**2. Method**

A case study of a descriptive type has been designed based on a qualitative approach [46]. Stake [47] described case studies as "the study of the particularity and complexity of a single case, coming to understand its activity within important circumstances" (p. xi). To obtain the data, the professional practice of a mathematics education university lecturer at the University of Girona (Spain) was observed. Specifically, the analysis was made in the subject "Learning Mathematics" of the Degree in Early Childhood Education, which includes 30 class sessions (10 theoretical and 20 practical). It is the first subject of mathematics education that students attend and it is located in the first semester of the 2nd year, after students have received basic training in psychopedagogical bases. In total, there are 48 students enrolled.

All class sessions were recorded on video, since this technique provides data from a more objective perspective; allows the analysis of a large quantity of data from a variety of perspectives and with di fferent analytical frameworks; and also allows the integration of microanalysis and macroanalysis [48], among other advantages.

An audio recorder was attached to the teacher's neck and a video camera was used to record the class sessions. The camera was located in a corner of the room so that the blackboard and as many students as possible could be included in the frame. During the recording, the person filming was attentive at all times to what was occurring, to be able to move the camera if necessary or zoom in to better record what was happening. The camera constantly followed the teacher (who is the focus of the study), covering as many students as possible and zooming in to record interactions related to the reflective dialogues, or to focus on the written productions resulting from the shared experience.

For the analysis, MediaNotes was used, which is a software-based video annotation system [49] that allows users to upload, tag, and annotate segments of the video. According to ref. [50], video annotation software systems provide users with the ability to synchronously or asynchronously watch and code (or tag) the video. In some of the video annotation software programs, the user can pre-define the criteria for tagging (analysis), whereas in other examples, the user is provided with pre-established tags. In most of the video annotation systems, the user can highlight certain parts of the video and add commentaries. MediaNotes was chosen over other video analysis tools because the coding and search options make it easier to data mine [51,52]. In addition, it has also been used as a video-based method of observation to promote teacher reflection [50,53].

To analyze the data, the authors used a deductive application of categories. Deductive category application works with prior formulated, theoretical derived aspects of analysis, bringing them in connection with the text. The qualitative step of analysis consists of a methodological controlled assignment of the category to a passage of text. Even if several procedures of text analysis are processing that step, it is still poorly described [54]. More specifically, the authors used the twelve self-regulation traits described by Alsina et al. [38]. As indicated in the section above, these traits are the pre-service teacher elements that promote the transformation of prior knowledge, experiences, and belief systems about teaching into scientific and professional competence (see Figure 1). To conduct the analysis, MediaNotes' tagging features were used to mark specific segments according to our own framework and the commenting features used to provide detailed insight that provided evidence of the university lecturer's practice.

To complete the analysis and investigate the lecturer's practice and its effect on student performance in more detail, the authors also analyzed the lecturer's evaluation forms completed by students enrolled on the "Learning Mathematics" subject. In a recent study, Ayllon, Alsina, and Colomer [55] assessed the effect of these on the three dimensions of need-supportive teaching (NST): autonomy support, structure and involvement [56], and students' self-efficacy [57], in order to gain new knowledge about students' achievement in higher education. In this context, and within the Self-determination Theorry (SDT) framework [56], NST is a powerful instrument to motivate students and help them achieve better results. From this point of view, these authors analyzed 86,038 complete evaluation forms (27,216 for 2014; 29,946 for 2015; and 28,876 for 2016) and confirmed their ability to inform higher education teaching and learning through the university lecturer's teaching practice. Thus, this paper considers that it is important to analyze these forms that can provide complementary data about how the lecturer's teaching practice can contribute to the professional development of pre-service teachers in the area of mathematics education for sustainable development, taking into account the vision that students have about their lecturer's teaching practice. The analysis of the forms also allowed the triangulation of results, thus enhancing the research validity.

The process for students to complete the evaluation forms is as follows: three weeks before the end of classes, students answer a brief online questionnaire via the Moodle platform: this means that students do not know their final mark when completing the questionnaire. The questionnaire is not compulsory, but students receive messages encouraging them to answer. They can do so at any time of day during the seven days that the questionnaire is online, and it is completely anonymous. In total, the lecturer obtained 12 complete evaluation forms, which represents 25% of the total number of students who attend class.

The evaluation form consists of two main parts. In Part A, students are asked the following six questions (in this order):


The authors are aware that through their researcher positionality and as qualitative researchers, they can influence the research they do and shape the knowledge produced [58]. For this reason, reflexivity has been an integral aspect of this study and has been put into practice through informal descriptions, memo-ing, and an iterative process of questioning the methods used and results emerged. Following Bourdieu [59], when conducting the research and analyzing the data, the authors took into account their social origins (class, gender, background, etc.), the position they occupy in their academic fields (maths education and sustainability), and their intellectual bias associated with how they see and interpret the world.

Regarding the ethical considerations, before initiating the study, the informed consent of the lecturer and the students was obtained. In addition, the lecturer agreed to provide the results of the evaluation forms, but all data that could identify the lecturer was intentionally deleted.
