Xingfeng Huang
email: xfhuang@shnu.edu.cn

Rongjin Huang

Teachers' learning from addressing the challenges of online teaching in a time of pandemic: A case in Shanghai

Keywords: Teacher collaboration, online teaching, Teaching Research Group, teacher professional development, teachers' resource

Due to the COVID-19 pandemic in Shanghai, China, all school classes were delivered through an online environment from February 24 to May 22, 2020. To support this transition, the Shanghai Education Commission led expert teachers and specialists to develop a series of online video lessons based on the Shanghai unified curriculum, and suggested students watch the online video lessons individually from home, followed by an online synchronous lesson supported by class teachers. This study investigated what primary mathematics teachers learned from addressing these challenges through a case study. By following two purposefully selected teachers over two weeks during the transition, multiple data sets including online video lessons, online synchronous lessons, daily reflections, and post-online teacher interviews were collected. A finegrained analysis of the data from the lens of the Documentational Approach to Didactics found that teachers adaptively used online video lessons as important resources for their online synchronous lessons and virtual Teaching Research Groups as a teachers' collaboration mechanism supported them to develop online video lessons and address various technological constraints. Finally, implications of this case study for mathematics education globally are discussed.

Introduction

Since early 2020, the COVID-19 pandemic has forced physical closure of schools and migrated the physical learning environment to online platforms. Responses to this emergency varied around the world. For example, the Chinese government launched the 'School Out but Class On' campaign (i.e., suspending classes without stopping learning) to provide platforms and models for online learning at home, while local educational practitioners adapted their own models and support systems based on local need for and availability of online teaching [START_REF] Yao | What role should teachers play in online teaching during the COVID-19 pandemic? Evidence from China[END_REF]). Yet, the pandemic is an unprecedented context in which millions of students have needed to adapt to a new medium for learning mathematics while teachers have had to provide teaching using online resources without any preparation [START_REF] Hodges | The difference between emergency learning remote teaching and online learning[END_REF]. This unique situation provided valuable insight into teachers' learning in response to the changing condition. Researchers have called for curating what 'people have learned or are going to learn about the current situation, for the benefit of future crises and times of stability' (Bakker & Wagner, 2020, p. 4).

To support the shift to online teaching in Shanghai, China, the Shanghai Education Commission led a team of expert teachers, who have rich school-teaching experience and who have been honored by local District Education Bureaus, to develop a series of online video lessons based on the Shanghai unified curriculum. The process of developing online video lessons was supported by the teaching research system [START_REF] Yang | How a Chinese teacher improved classroom teaching in Teaching Research Group: A case study on Pythagoras theorem teaching in Shanghai[END_REF] featuring Teaching Research Groups (教研组) at the school level for teachers' collaborative work, such as implementing curricula, mentoring teachers' research activities, organizing public lessons, and designing standardized examinations [START_REF] Huang | Teachers' learning through an online lesson study: an analysis from the expansive learning perspective[END_REF], and by Teaching Research Offices at district and municipal levels (Mourshed et al., 2010, p. 89-90). The online classes consisted of students watching the online video lessons individually and then participating in an online synchronous lesson. In Shanghai, unified textbooks with associated student books and teacher guidebooks are the primary materials used for supporting teachers' teaching [START_REF] Li | How Chinese teacher acquire and improve mathematics knowledge for teaching[END_REF]. In the pandemic period, the online video lessons were suddenly added into the teacher's resource system. In such a context, it was unknown how the teachers would move from face-to-face to online teaching rapidly without any preparation.

Several studies have identified the difficulties and challenges in online teaching and learning in response to the emergency of the pandemic [START_REF] Aldon | Teaching Mathematics in a Context of Lockdown: A Study Focused on Teachers[END_REF][START_REF] Cao | Post-pandemic reflections: lessons from Chinese mathematics teachers about online mathematics instruction[END_REF][START_REF] Chirinda | Teaching mathematics during the COVID-19 lockdown in a context of historical disadvantage[END_REF][START_REF] Clark-Wilson | Teaching with digital technology[END_REF][START_REF] Engelbrecht | Will 2020 be remembered as the year in which education was changed[END_REF][START_REF] Kalogeropoulos | Learning mathematics from home during COVID-19: Insights from two inquiry-focused primary schools[END_REF][START_REF] Mailizar | Secondary school mathematics teachers' views on e-learning implementation barriers during the COVID-19 pandemic: The case of Indonesia[END_REF][START_REF] Yılmaz | Are we all in this together? Mathematics teachers' perspectives on equity in remote instruction during pandemic[END_REF]. Teachers encounter challenges in four areas: technology, students, mathematics, and resources. These are: (1) unfamiliarity with online teaching platform functions which are needed for online synchronous teaching, and lack of access to Internet and computers in developing countries [START_REF] Aldon | Teaching Mathematics in a Context of Lockdown: A Study Focused on Teachers[END_REF][START_REF] Cao | Post-pandemic reflections: lessons from Chinese mathematics teachers about online mathematics instruction[END_REF][START_REF] Chirinda | Teaching mathematics during the COVID-19 lockdown in a context of historical disadvantage[END_REF][START_REF] Clark-Wilson | Teaching with digital technology[END_REF][START_REF] Engelbrecht | Will 2020 be remembered as the year in which education was changed[END_REF]; (2) difficulties in communication with students online [START_REF] Kalogeropoulos | Learning mathematics from home during COVID-19: Insights from two inquiry-focused primary schools[END_REF][START_REF] Mailizar | Secondary school mathematics teachers' views on e-learning implementation barriers during the COVID-19 pandemic: The case of Indonesia[END_REF]; (3) constraints in carrying out manipulative activity and grouping activity online [START_REF] Aldon | Teaching Mathematics in a Context of Lockdown: A Study Focused on Teachers[END_REF]; and (4) lack of time and support for identifying or designing appropriate electronic resources for students' learning from home [START_REF] Chirinda | Teaching mathematics during the COVID-19 lockdown in a context of historical disadvantage[END_REF]. Although in some countries, governments provided unified and systematic online support resources such as video lessons, television programs, and other resources for teachers and students to use, it is hard for teachers to adapt them in online synchronous instruction to meet various students' learning needs (e.g., [START_REF] Yılmaz | Are we all in this together? Mathematics teachers' perspectives on equity in remote instruction during pandemic[END_REF]. These studies presented benefits, strategies, and challenges of online mathematics teaching in the pandemic.

Yet, the teaching of mathematics online in a hybrid mode, both synchronously and asynchronously, in large scale is still underexplored [START_REF] Di Pietro | The likely impact of COVID-19 on education: Reflections based on the existing literature and recent international datasets[END_REF][START_REF] Ferdig | Teaching, technology, and teacher education during the COVID-19 pandemic: Stories from the field[END_REF]. Therefore, it is urgent for policymakers, school administrators, and teachers to work together systematically to ensure that all students have access to high-quality mathematics during the pandemic.

Building on previous relevant research in mathematics education which explored the potentials and challenges of online mathematics teaching, this research aimed to develop a deep understanding of what teachers struggled with and how teachers addressed various challenges during online teaching. Thus, it may provide insight into post-COVID mathematics teaching in an emerging hybrid learning environment in general, and how teachers learn to teach online through interacting with multiple resources and collaborating with colleagues within a virtual teaching and professional learning environment in particular.

Theoretical considerations

In this section, we first present the theoretical framework grounding our research, then state our research questions.

The Documentational Approach to Didactics

This paper is centered on studying teachers' learning through their interactions with resources, to address critical issues-in this case the pandemic situation. The interest in research related to teachers' resources appears clearly at the beginning of this century, resources being understood as what is provided by the curriculum [START_REF] Remillard | Examining key concepts in research on teachers' use of mathematics curricula[END_REF] and, more generally, what has the potential of 're-sourcing' teacher activity [START_REF] Adler | Conceptualising resources as a theme for teacher education[END_REF] at the core of instruction [START_REF] Cohen | Resources, instruction and research[END_REF]. This interest in resources has been strengthened with an evolution of the field of technology in mathematics education, 'rethinking the terrain' [START_REF] Hoyles | Technology and mathematics education: Rethinking the terrain[END_REF], towards a view of resources beyond the technology itself. In particular, [START_REF] Ruthven | Towards a Naturalistic Conceptualisation of Technology Integration in Classroom Practice: the example of school mathematics[END_REF] developed a structuring feature of classroom practice framework, which includes working environment, resource system, activity format, curriculum script and time economy. The concept of resource system focuses 'on the combined operation of the mathematical tools and curriculum materials in classroom use, particularly on their compatibility and coherence of use, and on factors influencing this' (Ruthven, 2009, p. 136). Ruthven suggested that the resource system consists of aspects of the artifacts in uses that are directly and deliberately related to the teaching and learning of the specific topic or subject.

To capture the features of mathematics teacher professional development through the lens of their interactions with resources, a specific conceptual framework, Documentational Approach to Didactics (DAD), has been developed over the past decade [START_REF] Gueudet | Towards new documentation systems for mathematics teachers[END_REF]Trouche, Gueudet et al., 2020). This frame extends the notion of resource given by [START_REF] Ruthven | Towards a Naturalistic Conceptualisation of Technology Integration in Classroom Practice: the example of school mathematics[END_REF], encompassing wider aspects of the artifacts that a teacher could use such as a textbook, a piece of software, a student's worksheet, an Internet resource, or a discussion with a colleague.

DAD calls the teacher's work with resources for achieving a given teaching goal his/her documentation work and identifies the result of this work as document: a document developed by a teacher is a hybrid entity made of resources resulting from the productive dimension of his/her activity and of the knowledge resulting from its constructive dimension. Knowledge is understood here in a broad way, following [START_REF] Gueudet | Towards new documentation systems for mathematics teachers[END_REF] proposition: 'In his conceptualization of teachers' knowledge, [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] distinguishes between subject matter content knowledge, pedagogical content knowledge, and curricular knowledge …In our study, we consider teachers' knowledge as a whole, without sorting out its different kinds ' (p. 201).

Three main dialectical relationships underpin the DAD: Concerning the teacher and the resources, specifically the relationship between two processes: the teacher permanently adapts the resources s/he is working with (instrumentalization process), while these resources and their constraints and affordances contribute to configuring the teacher's activity (instrumentation process). These two processes are indeed interrelated: the adaptation of resources (instrumentalization) conditions their integration into the practice of teachers (instrumentation);

Concerning the teacher's activity, specifically the relationship between two dimensions: There is a productive dimension (the teacher produces new resources for achieving a given teaching goal) and a constructive dimension (the teacher develops new knowledge in searching for, adapting and implementing resources). these two dimensions are indeed interrelated: the production of new resources leads to the development of new knowledge, and this new knowledge influences the production of resources.

Concerning the individual and collective dimensions of teachers' interactions with resources, which reciprocally impact each other: [START_REF] Gueudet | Communities, documents and professional geneses: interrelated stories[END_REF] evidence how the development of teachers' communities and their resources goes hand in hand. From the lens of DAD, studies have shown that teachers' collaboration has an important role in their use and design of resources (e.g., [START_REF] Gueudet | Collective work with resources: an essential dimension for teacher documentation[END_REF][START_REF] Pepin | Developing a deeper understanding of 'mathematics teaching expertise': Chinese mathematics teachers' resource systems as windows into their work and expertise[END_REF]. The study by Trouche et al. (2020) provided a window into the use, design, and transformation of resources by mathematics teachers in their teaching, which can help us better understand teachers' professional learning and the way collaborative contexts shape their professional work and learning. Through examining three Shanghai teachers' resources connected to their practice, their perceptions of mathematics teaching expertise, and their ways of developing their expertise, [START_REF] Pepin | Developing a deeper understanding of 'mathematics teaching expertise': Chinese mathematics teachers' resource systems as windows into their work and expertise[END_REF] found that the context of collective working enhanced mathematics teachers' expertise and capacity building that forms a strong workforce for promoting pupil learning. This finding supports that localized collaborative design efforts have an important role in generating resources appropriate for specific school settings and enhancing teachers' capability in effective use of these resources [START_REF] Ruthven | From design-based research to re-sourcing 'in the wild': reflections on studies of the co-evolution of mathematics teaching resources and practices[END_REF]. However, [START_REF] Gueudet | Collective work with resources: an essential dimension for teacher documentation[END_REF] found although teachers' routine work includes many collaborative aspects and teachers' collaboration surrounding resources is crucial for teacher professional development, productive communities of practice which include a shared interesting domain, pursuit of their common interest, and use of in their repertoire of resources [START_REF] Wenger | Communities of practice: Learning, meaning, identity[END_REF], are scarce. [START_REF] Pepin | Re-sourcing teachers' work and interactions: a collective perspective on resources, their use and transformation[END_REF] argued that how teachers work for adopting or adapting resources in collective contexts, and how to support teachers' collective work with resources for/in practice is needed for further investigation.

Research questions

In this study, due to the importance of both traditional resources (e.g., textbook, student exercise book and teacher guidebook) and the newly added ones (e.g., online lesson videos and online materials, and platforms) and due to the usefulness of the DAD utilized to examine teachers' work with resources, we adopted DAD as the theoretical framework for addressing our research questions. Specifically, this study aims to investigate the following research questions (RQ):

RQ1: What are the strengths and challenges of teachers' documentation work on online video lessons and platforms? RQ2: To what extent do teachers' collaborations support their documentation work?

Methodology

In this section, we first present our methodological principle, then describe the criteria of selecting participating teachers and rationale of focused content, and further explain data collection and analysis strategies.

A reflective investigation methodology

The study is based on the reflective investigation methodology proposed by DAD, (Trouche, Gueudet et al., 2020). This methodology is based on stimulating teachers' reflectivity on their interactions with resources: teachers are asked to show and describe their resources, their structure (related to their activity), their implementation and their evolution.

To address RQ1, we focus on the appropriation of online video lessons and platforms, analyzing how they affect teachers' documentation work, in terms of instrumentation and instrumentalization;

To address RQ2, we focus on the teachers' interactions in the Teaching Research Group and how these interactions support their documentation work.

Participants

Considering multiple factors impacting teachers' online teaching, the following three criteria were taken into account: (1) Teachers' teaching experience at school (experienced or novice); (2) Teachers' responsibilities and roles in the process of designing online video lessons (Yes or No); (3) Teachers' experiences in the Teaching Research Group during the online teaching period. Prior to the pandemic, Teaching Research Groups typically held activities once weekly in schools. However, during the pandemic different practices in terms of frequency of activities held emerged which include three levels: Remain (once a week as usual but online), strengthen (once a week online and frequent interactions on social media), and weaken (once online or none during the two weeks). Therefore, twelve mathematics teachers representing different situations were purposefully selected from different schools. The five teachers from a suburban district were chosen by the director of the district Teaching Research Office, and the other seven teachers from urban districts were chosen by the first author, who is a mathematics educator at a local university and has extensive collaboration with mathematics teachers in schools due to leading a longitudinal UK-China mathematics teacher exchange program since 2014 (Huang, Huang et al., 2021).

To address our research questions, we decided to focus on two contrasting cases to represent the wide spectrum of the data (Table 1). Zi, a designer and user of the online video lessons in this study, is the most experienced teacher in the teacher group. Wei, a novice teacher, generally needed more instructional support from colleagues or resources. Therefore, Zi and Wei were considered as two extreme cases among the twelve cases.

Teaching content

According to the Shanghai mathematics textbooks [START_REF] Huang | Real Shanghai mathematics: Textbook 4[END_REF], the online video lessons were organized, regarding decimal content, in four parts: (1) the meanings of decimals (13 online video lessons), (2) the properties of decimals (5 online video lessons), (3) movement of the decimal point (6 online video lessons), and (4) the addition and subtraction of decimals (9 online video lessons). The online classes included two parts (20 minutes for each): students watched the online video lessons individually and then participated in an online synchronous lesson. The eleven lessons (numbered from L1 to L11) were chosen from the properties of decimals, and movement of the decimal point, involving some difficult topics, e.g., proportional reasoning and connections with fractions [START_REF] Huang | Teacher guide: 4[END_REF][START_REF] Lamon | Rational numbers and proportional reasoning: Toward a theoretical framework[END_REF], which are big challenges for student learning [START_REF] Deliyianni | Representational flexibility and problem-solving ability in fraction and decimal number addition: A structural model[END_REF][START_REF] Tempier | New perspectives for didactical engineering: an example for the development of a resource for teaching decimal number system[END_REF][START_REF] Takker | Knowledge demands in teaching decimal numbers[END_REF]. The Shanghai Mathematics Curriculum Standards specify the learning goals of this content, based on visual representations and practical measuring: intuitively understanding of one to three decimal numbers (0.1…0.9; 0.001 …0.999); knowing the relationships between fractions and decimals; adding or removing 0 at the end of a decimal; and mastering the decimal size (Shanghai Education Commission, 2004).

Data collection

To address the research questions, we first collected three online video lessons (data named D0), corresponding to L5, L6 and L11, which were recommended by Zi as the designer. As Zi explained, L5 is a review lesson on the properties of decimals, providing exercises for consolidation, and L6 and L11 introduce new knowledge which is difficult for student learning (in L6 exploration of the change-pattern when moving the decimal point and in L11 using the Wan (万) unit (ten-thousand) or Yi (亿) unit (hundredmillion) to express large numbers). During the two weeks of online teaching, we also collected the following data sets: (1) videos of online synchronous lessons during L5, L6 and L11 (D1); (2) teacher daily reflections upon their teaching in response to question prompts (D2); and (3) audio or written records of interactions in the Teaching Research Groups (D3). In addition, immediately after completion of the two weeks of online teaching, semi-structured interviews with the two teachers were conducted and audio recorded (D4).

Data analysis

Based on the research questions and preliminary examination of the five types of data collected for drawing our results which were triangulated by other data. We employed grounded theory [START_REF] Corbin | Basics of qualitative research (3rd[END_REF] to generate codes from Zi's interview, including four categories: resources used in the episodes of online teaching, connection with the online lesson videos to improve student learning, tools of the platforms used to support the interactions in the online synchronous lessons, and teachers' interaction and collaboration in Teaching Research Groups. We used these codes to analyze the teacher's other types of data. When any data could not be encoded with the existing codes, we modified or adjusted the codes for adapting to the data. By iterating back and forth in this way, an initial coding system was formed. Next, we used this coding system to analyze Wei's data. We provide the prompts for interview and reflection, and the coding system as electronic Supplementary Material A and B.

Data processing and analyzing included three steps:  Transcription of data (both audios and videos). The three selected online video lessons were transcribed in Chinese verbatim. Then, English subtitles were added to the online video lessons for the third author's review. The teachers' videos of online synchronous lessons corresponding to the three selected online video lessons were transcribed in Chinese and then translated into English. The teachers' audio records of interviews were transcribed in Chinese verbatim;  Data coding and analysis. Regarding teaching materials, and student texts, the first two authors used the codes of Textbook, Student book, and Teacher guidebook in the first category of coding system to identify connections among the three selected online video lessons (D0) independently. Inter-rater reliability was checked (around 90%), and the disagreements were resolved through discussions. Both the first and second authors read the transcription of Zi's four types of data (D1, D2, D3 and D4) carefully, and coded the data independently with the coding system (See Supplementary Material B). Inter-rater reliability was checked (around 80%), and the disagreements were discussed and resolved. After that, two research assistants were trained to code, and they collaboratively coded Wei's data (D1, D2, D3 and D4). The first author then checked the codes by the research assistants and resolved some disagreements through discussion;  Categorization and illustration. Comparing the codes across teachers, the first two authors collaboratively identified patterns, and illustrated them using original data.

After that, the findings were translated into English by the first author, checked by the second author, and then sent to the third author for review. Some questions raised by the third author were discussed and clarified through weekly online meetings.  To answer RQ1, we analyzed the three selected online synchronous lessons (D1), teachers' diaries (D2), audio or written records of interactions in the Teaching Research Groups (D3), and teacher' interviews (D4), looking for the way they considered the online video lessons as new resources in the three online synchronous lessons in terms of instrumentation and instrumentalization. To answer RQ2, we analyzed the records kept by the two teachers about their interactions in the Teaching Research Groups (D3) and the interviews with teachers (D4) to identify how teacher collaboration in the Teacher Research Group supported their design or use of the online video lessons and online synchronous lesson implementation.

Results

We described each case in terms of what documentation works (including strengths and weaknesses) and how the Teaching Research Group may support the teacher's documentation works. At the same time, the three dualities of dialectic relationship are the core complements to highlight. We present the case of Zi first and then the case of Wei, followed by a summary of the two cases.

The case of Zi as an expert teacher

The case is described in two subsections with respect to our two research questions.

Documentation work for interacting with platforms and the online video lessons as a designer

The dialectical relationship between instrumentalization and instrumentation is featured by the ways of making use of students' mistakes in homework and content from online video lessons as teaching resources in her online synchronous lessons. In her online teaching [D1], Zi connected students' mistakes that occurred in homework with the online video lessons. In L5, she taught the contents summarized in online video lesson 5 and elicited the students to share what they had learned. She then had an interactive discussion of the most difficult task in the homework assigned in online video lesson 5. In L6, she employed the place value system to discuss the question, which had been explained with fractions in online video lesson 6, but it was hard for students to understand why multiply (or divide) by 10, then the decimal point moves one place to the right (or left). In L11, she quickly reviewed the key points of online video lesson 11 and asked students to summarize the procedure for rewriting numbers in the unit of wan (i.e., a ten-thousands) or yi (a hundred-millions) with an accurate language: to express a number in the unit of wan (yi), one should first find the digit in the wan (yi) place and then place a decimal point in the lower right corner of it.

She recorded related events explicitly in almost all her daily reflections. For example, in the L5 daily reflection, she wrote:

Zi: Today, I reviewed the students' mistakes, which were found through watching the video and marking students' homework.

[D2] In the L11 daily reflection, she wrote:

Zi: I asked students to explain what they have learned in the video, and then assigned related tasks to elicit the students' thoughts about their errors in their homework.

[D2] The fact that Zi is so comfortable linking online video lessons to her online synchronous lessons is a result of her involvement in designing and producing these videos. Zi's experience in producing the resource (e.g., online video lessons) contributes to her constructing knowledge about using the resource such as using students' mistakes in homework and appropriately addressing the difficult contents in synchronous online teaching. In the interview, she talked about how she knew the objectives of the video lessons and was familiar with the content, so she focused on student difficulties and online interactions in her synchronous online lessons.

Zi: For the content which are not introduced well in the videos, I will make it clear in the online lessons. For the contents that are clear in the video, I will focus on students' feedback on the exercises and provide specific instructions for students.

[D4] The dialectical relationship between instrumentalization and instrumentation can be also revealed by the ways of using the platform in her online teaching. From the three videos of online synchronous lessons [D1], we also find Zi had many online interactions with the students, using the various functionalities of the platform, efficiently instrumenting her teaching. On the platform, she constantly used the name list to find students who made mistakes in their homework and asked them to answer questions. For example, she required students to use the hands-up icon to show when they completed their classroom exercises.

In using the functions of the platform, Zi found that there were some problems with the platform that prevented her from interacting with the students. This led her to try different strategies to instrumentalize the functions of the platform. For Zi, the first problem was that the teacher could not see her students, which prevented her from getting immediate feedback from students' body language about their learning.

Zi: In the previous face-to-face classroom, I could observe students' eyes, expressions, and behaviors to understand how they were learning. But now it's impossible.

[D4] She tried to get more feedback from what students did in their homework and provided the students who made mistakes with more opportunities to communicate. Due to the design of the platform, when students used the hands-up function, their names were rearranged in the list in the order in which the students pressed the icons. This made it difficult for the teacher to immediately find the target students displayed on the platform and remove their mute restrictions to engage with them in conversation. Zi, later in the interview [D4], told us that she had taken two approaches to solving this problem. The first was to improve the student-teacher relationship: before officially starting online synchronous lessons, she logged onto the platform and chatted with the students to warm them up, thus bringing them closer to the teacher. The second was to give the initiative to the students: she let the students turn on the microphone by themselves before starting online synchronous lessons; then when the teacher called on a student, she could talk directly with him/her.

Collaboration to support documentation work

Zi, as the leader of her Teaching Research Group (9 teachers), held only one online formal activity of the Teaching Research Group in the two weeks [D3], but she had a very intensive interaction with her Teaching Research Group during the process of designing the online video lessons. For each video lesson, she first designed a specific lesson plan independently, which she submitted to the specialists from the Teaching Research Office for review and subsequently revised in response to the changes suggested. This went back and forth until a final script was created. Zi, with her young colleagues at the school, designed the lesson slides based on the lesson plans and submitted them to the specialists for review and revision. Indeed, this constituted a collaborative design team, involving the Teaching Research Group at the school coordinated by Zi and the specialists from the Teaching Research Office. Zi mentioned that specialists provided her with advice for improving lesson plans or slides.

Zi: They have higher professional capacities and think more comprehensively than me. I designed the lesson plans based on my previous experience and personal understanding. But the specialists have rich experience in and deep understanding of the curriculum, so they provided some suggestions on the improvement of the lesson plans or slides. [D4] Because she had no experience in creating slides for the online video lessons, she got support from her colleagues:

Zi: I invited a couple of young teachers from my Teaching Research Group to help with the lesson slides. Actually, I'm sure I couldn't do this alone. Some of the slides may involve more than a dozen animations. During this process, I need to design the teaching script, and then communicate with them what kind of visual effect I want to achieve. [D4] Due to the complexity of the process of designing online video lessons in the lockdown phase, Zi was very stressed. However, colleagues' collaboration encouraged her:

Zi: A lot of tasks we had to get done in a short period of time, so we cheered each other on. Sometimes I think it's almost done, and then they [the specialists] tell me no, I have to start over. I was desperate. But when I think of my colleagues who are so motivated to do it with me, I have no reason to quit. I feel that through the completion of this task, we deepened our understanding of each other and felt like comrades. [D4] The evidence above clearly shows the dialectic relationship between individual effort and collective collaboration with the specialists and her colleagues when producing the online video lessons. The teacher also claimed that she learned various knowledge from the specialists and her colleagues, including the understanding of the curriculum and the use of technology. At the same time, we argue that what the teacher learned would have a positive effect on her subsequent use and production of resources.

Based on the evidence Zi provided about the Teaching Research Group, they rarely used social media among colleagues to communicate about their work when they implemented synchronous online lessons. They did, however, hold one formal online Teaching Research Group event (about 56 minutes) in the two weeks through a platform. They shared how to use the platforms and online teaching practices in other schools. However, they did not share any experience of using online video lessons in their practice. Zi indicated that the biggest problem they encountered was technology, not related to content or instructional approach. So naturally, the issue of their Teaching Research Group focused on how to use the platform:

Zi: Online teaching is not something that teachers are familiar with at first. Even though we were trained how to use the platform before we started, when using it, a lot of problems occurred. For example, how to upload materials, how to show slides on the platform, how to interact with students on the platform became the most pressing issues for everyone. [D4]

The case of Wei as a novice teacher

The case is described in two subsections, with respect to the two research questions. 4.2.1 Documentation work for interacting with platform and with the online video lessons as a user Due to technology issues Wei was not able to video record her L6. From L5 and L11, we found that the major resources Wei used in her lessons were provided by her Teacher Research Group or designed by herself. She did not directly use the online video lessons. Through analyzing the diary reflection on the two days, we found that Wei designed tasks for her online synchronous sessions after watching the online video lessons. For example, Wei found number lines were used in both the textbook and the video to help students understand 'multiplying a decimal by 10, decimal point moving one place to the right.' But she felt that the number lines were too complicated, so she supplemented the use of the cube model, which she used to help with her students' understanding of the base-10 system in whole numbers.

Wei: In the video, the teacher used a number line to represent the relationship between 0.1, 0.01, and 0.001. I used a cube model to represent 1. When divided equally into 10 parts to make planes, each plane would represent 0.1. Then dividing a plane equally into 10 parts to make bars, each bar would represent 0.01, and dividing a bar equally into 10 parts to make small cubes, each small cube would be 0.001. [D2] Nonetheless, in the interview, Wei also described how the online video lessons had been one of the new resources she referred to for practice.

Wei: I will read the textbook first, then combine it with the exercise books to form initial slides based on the materials from my Teaching Research Group, and finally make adjustments and additions in conjunction with the online video lessons, possibly cutting or adding some content. [D4] In the interview, Wei appreciated the quality of the online video lessons and believed they were quality resources for teachers to promote reflection on their own teaching.

Wei: The teachers in the video are able to highlight the key points of teaching, elicit and motivate students to think and focus on mathematical ideas and methods, which I believe are very worthwhile. Through the videos, I feel that my practice needs to be polished. Particularly, I actually don't do well in the classroom discourse. I often record the teacher's language in the online video lessons to learn by myself.

[D4] By viewing the online video lessons as a new teaching resource Wei was adapting and modifying her teaching practice to incorporate these materials, and the teachers' approaches in the online video lessons also facilitated her reflection and stimulated her to improve practice. This process exemplifies the teacher's instrumentation and instrumentalization by the online video lessons as a component of her resource system. But Wei's documentation work for the online video lessons was not only in a oneway direction. She reflected upon and changed her ways of using the online video lessons, which constitutes the alternative process of instrumentalizing the online video lessons. In her diaries, she offered different ideas and advice for specific lessons. For example, for online video lesson 3, she commented 'when comparing decimals, we can arrange them vertically by aligning the decimal points to make the comparisons more intuitive' [D2]. In online video lesson 6, a task (0.1 × 10 = 1, 0.01 × 10 = 0.1, 0.001 × 10 = 0.01) was designed to generalize the pattern of the movement of the decimal point (multiplying by 10 results in moving the decimal point one place toward the right). Wei advised to design a variation task where the same decimals were divided by 10 [D2]. In the interview, she also addressed a more general question about the design of online video lessons.

Wei: In the online video lessons, the dialogues are well designed by the teachers and are expressed in a standard and concise manner, but those may not be easy for students to understand. I sometimes have to listen to those several times to understand what they were saying. I would recommend that subtitles be added to this section and the dialogues be designed to be more verbally adapted to students.

[D4] When interacting with the online video lesson, the dialectical relationship between the instrumentalization and instrumentation further results in the dualities of productive and constructive activities. The teacher claimed that she learned a lot from watching and adopting the online video lessons (constructive activity). She used different methods from those presented in the online video lessons (productive activity) based on internet resources and self-analysis of the content in the online video lessons and self-reflection on synchronous online teaching.

In the two videos of the online synchronous lessons [D1], Wei used the supplementary exercises to carry out teaching. She presented tasks on slides. These tasks were multiple-choice questions, true-false questions, and fill-in-the-blankquestions. Then she required students to use the answer-card function to respond. Wei used the data-collection function which shows the distribution of all students' answers. Then she required students to use the answer-card function to respond. Thus, Wei knew the overall situation of students learning, and then provided corresponding feedback and explanation. In the interview she talked about why she used such a teaching method, mainly because the network was slow and there was a time delay if students and teachers talked directly. Additionally, she felt the same way as Zi about the handup function of the platform.

Wei: After the students raise their hands, the platform only shows the icon of 6 children raising their hands. Because of the speed issue, it often shows the same students, but slower children are not shown. [D2] Wei felt that the platform limited her practice. When she used the slide-presentation function for online teaching, it was difficult to adjust some contents to student learning. When she used the shared interface function, it occupied part of the original interface and affected online interaction. Comparing these two methods, Wei gradually preferred to use the sharing screen to interact with her students.

Wei: If I use the slide-presentation function, I have to follow the content designed. It is difficult for me to present different ways that students come up with in the class. If I use the sharing screen, I can write on the pad, which is a little better. [D4] In Wei's process of using the platform's functions, we can see the instrumentalization and instrumentation processes regarding the platform. That is, she would first try to use a function of the platform, and when she found that it was not conducive to her online interaction with students, she would choose another function to improve that. 4.2.2 Collaboration to support documentation work Wei's Teaching Research Group, which included four novice and one experienced teacher, had an online meeting almost every day to discuss technical and didactical issues [D3], as well as to appease teachers' emotions. Wei indicated that the online Teaching Research Group meeting was not restricted by time/venue and made her to be 'brave behind the screen'; 'I have strengthened the confidence in teaching through communication and sharing.' However, 'we did not have enough time for a deep discussion' [D4].

Wei's Teaching Research Group rarely discussed how to integrate the online video lessons with their online teaching [D3]. Both in her interview and diary, Wei detailed a problem she encountered with online video lesson 8.

Wei: I was impressed with the lesson on the use of moving decimal points to perform conversions between different units. Because in this lesson, the teacher in the video wrote out the solution process in detail for the previous tasks but gave the direct answer to the final exercises. I think there is a gap. Some students may not know how to make the transition. [D4] She said that in the Teaching Research Group they also discussed this issue, but there was no consensus, and each teacher used a different approach.

Wei: My colleagues in the Teaching Research Group employed different approaches. Some colleagues felt it was completely fine, and the solution in the complete process was not emphasized in their lesson. Some colleagues required students to write the complete process, such as multiplying or dividing decimals by 10 or 100 to gain the outcome, while I focused on moving the decimal point on the original decimal to get an outcome. [D4] Wei found that her approach caused many errors or mistakes in the students' homework. In converting units, some students neither understood whether to multiply or divide, nor knew what to multiply or divide by [D2]. Wei needed more support from her colleagues. In Wei's case, although more individual effort was made, more strong collective collaboration is needed.

Summary addressing the two research questions

By comparing the two extreme cases with a focus on three dialectical relationships in the DAD, we were able to make the following observations in line with research questions: (1) The videos and platform as resources played a developmental role for the documentation work of both expert and novice teachers. The expert teacher developed technological skills through her online teaching experience, while she had a deep understanding of the videos because of her team's great effort in creating the videos. The novice teacher seemed to have fewer issues with the use of technology, but she developed her understanding of the content through watching and adapting the online videos which became an important source for her learning. (2) Teacher collaboration provided important support when the expert teacher designed and created the online video lessons. The novice teacher used social media to maintain rich interaction with her colleagues, for technological, didactical, and affective reasons in the lockdown period. However, she was not able to receive more support from her colleagues for adapting the online videos and supporting her synchronous online teaching.

These conclusions concerning one expert teacher and one novice teacher based on a contrasting case study may not be generalized to other situations. Yet, our observations over the whole sample of twelve teachers seem to confirm such patterns concerning the role of instructional expertise, involvement in development of online video lessons, and capacity in using technology in such complex periods where teachers have to face the uncertainty of a new mode of teaching requiring an immediate integration of new resources.

Discussion

This study presented various ways to address the challenges in online teaching as literature indicated [START_REF] Aldon | Teaching Mathematics in a Context of Lockdown: A Study Focused on Teachers[END_REF][START_REF] Cao | Post-pandemic reflections: lessons from Chinese mathematics teachers about online mathematics instruction[END_REF][START_REF] Chirinda | Teaching mathematics during the COVID-19 lockdown in a context of historical disadvantage[END_REF][START_REF] Kalogeropoulos | Learning mathematics from home during COVID-19: Insights from two inquiry-focused primary schools[END_REF][START_REF] Mailizar | Secondary school mathematics teachers' views on e-learning implementation barriers during the COVID-19 pandemic: The case of Indonesia[END_REF][START_REF] Yılmaz | Are we all in this together? Mathematics teachers' perspectives on equity in remote instruction during pandemic[END_REF]. Specifically in this study we found the following: (1) Limited 'face-to-face' communication online is still a pressing technological constraint. However, the teachers were able to use their own strategies to deal with receiving students' immediate feedback. In Zi's case, she addressed students' mistakes or errors from targeted students' homework, which was marked carefully before online synchronous lessons, while Wei used technology functions to obtain students' responses immediately. (2) Learning to use unfamiliar technology was not a big challenge in this case. For example, Zi was originally afraid of technology, but through collaboration with colleagues and her attempts at, and reflection on, online teaching, she developed technology capacity. (3) The online video lessons, as resources for students to learn from home, were integrated into the online synchronous lessons by teachers in two dialectical ways (instrumentation and instrumentalization) from the DAD perspective, to promote students' learning. (4) Teachers were encouraged to participate in the development of the resources so that they could adapt them for their students. Specially, regarding the online video lessons as resources for teachers, our study revealed that: for the novice teacher, the instrumentation process was salient, which helped her to develop deep understanding of the content and pedagogy; and for the experienced teacher, the instrumentalization processes involved using online video lessons to connect with students' difficulties. However, exploration of how to develop video lessons that can promote student engagement and interaction is needed in the future. Our study also evidences the critical role of the Teaching Research Group for addressing the challenge of online teaching. The Teaching Research Group system as a well-established infrastructure of teacher collaboration had already been analyzed as a powerful means for enhancing teacher documentation work [START_REF] Pepin | Developing a deeper understanding of 'mathematics teaching expertise': Chinese mathematics teachers' resource systems as windows into their work and expertise[END_REF]. During the pandemic, the usual form of the Teaching Research Group was adapted by teachers through technology (e.g., in using social media, or a meeting platform) for addressing the challenges of online teaching. Virtual Teaching Research Groups played different roles in teachers' documentation work. In the development of online video lessons, through the events in virtual Teaching Research Groups, specialists, experienced teachers, and novice teachers were closely connected to play different roles and develop the resources through division of labor and collaboration. The Teaching Research Group looks like a project team [START_REF] Bond-Barnard | Linking trust and collaboration in project teams to project management success[END_REF], where the interdependent individuals work together to meet their goals and share responsibility for specific outcomes in an organized context by the Teaching Research Office in Shanghai. When teachers used online video lessons and implemented their own online synchronous lessons, the virtual Teaching Research Groups opened new opportunities for the novice teacher (e.g., making it easier for her to 'be brave' in asking sometimes naïve questions) while providing new challenges for experienced teachers (e.g., addressing diverse and complex questions from novice teachers and supporting them). Due to the complexity and novelty of adapting to teaching online during this pandemic period, the collaborative work of Teaching Research Groups was focused on figuring out how to effectively use the platform, rather than discussing the content of online video lessons and the pedagogical issues in teaching, which novice teachers need as supports from their Teaching Research Groups. The Teaching Research Group in Wei's case did not form a true community of practice [START_REF] Gueudet | Collective work with resources: an essential dimension for teacher documentation[END_REF], because without opening a dialogue between the inside and outside perspectives [START_REF] Wenger | Cultivating communities of practice: A guide to managing knowledge[END_REF], the novice teachers' interests were not shared and pursued by the community. Unfortunately, the activities in the Teaching Research Group in Zi's case also weakened during the phase of implementing synchronous online lessons. Thus, during the pandemic, when all the contexts suddenly changed, teacher collaboration in virtual Teaching Research Groups, without strong organization and coordination, faced a huge challenge.

Actually, communities of practice are emphasized by DAD, when teachers interact with resources through their collaborative work (Trouche et al., 2020a). According to the principles for cultivating a community of practice [START_REF] Wenger | Cultivating communities of practice: A guide to managing knowledge[END_REF], to strengthen teachers' collaboration to help teachers address unexpected challenges, the following two aspects need to be strategically addressed: first, participants with different roles are encouraged to work together and have open dialogues between insider and outside perspectives. In this case as teachers implement online synchronous lessons, experienced and novice teachers in a school-based virtual Teaching Research Group can mutually support each other if they share the difficulties and successful experience using the platform technology and online video lessons. If expert teachers and specialists who design online video lessons can share their experiences of designing online video lessons in cross-school virtual Teaching Research Groups and engage in dialogue between designers and users, then the difficulties experienced by teachers like Wei can be resolved. Second, all teachers should be aware of the value of teachers' collaboration for helping them address new challenges. During the pandemic, although the traditional face-to-face events disappeared in the virtual Teaching Research Group community, the established traditional Teaching Research Group community values and norms are still important. When teachers developed resources, the virtual Teaching Research Group played a great role in linking the different level participants to work together. While the novice teacher used the teaching resources from the Teaching Research Group for her synchronous online practice, she indeed wanted more support from the virtual Teaching Research Group. In order to establish a community of practice, teachers' collaboration needs to create dialogue, activities, and relationships that share and address the diverse issues and problems emerging in the community.

Additionally, this study of mathematics teachers in Shanghai features two aspects that are specific to mathematics teachers. First, the Shanghai primary mathematics teachers are specialists who mainly teach mathematics. So, they may have a profound understanding of mathematics knowledge [START_REF] Ma | Knowing and teaching elementary mathematics: Teachers, understanding of fundamental mathematics in China and the United States[END_REF]. Second, the content of decimals requires the use of multiple, visual and dynamic representations to support students' understanding and the use of multiple approaches to solve a problem. These features may result in mathematics teachers needing more resources, as well as being adept at using them. Thus, the findings of this study could provide specific implications for mathematics teacher professional development in the post-COVID era.

Due to the difficulty following up on the quick changes in teaching modes during the period, we probably missed important data (e.g., following the online video lessons' design process itself). In addition, our analysis does not consider the students' written work due to the space limitation of the paper. The whole set of data could serve indeed for further analysis.

Conclusion

As a conclusion, we would highlight the potential of the documentational approach to didactics for theoretically grounding our study. The three dialectical relationships at the heart of this approach appeared particularly relevant for developing our analysis:  The duality instrumentation-instrumentalization helped us to understand the way that the online video lessons and platform structured teacher work in a time of uncertainty, and, the way that the teachers adapted these resources as well;  The duality of productive-constructive activity helped us to understand the link between resource design and professional development. The case of Zi as an online video lesson designer was particularly illustrative of this duality, leading to the notion of design capacity [START_REF] Pepin | Refining teacher design capacity: Mathematics teachers' interactions with digital curriculum resources[END_REF];  Finally, the duality collective-individual documentation work clearly appeared in the pandemic, where the teachers facing the complexity of online teaching developed a new form of collective work (virtual Teaching Research Groups combined with various usages of social networks). Our study and relevant literature ( § 2.2) evidenced how the pandemic brings the issues of resources to the fore. This focus creates new theoretical and methodological needs, aiming to deepen the processes involved in the design, use and sharing of resources for teaching mathematics. These theoretical and methodological needs call for a networking of the different scientific fields ( § 2.1) and cultures studying teachers interacting with resources [START_REF] Choppin | Reflections on the documentational approach to didactics[END_REF]Fan et al., 2018;[START_REF] Remillard | Teachers' use of Mathematics Resources: a look across Cultural Boundaries[END_REF]Ruthven, 2019), towards a domain extending the initial scope of DAD: a 'resource' approach to mathematics education (Trouche et al., 2019). 021-10060-1.

Table 1 .

 1 The two teachers' profiles

	Name Teaching	Professional	Position	Designer	Collaboration
	experience	position	in	for online	in	Teaching
			Teaching	video	Research
			Research	lessons	Groups
			Group

Shanghai Mathematics Education under the Condition of Online Interaction (2019-2028) supported by the Teacher Education Center under the Auspices of UNESCO. Conflicts of interest/Competing interests: No Availability of data and material: Yes Code availability (software application or custom code): No


Petra Scherer
email: petra.scherer@uni-due.de

Katrin Rolka
email: katrin.rolka@rub.de

Facilitator educators'

Keywords: facilitators, facilitator educators, professional development, inclusive mathematics

published or not. The documents may come

Facilitator educators' professional development and professional roles

Within the complex field of teacher education, facilitators and their professional development (PD) are of great importance. Facilitators as such are an extremely heterogeneous group: They have different backgrounds, tasks and functions in different working contexts, teach different subjects and types of learners [START_REF] Dengerink | The Professional Teacher Educator: Six Roles[END_REF]. We focus here on persons responsible for providing PD courses for teachers (facilitators), and providing PD courses for facilitators (facilitator educators).

For qualifying facilitators, the Three-Tetrahedron Model (3T-Model, see Figure 1, [START_REF] Prediger | Which research can support PD facilitators? Strategies for content-related PD research in the Three-Tetrahedron Model[END_REF] is of major importance. For the professionalization process, all three levels of the 3T-Model are relevant [START_REF] Prediger | Conceptualizing content-related PD facilitator expertise[END_REF]: The tetrahedron on the classroom level -comprising students, teachers, classroom mathematical content, and classroom resources -will be transposed to the teacher PD level as well as to the facilitator PD level. On the facilitator PD level, facilitators are learners, the facilitator PD content comprises the whole tetrahedron on the teacher PD level which, again, comprises as teacher PD content the complete tetrahedron on the classroom level. The main level of acting concerns the facilitator PD level, but teacher PD level as well as classroom level are also touched.

Subject specific PD course 'Coping with heterogeneity in inclusive settings'

Within the German Centre for Mathematics Teacher Education (DZLM) in cooperation with a statewide agency for teacher education, a subject specific PD course 'Coping with heterogeneity in inclusive settings' has been developed, aiming at qualifying a group of 15 facilitators (working in primary and lower secondary mathematics and special education). Exchange and cooperation of the different professions was intended for the whole course. Planned as a scaling up process, the qualified facilitators should function as facilitator educators and offer the course to other facilitators or teachers for qualifying them as facilitators. The research-based course design was grounded on various evaluated course concepts, done by the authors before. The mathematics modules were embedded in a broader concept, starting with a module, focusing on the general role of facilitators and adult learning, and followed by a module, given by the statewide agency, representing their general objectives and strategies with respect to subject specific PD. The four mathematics modules included a basic module as well as three thematic modules for inclusive mathematics, each module lasting 2.5 days. In particular: 'Basic module for deepening didactics of mathematics in the context of inclusion', 'Learning difficulties/learning disabilities/learning potentials in mathematics', 'Diagnosis and support in mathematics instruction', and 'Learning mathematics in inclusive settings'. Beyond the meaningful content selection and consideration of design principles (cf. [START_REF] Prediger | Which research can support PD facilitators? Strategies for content-related PD research in the Three-Tetrahedron Model[END_REF], the course concept put the main stress on addressing the role as facilitator educator, for example by integrating reflections as well as simulations and at the same time connecting facilitator PD level, teacher PD level, and classroom level [START_REF] Prediger | Which research can support PD facilitators? Strategies for content-related PD research in the Three-Tetrahedron Model[END_REF]. Connecting the levels was also realized with different forms for evaluating the course quality and acceptance. One of these evaluations asked for addressing the different roles on the different levels: At the end of the mathematics modules, the facilitators had to rate the relevance of each module on a six-point likert scale, differentiated according to their role as teacher, facilitator, and facilitator educator (1 indicated 'not at all relevant' and 6 'very relevant').

Exemplary evaluation results

The overall evaluation showed that all mathematics modules mainly were assessed as very relevant or relevant for their various roles (teacher, facilitator, facilitator educator), and that the three levels (classroom, teacher PD, facilitator PD) were thus addressed (cf. [START_REF] Prediger | Which research can support PD facilitators? Strategies for content-related PD research in the Three-Tetrahedron Model[END_REF]. The primary course goal was the qualification as facilitator educator. Hence, it was expected that this level would be addressed in particular. However, all levels were addressed throughout the course: Exemplary for the module 'Learning mathematics in inclusive settings', the participants assigned greater relevance to their role as facilitator educator and as facilitator (12 out of 13) than to their role as teacher. This is plausible and also desirable insofar as their qualification as facilitator educator and their subsequent activity as facilitator educator was intended. Nevertheless, for 9 out of 13, this module was also relevant or very relevant with regard to their role as teacher. Among other things, facilitator educators' ratings could be attributed to their individual prerequisites and acquaintance of a specific content.

Figure 1 :

 1 Figure 1: Three-Tetrahedron Model for content-related PD research (Prediger et al., 2019, p. 410)


Cecilia Kilhamn
email: cecilia.kilhamn@gu.se

Tinkering in algebra -the case of John

Keywords: Algebra, figural patterns, computational thinking, programming, tinkering

Finding a general rule for a figural pattern is a common type of task in early algebra, intended to enhance the ability to express generalization. In light of the incorporation of programming in school mathematics, this paper reports on one teacher's experience of using tinkering as a didactic strategy for patterning tasks, in comparison with a traditional approach. The discussion centers around possible benefits of integrating programming and algebra. An affordance of working with a computer program was that the general expression became relevant for the students and changed from being the end point of a patterning task to function as the starting point for mathematical tinkering.

Introduction

The current world-wide incorporation of programming in school curricula [START_REF] Brown | Restart: The resurgence of computer science in UK schools[END_REF][START_REF] Mannila | Computational thinking in K-9 education[END_REF] raises questions of how these new ideas and technologies could enhance mathematics education. In Sweden, as in many other countries, programming has been included in the mathematics syllabus [START_REF] Bocconi | The Nordic approach to introducing computational thinking and programming in compulsory education[END_REF]. Uniquely for Sweden, programming has also specifically been connected to algebra, although it is not a well-defined concept in the curriculum documents [START_REF] Kilhamn | Algebraic thinking in the shadow of programming[END_REF]Swedish National Agency of Education, 2018). Programming could be broadly interpreted as a pedagogical tool for developing students' digital competence and computational thinking, or in a narrow sense as a set of computer coding activities. Many Swedish teachers struggle to understand and shape what programming in school mathematics is and what it might be in relation to mathematics education [START_REF] Kilhamn | Teachers' arguments for including programming in mathematics education[END_REF][START_REF] Misfeldt | Surveying teachers' conception of programming as a mathematical topic following the implementation of a new mathematics curriculum[END_REF]. In this paper programming is seen in the broad sense, defining computational thinking (CT) in line with [START_REF] Aho | Computation and computational thinking[END_REF] as "the thought processes involved in formulating problems so their solutions can be represented as computational steps and algorithms" (p. 832). The aim of this paper is to contribute to a discussion about how algebra learning could benefit from programming activities and computational thinking practices such as tinkering. By describing a teacher's experience of a lesson on figural patterns based on tinkering with code, the paper seeks to induce a discussion in the spirit of [START_REF] Gadanidis | Computational thinking, grade 1 students and the binomial theorem[END_REF], who write that "We need many more cases of what might be in mathematics and CT integration to better understand the role CT affordances might play in disrupting and improving mathematics education" (p. 94).

Figural patterns

Algebra is sometimes described as the study of structure, where algebraic thinking emphasizes relations and structure over processes. [START_REF] Hewitt | Never carry out any arithmetic: the importance of structure in developing algebraic thinking[END_REF] defines algebraic structure as a combination of recognizable parts and recognizable patterns connecting the parts. Structure is important to discern in order to describe generalizations across and between specific instances. Patterning tasks, i.e. working with figural patterns, can be used to enhance students' sense of structure [START_REF] Hewitt | Never carry out any arithmetic: the importance of structure in developing algebraic thinking[END_REF], and to promote students' understanding of functional relationships [START_REF] Friel | A framework for analyzing geometric patterns[END_REF]. In this paper, figural patterns are seen as a way into generalization.

A traditional teaching trajectory for patterning tasks starts with a figural pattern where the first three figures are visually exposed. The students are then asked to find the number corresponding to the next figure, then perhaps to figure number 7, then 10, then 20, then 100. Typically, students insert the numbers in a table to find a functional relationship between a figure number and the total number of elements in the figure, and finally write a general expression for figure n.

This teaching trajectory has been widely reported in research literature (e.g. [START_REF] Blanton | The development of children's algebraic thinking: The impact of a comprehensive early algebra intervention in third grade[END_REF][START_REF] Kieran | Teaching and learning algebraic thinking with 5-to 12-year-olds[END_REF]. Some researchers stress the importance of drawing [START_REF] Friel | A framework for analyzing geometric patterns[END_REF] or using words and gestures [START_REF] Radford | The progressive development of early embodied algebraic thinking[END_REF] before symbolizing. However, sometimes the teaching trajectory quickly moves away from the visual figures into a table of values, where a total number of elements for each figure is calculated and students are asked to look for patterns in the table. [START_REF] Hewitt | Never carry out any arithmetic: the importance of structure in developing algebraic thinking[END_REF] describes the danger of students losing touch with the context of the figural pattern by spending time calculating and looking at the table of values created from the figures. In order to see structure, he suggests students should stay focused on the figures and avoid doing the arithmetic, since the total number hides the structure that reveals the generalization. He argues that "learners should never do any arithmetic, just write down the arithmetic they would do"(ibid, p 563). El Mouhayar (2018) showed that students who focused on numbers were more likely to take a recursive approach, going from one figure to the next, whereas [START_REF] Strømskag | A pattern-based approach to elementary algebra[END_REF] found that students who stayed within the figural context more often found a general expression In contrast to the traditional learning trajectory for figural patterns, a more challenging problem could be to present a pattern and ask for the figure number of a large number of parts. One such problem about a matchstick figure, taken from TIMSS07, was used in a study of problem solving in small groups [START_REF] Kilhamn | The role of justification in small group discussions on patterning[END_REF]. When the study was presented at a conference in Cambridge, participants in the audience criticized the problem, advocating instead the traditional step-by-step instruction described above, claiming it to be the 'right way to teach patterns'. The reaction pinpointed the fact that this type of instruction is a well-established didactic strategy. However, as the case of John presented below will show, it may not always be the most prosperous strategy.

Computational thinking and tinkering

Computational thinking (CT) is a fairly new concept in educational research, first introduced by Papert in 1996. The term involves the kind of thinking skills needed to understand and capitalize on digital technology, and practices used by programmers. In recent years, researchers in computational science as well as mathematics education have attempted to define CT or create frameworks that describe it (e.g. [START_REF] Aho | Computation and computational thinking[END_REF][START_REF] Grover | Computational thinking in K-12: A review of the state of the field[END_REF][START_REF] Kotsopoulos | A Pedagogical Framework for Computational Thinking[END_REF]. One of several commonly accepted elements of CT is pattern generalization [START_REF] Grover | Computational thinking in K-12: A review of the state of the field[END_REF], which is also an important ingredient in algebraic thinking. In this paper we will focus on the practice of tinkering brought up more or less explicitly in all CT frameworks. Dictionaries often define tinkering as the act of improving something by making changes to it, and a tinkerer as a person who enjoys experimenting with and repairing machines. In a CT perspective, tinkering experiences promote engagement in changes and modifications of existing objects [START_REF] Kotsopoulos | A Pedagogical Framework for Computational Thinking[END_REF]. [START_REF] Brennan | New frameworks for studying and assessing the development of computational thinking[END_REF] describe several CT practices such as testing, debugging, reusing and remixing, that could be seen as aspects of tinkering. Tinkering involves exploration, modification and reflection, it is about trying, adjusting and trying again. Fundamentally, it builds on learning from failures, turning mistakes into triumphs in the spirit of Thomas Edison, who is said to have uttered "I have not failed, I have just found 10 000 ways that won't work". 1 In mathematics education, tinkering could mean to explore a mathematical idea or mathematical relationship by making small and purposeful changes and reflecting on these. It is not the same as a guess-and-check problem solving strategy, since the main objective when tinkering is not to find a correct answer but to explore and change something already present. In a description of a teaching intervention in a Grade 6-7 algebra class, Boaler and Sengupta-Irving (2016) used the term mathematical tinkering to describe how the students played around with a problem and its solution, going beyond the problem at hand to explore the effect of small changes, challenging themselves to make the problem harder.

Method

The case study presented here emerged from a larger set of data collected when studying the transposition of knowledge that followed the recent inclusion of programming into school mathematics in Sweden [START_REF] Bråting | Integrating programming in Swedish school mathematics: description of a research project[END_REF]. A research question for the larger study was: What opportunities, challenges and pitfalls related to the learning of algebra can be identified in the didactical choices teachers make when implementing programming in mathematics? Interviews were initially made with 20 teachers identified as early adopters [START_REF] Kilhamn | Teachers' arguments for including programming in mathematics education[END_REF]. They were all enthusiastic about the challenge, had some previous experience in teaching programming, and many were responsible for implementation of digital technology in their schools. The audio recorded interviews took approximately 30 minutes and were semi-structured around questions that had been supplied in advance. Following four background questions the interview guide included the following topics: What is the role of programming in mathematics? Where do you find inspiration and ideas? Can you give an example of a good programming activity that you have tried? What programming concepts are important to bring up in mathematics? Many of the teachers described programming activities with little connection to mathematics, and when there was a mathematics content, it was most commonly geometry, arithmetic or probability. One teacher was different in that he said he tried to incorporate some Python programming into every topic, in almost every lesson. When asked to describe a good lesson he chose an algebra lesson. He claimed that the incorporation of programming had changed his teaching and created better learning opportunities for his students. His story is reported here as the case of John.

The case of John

The interview with John was made in October 2019. John had by then been teaching mathematics, science and technology in grades 7-9 for over 25 years. In 2017 he was appointed head teacher with a specific responsibility to coach his colleagues in the use of digital tools and programming, mainly in technology. In mathematics he used an interactive whiteboard, but struggled to find ways to engage students in programming activities that were compatible with the mathematics curriculum. In 2018, when programming was officially included in the mathematics curriculum, he started to use the netbased platform Google Collaboratory, coding in Python. John described a lesson about figural patterns in Grade 7 (students' age 13), claiming that it was the first time ever that his students truly engaged in finding a general expression for a pattern. When he started working with these students two months earlier, they had no previous coding experience through school. Every student in his classroom has a personal laptop, so he can use the platform both in class and for homework, where he can see what the students do, give personal feedback and pick up student solutions for whole class discussions. John's ambition is to get students into the habit of using Python code every lesson so that they become accustomed to programming, learning syntax through experience more than direct teaching. John stood out from the other early adopters in that he included programming in almost every content topic, rather than making it a topic in itself. He said he did not teach programming per se, but he told his students to use programming, treating it as a mathematical tool. Typically, he would give them an example code to tinker with and modify. In addition to the interview, python codes as well as written reflections that John shared with his colleagues were collected.

The lesson

Below, the lesson is described in three parts and analysed in relation to aspects that were in some way different from the traditional teaching trajectory on figural patterns described above.

Part 1: Introducing the general expression

During the previous lesson, the class had worked with a figural pattern where a number of markers in a figural arrangement increased by three for each figure, starting with five. They had explored and discussed what features of the pattern were important in order to find a general expression: change and starting number. Together they now wrote the following basic code in Python, which everyone copied into their personal computer and tried out.

The code gives a sequence of instructions that computes and prints the figure number and its corresponding number of markers. John expressed that he did not expect all students to comprehend the code in detail, but for this part of the lesson it was enough to copy the code and run it to confirm that it worked for that pattern. John wrote the following reflection: "Many students found this difficult and did not seem to see the benefit of this program. The next step is to get the students to program."

Part 2: Transferring to other patterns

In contrast to a standard lesson where the general expression for the pattern comes at the end, the programming activity used the general expression as the starting point for exploration through tinkering. After introducing the above piece of code, the students worked with similar figural patterns presented as tasks in their textbook, all visual configurations of growing patterns with linear solutions. The students were encouraged to make use of the code they had produced but with access to sticks and markers if they needed. All students chose to work with the code. When they knew that one pattern was described by figure_number*3+2 they could start to tinker with the expression to explore the effect of small changes. A further challenge came when the question was posed the other way around, for example "what figure number would need 64 sticks?" The code did not produce the answer and needed to be modified. John explained: "Then I showed the class how to modify this code so that instead of asking for a certain figure it prints the first 50 figures with the number of sticks in. And then you can see that: 'Well there are 55 sticks in figure number 17'. And thiswhen they suddenly sawthey had it all there! It was like 'wow' they could do anything, all information was there in this table that was printed."

In this section of the lesson the power of the computer was put to use, so that patterns could be investigated using larger numbers and more data without having to do all the calculations. The students discussed large numbers and compared different outputs. The most confident students started to think about what needed to be changed in the code to get the inverse operation instead.

1.

2.

3.

Figure 1: Figural pattern used for homework: the first three figures.

All students managed to solve the task although the pattern on surface value looked quite different from previous patterns. Many explained that they had copied the original code and tinkered with it until they got the correct result when they ran it (figure_number*3+1), checking with the given Figures 123. Some modified the code, adding features. Some came up with quite different solutions, with more or less efficient codes. Others came up with creative expansions, such as making a loop that kept asking for another figure number or commands that could handle an invalid input.

In the interview, John expressed that his students better understood the importance of finding the formula for a pattern because it is used in the program. It was the first time he saw that students found the general expression useful. In traditional patterning tasks many students would do the arithmetic for the first three figures and perhaps a few more, but they would lose interest before they got to the general expression, not seeing the point of finding it. Now they were dazzled with what the computer could do once they inserted the expression and were curious to see what happened with the numbers produced when small changes were made. It was, John noted, easier for them to relate to the figure n, because it was there, in the program, with the descriptive name figure_number.

Discussion

The case of John brought out several possible affordances of introducing programming when dealing with figural patterns. One was that students, as suggested by [START_REF] Hewitt | Never carry out any arithmetic: the importance of structure in developing algebraic thinking[END_REF], could explore the pattern without doing a lot of arithmetic. In fact, the power of using a computer code to do the arithmetic enabled students to look for patterns in larger numbers and to find structure in a table with many more entries. Iteratively moving back and forth between the code, the printed output and the original pattern figures enhanced students' focus on structural aspects of the pattern while downplaying arithmetic procedures. However, according to John, the main benefit had a more psychological character addressing the fact that students found the general expression relevant and useful as a way to communicate with the computer. Instead of being the end point of the task, as in traditional patterning tasks, the expression became a starting point, something to tinker with that sparked curiosity. Furthermore, the expression became more visible and made more sense when the variable was represented by a descriptive word, not as the abstract letter n. This could potentially be a useful intermediate step before dealing with variables in symbolic algebra, well worth further investigation.

Tinkering proved to be a valuable practice for these students when looking for a general expression for a pattern. The algebraic question "What is the general expression for the nth figure of the pattern?" changed into a computational question "What do we need to change in the command given to the computer so that it describes or generates the pattern?", leading up to questions about the effects of different changes made in the code. While this is in line with how [START_REF] Aho | Computation and computational thinking[END_REF] defines CT, it also uncovers opportunities for students to rise to yet another level of generalisation that further develops their algebraic thinking, namely that very different patterns can be described by the same or a similar algebraic expression. Tinkering, or what [START_REF] Brennan | New frameworks for studying and assessing the development of computational thinking[END_REF] describes as reusing and remixing predefined code, empowered students to see similarities between patterns with quite different surface features.

Another benefit was the opportunity of expansion, providing more challenge for students who needed it. Many of the students in John's class started posing new questions, such as what to do if the input is invalid or how to write the code so that the figure number is the output when the total number is the input. Furthermore, it would be possible to tackle the pattern the other way round by asking students to change the expression in the code slightly and then try to create a pattern to match the new expression and the numbers printed when the program was run.

The various affordances of programming in John's algebra lesson suggest that we should be open to the use of programming is school mathematics. Further design research with lessons like this would therefore be valuable. However, a teaching approach that involves tinkering with code is only possible when the teachers' coding proficiency is high, otherwise the work is bound to get stuck on syntax issues. This case study describes a teacher with many years of experience and good programming skills, which is not always the case. Much work is still needed to help teachers develop necessary programming competence [START_REF] Kilhamn | Teachers' arguments for including programming in mathematics education[END_REF][START_REF] Misfeldt | Surveying teachers' conception of programming as a mathematical topic following the implementation of a new mathematics curriculum[END_REF], but in the meantime teachers could embrace a tinkering approach to teaching and start using mathematical tinkering, in the sense of [START_REF] Boaler | The many colors of algebra: The impact of equity focused teaching upon student learning and engagement[END_REF], for mathematics in general and patterning tasks in particular.

Why not let the lesson start with an intricate algebraic expression, perhaps graphically represented using dynamic software, and then explore what happens when small changes are made?

Part 3 :

 3 Homework assignment Finally, John the following task as homework: Make a program that calculates the number of white squares in the following pattern. The program should ask what figure you want to know the number of white squares for, and then show you the result. If you want, you can modify and improve the program.

https://www.goodreads.com/author/quotes/3091287.Thomas_A_Edison

Acknowledgment

This work was supported by the Swedish Research Council [Grant no. 2018-03865].