructability tasks in a dynamic geometry environment[END_REF][START_REF] Højsted | Signs emerging from students' work on a designed dependency task in dynamic geometry[END_REF], or developing frameworks to assess task quality (e.g. [START_REF] Trocki | The development of a framework for assessing dynamic geometry task quality[END_REF].

Less research has been conducted in relation to the role of the teacher in facilitating the implementation of DGE [START_REF] Komatsu | Task Design Principles for Heuristic Refutation in Dynamic Geometry Environments[END_REF]. The same applies to other digital tools, as described by Traglová and colleagues (2018), who reviewed the research output on digital technology in mathematics education coming from CERME conferences since 1999, "the awareness of the importance of the teacher dimension in research on technology in mathematics education [...] emerged slowly" (p. 154). Although there are several salient contributions on the role of the teacher (e.g., [START_REF] Bartolini-Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF][START_REF] Drijvers | The teacher and the tool: instrumental orchestrations in the technology-rich mathematics classroom[END_REF]) it is my impression, that this dimension still needs the most research attention. Suggesting the same in her PME plenary, Ana Sacristán proposed that teachers are the key players for the successful implementation of digital technologies, calling for "more teacher involvement in both professional development, and as co-constructors and collaborators in the design of technological implementations and resources" (Sacristan, 2017, p. 90).

Indeed, any designed digital mathematical resource may be adopted by their users (teachers and students) to suit their perceived needs in a particular classroom context (e.g., [START_REF] Trgalová | Model of Appropriation of a Curricular Resource: A Case of a Digital Game for the Teaching of Enumeration Skills in Kindergarten[END_REF]. The appropriation process may lead to a use of the resource that is not coherent with the educational intentions of the designer.

In a recent study (Højsted & Mariotti, in press), we found that collaborating with teachers about theoretical aspects of technology implementation can be a complex affair. This intricate issue requires reflecting on how to interface with teachers effectively to accomplish specific educational goals, while taking into account the variety of possible pedagogical paradigms that different teachers may adoptparadigms that may well be implicit.

Looking forward

Even if much has been accomplished in research on digital technology in mathematics education, it is evident that successful integration of digital technology into teaching and learning mathematics remains a difficult and complex issueakin to a gordian knot. Therefore, looking forward, I suggest that we reflect on these broad questions in the plenary debate:

1. How can we ensure that in another two decades, we are not still disappointed? 2. Which are the main issues hindering successful implementation of digital technologies in mathematics education and what are the solutions? (The teacher dimension? Design of resources?) 3. How can solid research findings on digital technologies in mathematics education find its way into praxis?


Anna Baccaglini-Frank
email: anna.baccaglinifrank@unipi.it

Big Questions in Mathematics Education: A "Representations in Mathematics Teaching and Learning" Perspective

Keywords: Digital extension, mathematical objects, representations

I summarize the perspective that I presented, as leader of TWG24 "Representations in Mathematics Teaching and Learning", bringing attention to a shift in how groups of researchers in Mathematics Education think and talk about mathematical objects and their representations. I argue that such a shift has theoretical and practical implications that should be taken into account in future research, especially when addressing some "Big Questions in Mathematics Education".

A shift in how we think about mathematical objects and their representations

In a recent paper (Baccaglini-Frank et al., 2022), we discussed what we see as a shift away from a Platonic conception of mathematical objects and their representations, characterizing various current lines of research in Mathematics Education. Such a shift opens new venues to how we think and talk about representations, which, in turn, influences how we use (and study) them in the teaching and learning of mathematics. In Plato's allegory of the cave, representations of mathematical objects are seen as reflections of natural things; such reflections inspire artificial objects, of which we only get to see shadows; such shadows are the imperfect forms that allow indirect access to the "real" perfect objects behind them. An implication is that whoever seeks mathematical knowledge should strive to obtain mental representations as close as possible to the ideal (non-physical) forms. The Platonic philosophical stance is at the basis of much research on mathematical representations. Many theories on learning mathematics claim the importance of understanding mathematical objects by somehow tapping on their "true" meanings that can be constructed by abstracting from their representations. However, researchers in Mathematics education, including members of the CERME12 community, have also started exploring different theoretical perspectives on the meanings and representations of mathematical objects (e.g., [START_REF] Miragliotta | Did you know you can draw a huge number of infinite heights? The students' realization tree of the heights of a triangle[END_REF][START_REF] Palatnik | Escape from Plato's cave: An enactivist argument for learning 3D geometry by constructing tangible models[END_REF].

A very important shift away from the Platonic perspective has been initiated by Anna Sfard in her Commognitive Framework (2008). Taking a Vygotskian socio-constructivist perspective, and following Wittgenstein, Sfard sees mathematical objects as no longer residing in some hyper-reality, but in discourse itself, being part of an autopoietic system, a system that defines its own objects. Hence, their meanings stem from the ways in which realizations of a mathematical object are used discursively. An implication of such a shift is that the term "representation" is inappropriate: Sfard rejects the Platonic view of mathematical objects existing "out there" and being re-presented in discourse; rather, for her, mathematical objects "come to life" as part of a discourse of certain human communities.

Another perspective, supported by cumulative data from various fields (neurobiology, robotics, kinesiology) is casting doubt on the Platonic view, and in particular on its implication that bodily experiences are separate from the ideal "mental representations" discussed above. Indeed, the "embodied" turn in cognitive science rejects the hierarchical mind-body separation and stresses that perception and action are formatively constitutive of our thinking. In the Mathematics Education field, the embodied paradigm has been taken to suggest that learning new concepts begins with discovering new ways to act in the environment, using new instruments to perform tasks on discovered affordances (e.g., [START_REF] Abrahamson | Making sense of movement in embodied design for mathematics learning[END_REF]. Working with the things themselves, students develop a capacity to act efficiently; they learn to describe the world mathematically to coordinate collaborative actions; they iteratively encounter more complex problems; and ultimately they modify the environments to solve emergent problems [START_REF] Abrahamson | Towards an ecological-dynamics design framework for embodied-interaction conceptual learning: the case of dynamic mathematics environments[END_REF].

A specific interest of mine concerns learning mathematics with or through digital tools. Turning away from the Platonic perspective, these can be conceived as "extensions" of our mind-and-body selves, and we can explore new ways of thinking, talking and using representations of mathematical objects. This is a line opened by posthuman discourse that describes the blending of human and technology as a "triumphant overcoming" the "natural" limitations of the human body, leading to the fascinating notion of "digital extension" (see the discussion on Merleau-Ponty in [START_REF] Dolezal | Disability as malleability: The prosthetic metaphor, Merleau-Ponty and the case of Aimee Mullins[END_REF].

My three questions

Based on the shift discussed above, on discussions in TWG24 over past meetings, and on my personal research interests, I am particularly interested in thinking about the three following questions.

Question 1: How can we (and will we) produce and share representations/realizations of mathematical objects to make teaching-learning processes truly inclusive? Indeed, producing and sharing representations/realizations of mathematical objects are fundamental processes to consider and study in the context of inclusive mathematics education. Research in this domain has highlighted the importance of using multi-modal channels of communication, perhaps also supported by Artificial Intelligence artifacts (e.g., [START_REF] Lew | Creating constructive interference between the 4th Industrial Revolution (+ COVID 19) and the teaching and learning of mathematics[END_REF]. Given the new perspectives on what representations/realizations of mathematical objects might be, we should explore ways to share and appropriate others' thoughts and personal experiences with such representations.

Question 2: How does (and will) learning occur (a) through making representations/realizations of mathematical objects or (b) through making artifacts that make these? A very interesting (to me) direction of research has been opened by research on "learning as making", in a constructionist perspective, where, for example, shapes in space are constructed using a 3D pen or a 3D printer (e.g., [START_REF] Ng | Drawing in space: doing mathematics with 3D Pens[END_REF][START_REF] Ng | Constructionist Learning in School Mathematics: Implications for Education in the Fourth Industrial Revolution[END_REF], or sketches of figures are produced on the plane by drawing robots that can be programmed by young children using a graphical block coding language (e.g., [START_REF] Baccaglini-Frank | Teachers' perspectives on the intertwining of tangible and digital modes of activity with a drawing robot for geometry[END_REF][START_REF] Baccaglini-Frank | Doing well" in the Teaching for Robust Understanding approach revealed by the lens of the semiotic potential of tasks with the GGBot[END_REF].

Question 3: Finally, since learning experiences can be very different and involve many different factors, I ask: How can we capture and study students' experiences with representations/realizations more holistically? I ask this because in most of the research studies I am familiar with, we attend to only a small part of the "whole picture", focusing, for example, either on cognitive aspects or affective ones, or on a certain small "bit", of a student's larger and more complex interaction with an artifact. However, I believe that it would be beneficial to have analytical tools that allow us to see more of the bigger picture. Perhaps we could work harder on trying to integrate results from studies that each looked at a small bit, but that together can provide new insights into students' learning processes.


Benoît Darquié

Vers une mesure de la brisure de la parité dans les molécules

Chapeau

Le vivant n'a sélectionné qu'une seule des deux formes, droite ou gauche, de molécules chirales aussi essentielles que les acides aminés ou les sucres. Les causes de cette asymétrie restent un mystère mais d'après les prédictions théoriques, l'interaction faible serait à l'origine d'une différence d'énergie, extrêmement faible, entre les deux formes. Les enjeux relatifs à une première mesure de cette différence d'énergie sont présentés ici. Elle apporterait une lumière précieuse sur un sujet qui touche à la fois à l'origine de la vie et à la physique fondamentale.

Introduction

Alors que de nombreuses molécules sont chirales et existent sous deux formes énantiomères images miroir, dites gauche et droite [START_REF] Collet | Molécules chirales -Stéréochimie et propriétés[END_REF], on observe un très fort déséquilibre entre ces deux populations dans les molécules du vivant : les acides aminés, qui composent les protéines de tous les êtres vivants, ainsi que toutes les molécules biologiques essentielles, n'y existent que sous l'une des deux formes. On parle d'homochiralité de la vie [START_REF] Hegstrom | The Handedness of the Universe[END_REF]. Il s'agit d'une condition nécessaire aux processus biomoléculaires tels que l'autoréplication et la reconnaissance moléculaire, et il est donc généralement admis que l'homochiralité sur Terre a précédé la vie. Presque tous les scénarios sur l'origine de l'homochiralité commencent par l'établissement d'un très faible excès énantiomérique dans une soupe prébiotique initialement racémique -c'est-à-dire contenant à part égale les deux formes énantiomériques -qui aurait ensuite été amplifié jusqu'à un état final de pureté chirale. Il existe de nombreux mécanismes d'amplification plausibles, mais la question de savoir ce qui a causé l'excès initial reste très spéculative et très débattue. Puisque la chimie est essentiellement régie par des interactions électromagnétiques qui agissent de façon identique sur les deux énantiomères, n'induisant pas d'asymétrie entre gauche et droite, comment se peut-il qu'il y ait une préférence pour un énantiomère plutôt que pour l'autre ? Or, l'une des quatre interactions fondamentales, l'interaction faible, ne respecte pas la symétrie par réflexion dans un miroir -on dit qu'elle brise la parité (voir article I.1, p.xxx). Elle est ainsi à l'origine d'une différence d'énergie extrêmement faible entre les molécules gauche et droite [START_REF] Jones | Tough science. Five experiments as hard as finding the Higgs[END_REF], de l'ordre de la centaine d'atto-électronvolts (∼ 10 -16 eV, soit ∼ 10 -11 J mol -1 ) pour la plupart des molécules ordinaires. Certains scientifiques ont émis l'hypothèse que ces différences d'énergie, et les différences qui en résultent entre les barrières énergétiques et les taux de réaction, pourraient avoir induit le biais nécessaire pour induire l'homochiralité de la vie alors que d'autres affirment par contre que cet effet serait bien trop faible [START_REF] Schwerdtfeger | The search for parity violation in chiral molecules[END_REF]. Notre objectif est de réaliser la toute première mesure de cette différence d'énergie. Cette mesure permettra d'alimenter le débat sur l'origine de l'asymétrie du vivant, mais constituera aussi un test très sensible des lois fondamentales de la physique. Nous présentons par la suite les enjeux expérimentaux liés à la mesure d'une grandeur aussi faible, qui concernent aussi bien la synthèse des molécules susceptibles d'avoir l'effet le plus important et leur refroidissement jusqu'à des températures de 1 K, que des développements de métrologie laser et l'amélioration de la sensibilité de la détection du signal.

Interaction faible et physique moléculaire

La chiralité moléculaire peut avoir diverses origines [START_REF] Collet | Molécules chirales -Stéréochimie et propriétés[END_REF] et le centre asymétrique en est sans doute l'exemple le plus familier. Il s'agit d'un atome qui, parce qu'il est substitué par quatre atomes ou groupes d'atomes différents, ne peut pas être superposé à son image dans un miroir (voir encadré de l'Introduction, p.xxx). Par exemple, les molécules 1 et 4 de la Figure 1 sont de géométrie tétraédrique, celle d'une pyramide composée de 4 faces triangulaires, et ont toutes deux un centre asymétrique, respectivement l'atome de carbone et l'atome d'uranium qui occupe le centre de la pyramide. La molécule 2 possède quant à elle deux centres asymétriques, les deux atomes de carbone marqués d'un astérisque. Mais il existe également d'autres configurations chirales comme celle de la molécule 3 dont la structure est celle d'une hélice de bateau. Comme prédit il y a près de 50 ans, l'interaction faible serait à l'origine de différences infimes entre les niveaux d'énergie des deux énantiomères d'une molécule chirale [START_REF] Jones | Tough science. Five experiments as hard as finding the Higgs[END_REF]. En effet, malgré la répulsion coulombienne, certains électrons de la molécule ont une probabilité infime de « se trouver dans le noyau » (leur fonction d'onde quantique, bien que s'étendant essentiellement à l'extérieur du noyau, a avec lui un recouvrement non nul). Or ces électrons interagissent avec les protons et les neutrons du noyau via l'interaction faible. Ceci a pour effet de perturber les niveaux d'énergies de la molécule, par ailleurs presque exclusivement déterminés par les forces électromagnétiques, et l'énantiomère droit voit ses niveaux d'énergie déplacés dans un sens, alors que les niveaux de l'énantiomère gauche sont décalés d'une même quantité, mais en sens opposé. De nombreuses idées d'expérience ont été proposées pour mesurer la différence d'énergie attendue entre les énantiomères d'une molécule chirale, mais aucune n'a encore atteint la sensibilité requise. La mesure la plus précise à ce jour a été réalisée il y a 20 ans dans notre laboratoire. Il s'agissait de mesurer la fréquence de vibration de la liaison C-F des deux énantiomères de la molécule chirale CHFClBr (Figure 1, molécule 1) par spectroscopie d'absorption de très haute précision d'une vapeur confinée dans une cellule à température ambiante. Cette fréquence est proportionnelle à la différence d'énergie des niveaux de vibration moléculaire entre lesquels a lieu l'absorption du photon et est égale à environ 30 THz. Chaque niveau étant affecté de façon distincte par l'interaction faible, on s'attend à une différence de fréquence entre énantiomères, de l'ordre de 2,5 mHz. La mesure de fréquence a été réalisée avec une précision de 8 Hz, soit ∼ 3×10 -13 en valeur relative, ce qui est un record, mais qui reste 3000 fois plus grand que la différence de fréquence prévue. La différence d'énergie entre énantiomères croit fortement avec la charge nucléaire et nous nous intéressons donc à des molécules possédant des atomes lourds. Pour la recherche de molécules candidates, nous avons établi des collaborations étroites avec des chimistes théoriciens, experts en calculs de chimie quantique relativistes incluant l'interaction faible, et avec des expérimentateurs, spécialistes de la synthèse de complexes chiraux contenant des atomes lourds, notamment métalliques. La molécule candidate idéale doit en effet satisfaire plusieurs critères qui sont autant d'enjeux, tant au niveau théorique qu'au niveau de la synthèse : elle doit (i) présenter une différence de fréquence importante induite par l'interaction faible, pour une transition vibrationnelle intense dans une gamme accessible aux technologies laser actuelles ; (ii) être disponible en quantité suffisante, à l'échelle de plusieurs grammes, et pouvoir être dédoublée (la séparation des deux énantiomères s'appelle le dédoublement) ; (iii) avoir une volatilité suffisante ou pouvoir être sublimée pour permettre des études en phase gazeuse. Cette quête a notamment débouché sur la synthèse de composés organo-métalliques solides d'oxorhénium ou de ruthénium, tels que les espèces 2 et 3 de la Figure 1, possédant des transitions vibrationnelles autour de 30 THz et des décalages de fréquence entre énantiomères atteignant ∼ 1 Hz, soit 100 à 1000 fois plus importants que pour CHFClBr. Plusieurs autres pistes sont actuellement explorées et d'autres molécules sont en préparation. Nos collaborateurs théoriciens ont en particulier identifié un composé chiral d'uranium (N≡UHFI, molécule 4 sur la Figure 1), avec un décalage de fréquence estimé à un niveau record d'environ 20 Hz. La synthèse et la séparation de ces énantiomères ne sont pas encore démontrées mais pourraient l'être dans le futur avec le développement du domaine encore peu exploré de la chimie de l'uranium. Afin de mieux appréhender l'instrumentation et le savoir-faire requis pour effectuer des mesures spectroscopiques précises sur des espèces aussi complexes, nous avons mené, en collaboration avec des collègues spécialistes de spectroscopie microonde, millimétrique ou moyen infrarouge, des études préliminaires sur certaines espèces candidates et sur certains de leurs précurseurs non chiraux comme le MTO (méthyltrioxorhénium, CH 3ReO3, représenté sur la Figure 1 en numéro 5). Sur la base de ces premiers résultats, nous avons démarré la construction d'un dispositif de nouvelle génération dédié à la spectroscopie vibrationnelle de précision de molécules chirales.

Les molécules considérées

Présentation du dispositif :

La Figure 2 illustre le principe de cette nouvelle expérience. Un jet lent (de quelques dizaines de mètres par seconde) et froid (quelques kelvins) d'une espèce chirale organo-métallique est obtenu en introduisant les molécules dans une cellule contenant un gaz d'hélium refroidi (voir encadré), puis en extrayant les molécules refroidies par collisions avec les atomes d'hélium. Ce type de source est l'une des technologies de jet moléculaire les plus récentes, avec à la fois une vitesse environ 10 fois plus faible et des flux 10 à 100 fois plus élevés qu'un jet traditionnel.

Les molécules traversent ensuite un interféromètre suffisamment sensible pour mesurer des décalages de fréquence vibrationnelle aussi petits que 100 mHz. L'interféromètre est basé sur une méthode utilisée en métrologie des fréquences, la méthode des champs séparés, introduite par N. F. Ramsey. Dans cet interféromètre, nous préparons ainsi chaque molécule dans une superposition quantique cohérente de deux états afin qu'elle oscille à une fréquence vibrationnelle naturelle et, pour mesurer les décalages de fréquence, nous comparons cette oscillation et celle d'un laser très stable, d'abord pour un énantiomère puis pour l'autre. Chaque niveau vibrationnel est constitué de plusieurs sous-niveaux correspondant aux énergies de rotation de la molécule. La transition qui nous intéresse couple donc deux états de rotation bien définis, l'un dans l'état vibrationnel fondamental, l'autre dans un état vibrationnel excité. Notre équipe a été pionnière dans l'exploitation de cette méthode pour la mesure de fréquences vibrationnelles avec un niveau de précision record de 14 chiffres significatifs [START_REF] Shelkovnikov | Stability of the Proton-to-Electron Mass Ratio[END_REF].

La fréquence requise se situe dans la région de l'infrarouge moyen du spectre électromagnétique, entre 10 et 100 THz (soit entre 3 et 25 µm). Les lasers, et en particulier les lasers stables, ont longtemps fait défaut dans cette fenêtre spectrale, mais la technologie des lasers à cascade quantique (QCL, de l'anglais quantum cascade laser), qui a émergé il y a 20 ans, est aujourd'hui suffisamment robuste pour pallier ce manque [START_REF] Capasso | Quantum cascade lasers[END_REF]. En collaboration avec le LNE-SYRTE (le Laboratoire National de Métrologie chargé de la réalisation des références dans le domaine du temps et des fréquences), nous avons récemment développé des QCL de très haute pureté spectrale et de très haute précision [START_REF] Miller | Precision spectroscopy comes to the mid-IR[END_REF]. Comme illustré sur la Figure 2, nous stabilisons le laser sur un peigne de fréquences optiques [START_REF] Amy-Klein | Des lasers à impulsions femtosecondes pour mesurer les fréquences[END_REF], lui-même verrouillé sur une référence de fréquence dans le proche infrarouge située au LNE-SYRTE. Cette référence est un signal laser ultrastable à 1,5 µm qui est transféré dans notre laboratoire par une fibre optique de 43 km de long, avec la correction du bruit de phase induit par la propagation [START_REF] Amy-Klein | La lumière des meilleures horloges accessible par les fibres du réseau Internet[END_REF]. Au LNE-SYRTE, ce signal est calibré sur des étalons de fréquence primaires qui sont les références ultimes en métrologie des fréquences, à savoir des horloges à fontaine de césium qui réalisent l'étalon de fréquence du Système International d'unités. L'excellente pureté spectrale de la référence de fréquence est alors transférée au QCL, dont la largeur d'émission passe de 1 MHz à 0.1 Hz, soit une amélioration de 7 ordres de grandeur. De plus, le lien aux étalons primaires du LNE-SYRTE permet de contrôler sa fréquence absolue avec une incertitude inférieure à 1 Hz. À notre connaissance, il s'agit du QCL le plus étroit et le plus précis existant.

Pour accéder au décalage de fréquence entre énantiomères gauche et droit, les molécules doivent être détectées avec une grande sensibilité. Les techniques de détection sensibles pour les molécules simples, comme l'absorption dans le moyen-infrarouge, la fluorescence, la photodissociation ou la photoionisation sont inadaptées aux molécules complexes. Nous construisons donc actuellement un nouveau détecteur, basé sur l'enregistrement du champ microonde émis par les molécules lorsqu'elles tournent. Dans l'interféromètre de Ramsey, un faisceau moyen-infrarouge transfère d'abord la population d'un état rotationnel bien défini de l'état vibrationnel fondamental (l'état 1) vers un état vibrationnel excité inoccupé (il s'agit de la transition d'intérêt pour notre mesure). Puis, en sortie de l'interféromètre, une impulsion microonde couple l'état 1 à un autre état rotationnel de l'état vibrationnel fondamental et c'est le rayonnement induit par cette mise en rotation des molécules qui est détecté. L'amplitude de ce signal microonde dépend de la déplétion de l'état 1 par le laser moyeninfrarouge dans l'interféromètre de Ramsey. Nous obtenons ainsi le spectre vibrationnel avec un rapport signal-à-bruit satisfaisant. La fréquence vibrationnelle de chaque énantiomère doit être mesurée avec ce nouveau dispositif. Comme il s'agit d'une mesure différentielle entre les deux énantiomères, de nombreux effets systématiques à l'origine de décalages en fréquence seront réduits, sans être forcément éliminés totalement, notamment ceux découlant de la vitesse et de la température finie des molécules. Nous prévoyons une précision de mesure inférieure à 0,1 Hz, soit une amélioration d'au moins deux ordres de grandeur par rapport à l'expérience sur CHFClBr.

Refroidissement cryogénique par un gaz tampon : [encart]

Nos travaux sur le refroidissement des composés organo-métalliques sont le fruit d'une collaboration avec M. R. Tarbutt et ses collègues du Centre for Cold Matter à l'Imperial College London, l'un des quelques groupes à travers le monde à avoir développé la technologie du refroidissement cryogénique par gaz tampon, jusque-là essentiellement limitée aux espèces diatomiques et aux radicaux légers. De nombreuses molécules polyatomiques sont peu volatiles aux températures facilement accessibles, ce qui rend difficile le refroidissement par des collisions en phase gazeuse avec une espèce plus froide. Grâce à l'ablation laser d'une cible solide contenant du MTO (composé 5 de la Figure 1) et placée à l'intérieur d'une cellule cryogénique, nous avons pour la première fois refroidi à quelques kelvins une espèce organo-métallique par collisions avec un gaz tampon d'hélium à 4 K (Figure E1). La température du gaz de MTO évaluée par la population des niveaux rotationnels est d'environ 6 K. Ces résultats montrent que les complexes de rhénium, d'intérêt pour mesurer la brisure de la symétrie droitegauche dans les molécules chirales, survivent au processus d'ablation et sont efficacement refroidis par collisions avec l'hélium, ce qui est très prometteur pour la production de jets moléculaires froids, lents et intenses.

Conclusion :

Inspirée des toutes dernières avancées sur les molécules froides et en métrologie des fréquences, l'exploitation combinée dans une expérience d'interférométrie de Ramsey d'échantillons moléculaires refroidis à 1 K et de lasers référencés aux meilleurs étalons de fréquences au monde devrait permettre de mesurer des décalages de fréquences vibrationnelles aussi petits que 100 mHz. La mesure de la différence de fréquence entre énantiomères induite par l'interaction faible, attendue à ∼ 1 Hz pour certaines des molécules envisagées, parait donc à notre portée. Au stade où nous en sommes, la capacité à produire des jets de molécules froides suffisamment intenses pour obtenir un rapport signal-à-bruit satisfaisant est le défi principal, et le développement d'un détecteur de haute sensibilité est primordial. Une mesure -même grossière comme un ordre de grandeur ou simplement le signe du décalage en fréquence -sera déterminante pour éclairer le débat sur l'homochiralité de la vie. Si la précision attendue est au rendez-vous, ces mesures constitueront aussi une source d'informations inédite sur l'interaction faible [START_REF] Jones | Tough science. Five experiments as hard as finding the Higgs[END_REF] qui pourrait permettre à terme d'explorer les limites du modèle standard de la physique. Il a en effet tout récemment été montré que les décalages en fréquence entre énantiomères étaient particulièrement sensibles à la présence hypothétique de matière noire [START_REF] Gaul | Chiral Molecules as Sensitive Probes for Direct Detection of P-Odd Cosmic Fields[END_REF].

Figure 1 :

 1 Figure 1: Classes de complexes organo-métalliques chiraux (molécules 1 à 4) et non chiraux (molécule 5) d'intérêt pour l'expérience. Molécule 1 : le bromochlorofluorométhane possède un carbone asymétrique ; molécule 2 : ce complexe dioxorhénium à ligand cyclohexane-1-olato-2-thiolato est un dérivé chiral de la molécule non chirale 5 et possède deux carbones asymétriques marqués d'un * ; molécule 3 : le tris-acétylacétonate de ruthénium(III) possède une structure chirale en hélice de bateau ; molécule 4 : ce composé chiral d'uranium de géométrie tétraédrique possède un centre asymétrique au niveau de l'atome d'uranium ; molécule 5 : le méthyltrioxorhénium (MTO) est le

Figure 2 :

 2 Figure 2 : Principe de l'expérience proposée pour mesurer la différence de fréquence vibrationnelle entre les énantiomères d'une molécule chirale. Le dispositif combine (i) une source intense de molécules froides et lentes issues d'une cellule cryogénique, (ii) un interféromètre de Ramsey utilisant des QCL calibrés sur des étalons de fréquence primaires pour sonder avec un grande précision la transition de rotation-vibration d'intérêt et (iii) un détecteur de haute sensibilité du signal interférométrique qui mesure le rayonnement microonde émis par les molécules en rotation. QCL : laser à cascade quantique ; LNE-SYRTE : Laboratoire national de métrologie et d'essais -Système de références temps-espace, le laboratoire chargé de la réalisation des références dans le domaine du temps et des fréquences.

Figure E1 :

 E1 Figure E1 : Gauche : dispositif expérimental pour le refroidissement du MTO (CH3ReO3, composé 5 sur la Fig. 1) par collisions avec un gaz tampon cryogénique. Une cible contenant du MTO est placée dans une cellule de cuivre remplie de ∼ 10 -2 mbar d'hélium et est refroidie à 6 K à l'aide d'un cryoréfrigérateur. La cellule est entourée d'écrans thermiques en aluminium à une température d'environ 40 K et placée dans une enceinte à vide. L'ablation de la cible est réalisée un laser Nd:YAG à 1064 nm. La lumière d'un laser à cascade quantique (QCL) émettant à ∼ 976 cm -1 (∼ 29.3 THz) traverse la cellule pour enregistrer le spectre d'absorption. Droite : Spectre du mode vibrationnel d'élongation antisymétrique Re=O obtenu. La structure observée résulte des différentes transitions entre niveaux de rotation pour la vibration étudiée. Chacun des pics est composé de plusieurs transitions que l'on ne résout pas. Le pic central correspond au signal produit par les molécules qui tournent initialement autour de leur propre axe de symétrie. La courbe en pointillés (rouge) est un ajustement des données permettant d'évaluer la température rotationnelle à 6 K ± 3 K.

Remerciements :

Nous tenons à remercier toutes les personnes suivantes ainsi que leurs collègues : R. Bast, A. Borschevsky, L. Pašteka, T. Saue et P. Schwerdtfeger (chimie théorique), J. Crassous et L. Guy (synthèse moléculaire), P. Asselin, B. Tremblay, T. R. Huet et P. Soulard (spectroscopies microonde, millimétrique et infrarouge), M. Abgrall, Y. Le Coq, R. Le Targat, P.-É. Pottie (métrologie des fréquences), et enfin A. Amy-Klein, B. Argence, N. Cahuzac, A. Cournol, M. Goncalves, A. Goncharov, A. Kaladjian, L. Lecordier,