**3. Results**

This section is organized according to the self-regulation and confrontation processes described in Figure 1 (deconstruction, co-construction, and reconstruction), and considering the 12 self- regulation traits defined by ref. [38]. It is considered that pre-service teachers' comments and productions are the result of their way of acting, which is why such evidence is indispensable to identify teacher practice and extract the main elements that characterize it. Although all sessions were recorded, a teaching sequence of one of the parts of "Learning Mathematics" is presented and analyzed as an example: algebra and logical-mathematical reasoning in Early Childhood Education.

#### *3.1. Phase 1. Deconstruction of Prior Knowledge, Experiences, and Belief Systems*

In this initial phase, the university lecturer puts into practice some procedures and skills to establish a relational climate that encourages the active participation of students. In our case, the lecturer uses previously thought-out questions and formulates challenges to the students. On the one hand, the questions that are posed to pre-service teachers when starting the topic of logical- mathematical reasoning are: "What do you think is logical-mathematical reasoning?"; "what is it for?"; and "what benefits do you think it contributes to effective performance in today's unsustainable context?" On the other hand, a structured logical material (Dienes' Logical Blocks) is presented to the students and they are asked to find out what knowledge related to algebra, children from 3–6 years old could learn with this material. These questions and challenges are formulated to reveal pre-service teachers' prior knowledge, experiences, and beliefs. The questions are linked to the socio-cultural context so that students can begin to understand the role that mathematics education can play as an agen<sup>t</sup> of change in accordance with sustainable development.

The procedure for discussing student responses takes into account the guidelines of ref. [60] to promote reflective dialogue: students are grouped into small groups (3–4 students) and debate the answers for a limited time, not exceeding 15 min. Afterwards, each group writes the agreements and a spokesperson communicates the answers to the others. While the lecturer writes and organizes the answers on the board, he does not make evaluative comments, makes sure that what is written matches what the students say, accepts all interventions, and maintains close contact with the participants through non-verbal language (gestures, looks, movement in the room, etc.). Table 1 presents some evidence about the prior knowledge, experiences, and beliefs revealed by students.

In the evidence provided in Table 1, it is observed that, through the questions formulated, the university lecturer has helped to uncover some pre-service teachers' intuitive knowledge of logical- mathematical reasoning. In summary, this knowledge shows that students know some of the general functions of logical-mathematical reasoning (organize and structure thought, enable thinking, internalize strategies, etc.), but do not specify the main content associated with logical- mathematical reasoning or algebra in the early ages, such as sorting, classifying, and ordering objects by size, number, and other properties; recognizing, describing, and extending patterns, such as sequences of sounds and shapes or simple numerical patterns, and translating from one representation to another; or analyzing how both repeating and growing patterns are generated [61]. The data also shows that pre-service teachers realize that math can be used to understand better the socio-cultural context and environment where they live, but do not generally explicitly see its connections with sustainability, such as predicting possible effects or searching alternative ways of thinking and acting.



#### *3.2. Phases 2 and 3. Co-Construction and Reconstruction of Knowledge*

After deconstructing pre-service teachers' knowledge, experiences, and system beliefs, the lecturer takes the pre-service teachers to the co-construction phase. First, he makes an anchor between prior knowledge and sustainability science knowledge, providing students with theoretical documentation in the form of the specific documents provided, and supported by their own bibliographical searches. In this context, he uses a type of neo-Vygotskian instruction called "Concept-based instruction" (C-BI), which is carried out through a series of stages that can be organized and implemented for a specific educational context, as determined by the educator in accordance with the specific context [62–65]. Generally, however, the initial stage from which all others emerge is the orientation stage, which determines the overall quality of a particular approach to a given situation. This stage begins with the students' pre-understanding of a certain topic. Hershkowitz and Schwarz [60] referred to pre-understanding as the Orienting Basis of Action, or OBA, because it is assumed that students base their actions, including language action, on their OBA. This knowledge can come from everyday experiences or from previous instruction, especially of the traditional type. The new conceptual knowledge is first explained and then represented imagistically to learners as a Schema for the

Orienting Basis of Action, or SCOBA. According to ref. [65], the remaining stages of the Gal'perin [64] educational model are designed to promote student internalization of the knowledge represented in a SCOBA, and to use that knowledge in practical activities, including spoken and written communication.

Based on the lecturer's guidelines, the pre-service teachers share their conceptual expansion or SCOBA, following the same procedure as in the first phase of the teaching sequence. Table 2 presents some evidence of the co-construction phase in relation to algebra, logical-mathematical reasoning, and sustainability, obtained from their pre-service teachers' portfolios.


**Table 2.** Co-construction of knowledge, experiences and belief systems.

As it can be observed in Table 2, using CB-Instruction, the lecturer not only encourages preservice teachers to acquire disciplinary knowledge, but also to learn strategies and resources to carry out effective teaching in considering sustainable development. According to ref. [61], effective teaching means identifying what students already know and what they need to learn, and then stimulating and helping them to learn it well. The association of American mathematics teachers develops this idea with the following three requirements: (1) teacher effectiveness requires mathematical knowledge and awareness that students are learning and must have adequate access to pedagogical strategies; (2) effective teaching requires a supportive and stimulating learning environment; and (3) effective teaching requires constantly striving to improve.

From this perspective, the SCOBA designed by the lecturer in the co-construction phase of the teaching sequence has allowed students to learn methodological resources and appropriate ways of acting according to education for sustainability principles in order to promote the learning of algebra and logical-mathematical reasoning at early ages.

In the next stage, the reconstruction phase, the lecturer encourages students to contrast their previous knowledge with new perspectives, understanding contrast to be a process that starts from the experience of each member of the group [60]. The lecturer shows di fferent structured logical materials and asks students to work in groups (3–4 students) to analyze them and design some activities based on a learning guide. In each activity, students have to describe: (1) the level and contents; (2) the managemen<sup>t</sup> of the activity; (3) the solution of the activity; and (4) the mathematical language and other aspects associated with communication, such as the questions that children are asked to promote understanding, etc.

The activities designed are presented to the other students and, after the lecturer has mediated and consensus has been reached by everyone through a process of interaction, negotiation, and reflective dialogue, activities are then implemented in a school. Specifically, each group of students applies the activities to a group of about 10–15 students from 3 to 6 years old. With this task, the teacher enables pre-service teachers to apply the theoretical knowledge acquired and to obtain conclusions regarding their own practice in a real situation. Considering education for sustainability criteria, the lecturer establishes links between the university and the community while creating a context that encourages the development of professional competences in this area. Table 3 shows the evidence collected from this phase.


**Table 3.** Reconstruction of knowledge, experiences and belief systems.

The evidence presented in Table 3 reveals the starting point from which students began, the process they followed, and the transformation experienced throughout the process. This contrast has caused some cognitive conflicts, according to ref. [32] (p.108) who state that "critically reflective learning is, in itself, disturbing, but also stimulating and demanding, potentially". In addition, students identify what they need to rethink about how to teach and learn in the area of mathematics education for sustainable development. This idea links with Esteve and Alsina [30], who point out that these

cognitive conflicts are necessary to begin to self-regulate one's knowledge, and also to seek answers individually and collectively in order to advance, improve, and learn.

From this perspective, these cognitive conflicts, managed successfully by the lecturer and the students themselves, help to reconstruct prior knowledge, experiences, and belief systems and to build new professional knowledge in a collective and consensual way. This is when it becomes clear that the pre-service teachers in our study have understood that knowledge and skills in mathematics and sustainable development cannot be acquired only through the simple transmission of information by the lecturer, and that they must assimilate it from their own practice.

If the evidence of Table 3 is analyzed in more detail, we can also observe that the experience has enabled endogenous, collective, and cooperative work, key to education for sustainability. Students come to the conclusion that everyone can learn from everyone: the students from the lecturer, the lecturer from the students and, above all, the students from themselves.

Finally, and in order to complement the data presented, we considered the results of the lecturer's evaluation forms and compared the results with the means of the department and of the degree in Early Childhood Education. For the seven questions posed, the lecturer received a better mark than the means of the department and the degree. He was marked with the highest rate in five of the questions (5 out of 5), including the final one that asks the students to evaluate the overall lecturer's performance. The analysis of students' qualitative comments shows that what they valued the most is the practical nature of the subject and the lecturer's teaching performance.

## **4. Discussion and Conclusions**

In this article, a link was established between reflexive learning and education for sustainability, and this was integrated within a teaching-learning process of the professional didactic and disciplinary knowledge needed to teach mathematics.

In accordance with Sanmartí, Jorba, and Ibáñez [66], when training teachers, we think it is necessary to teach them how to learn from within the discipline itself, if we want the teaching process to be successful. This is not so much because it is di fficult for pre-service teachers to apply general learning strategies to the learning of specific didactic knowledge—in our case referring to mathematics education—but also because teachers must design their didactic practice so that students can implement these strategies to learn.

More specifically, the initial data obtained tends to show that the teaching practice described proved to be an e ffective tool to promote the professional development of future mathematics teachers. In particular, we were able to extract various elements of the teacher's practice, typical of reflective learning and education for sustainability. For example, the formulation of questions and challenges when starting a teaching practice, which are established as a fundamental tool to encourage reflective dialogue and activate students' prior knowledge in maths and sustainable development; or the fact that encouraging interaction with others, with oneself, and with theory promotes the construction of meanings and the learning of knowledge, in addition to the contrast between the new perspective and the starting point, according to Hershkowith and Schwarz [65]. The lecturer uses education for sustainability pedagogical approaches such as participatory and action learning, as well as promotes closer links with schools in the community to work on mathematical and sustainability matters. Strategies are used to help train critical professionals who are willing to act and are prepared to adapt to di fferent situations, based on the pillars of learning to know, learning to do, learning to live together, and learning to be.

In addition, the data from the evaluation forms confirmed that the lecturer's practice has had positive e ffects on various aspects of the students. In particular, the elements of reflective learning and sustainability seem to have influenced the three dimensions of need-supportive teaching (NST): structure, autonomy, support, and involvement [67,68], and also student self-e fficacy [69,70].

From this perspective, Figure 2 shows the elements of the teacher's professional practice that have contributed to the transformation of prior knowledge, experiences and belief systems into professional competence, using reflexive learning in the framework of sustainability.

**Figure 2.** Lecturer elements for the transformation of prior knowledge, experiences, and beliefs into professional competence in teacher education.

Figure 2 shows that a vital previous element in mathematics education for sustainability is the teaching practice presentation through the formulation of challenges, problem solving, etc. In this formative setting, the first element is "uncovering students' preconceptions", that is, bringing to the surface students' prior knowledge about mathematics education and sustainability, and keeping this in mind since it can sometimes be a real obstacle preventing the construction of the individual's professional profile, as already indicated [39].

In order for the university lecturer managemen<sup>t</sup> to be effective, the next element is to systematize this prior knowledge, experiences, and belief systems, contrasting it with pre-service teachers' sustainability ideals, which can result in the emergence of some conflicts and contradictions. In order to manage emotions, it is important to build an anchor between prior knowledge and the students' ideals, which is to say that the value that intuitive knowledge has in the construction of one's own teacher profile must be acknowledged. As indicated, it is through this anchoring that pre-service teachers progressively incorporate and understand new concepts through Concept-Based Instruction [62–65]. Finally, in the last phase of the training, the lecturer incorporates new action plans, in the form of new methods of action that allow the co-construction and reconstruction of prior knowledge, experiences, and beliefs into professional competence. This allows the students to put into practice mathematics education for sustainable development in schools and reflect on their performance.

An important consideration to note is that, in this final phase, many students started to speak as a collective. This may mean that, through education for sustainability approaches and reflective learning, some of the students were able to leave behind a unidirectional vision of the teaching–learning process, in which the teacher transmits knowledge and the student receives it passively [31,69,70]. This statement is, however, a risky interpretation, since it is difficult to determine with exactitude if there has been a change in ways of teaching mathematical knowledge with the analysis of a single teaching sequence. However, some transcripts sugges<sup>t</sup> that the inquiry-based and collective scaffolding that the lecturer promoted may have generated this transformation in some students, in line with ref. [67].

From the data obtained in this study on teacher elements that promote the transformation of prior knowledge, experiences, and belief systems into professional competence, together with the student elements indicated in Figure 1 [34], a first definition is proposed to advance towards a transformational

knowledge model in teacher education based on education for sustainability principles and reflective learning approaches (Figure 3).

**Figure 3.** Elements for the transformation of prior knowledge, experiences, and belief systems into professional competence in teacher education.

Figure 3 shows, first of all, that one of the main purposes of teacher education and sustainability is to progressively transform students' prior knowledge, experiences, and beliefs into competences that contribute to their professional development. A second issue that is essential to consider is that knowledge is transformed into various phases that must necessarily be developed symmetrically in the two agents involved: pre-service teachers and lecturers. This is a matter of grea<sup>t</sup> importance, because if pre-service teachers and lecturers are not synchronized during a teaching sequence, it is very difficult for teacher education to contribute to the transformation of students' knowledge and values.

Some of the main limitations of the study have been the following: first, the data was obtained from the analysis of a single didactic sequence, which is why it is difficult to determine with exactitude if future teachers transform their knowledge about ways of teaching mathematics with sustainability criteria in mind. In addition, the teaching practice of a single teacher was analyzed, so the results cannot be generalized; second, we consider that it is not possible to achieve the effective transformation of university teaching and learning practice from a single subject. Therefore, it seems necessary that the university should adopt a whole-institutional approach to sustainability; third, some students indicated that the continued use of the techniques mentioned in our study, for example those aimed at promoting reflective dialogues, may cause monotony; and, finally, the data of the teaching evaluation is limited, since only 25% of the students answered, so it is not possible to obtain firm conclusions about the effect of the lecturer's practice on the three dimensions of need-supportive teaching (NST) and student self-efficacy. Therefore, in the future it would be useful to design new studies with larger samples to refine the transformational knowledge model described, and to incorporate other instruments and techniques to continue advancing towards teacher education based on education for sustainability principles and reflective learning approaches, with the purpose of improving the professional development of pre-service teachers.

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**Author Contributions:** A.A.—contributions on reflective learning; I.M.—contributions on education for sustainability. **Funding:** This research received no external funding and the APC was funded by the Universitat de Girona. **ConflictsofInterest:**Theauthorsdeclarenoconflictof interest.
