Learning the Mathematical Discussion (MD) Pedagogical Model in and from Practice: The Professional Development of Mathematics Teachers in a Technological Environment

Abstract

On the basis of the hypothesis that has emerged from recent research studies, this article discusses the new pedagogical training model for primary school pre-service teachers (PSTs), namely the Meta Discussion on a Pedagogical model (MDPm). The study was conducted with 400 PSTs enrolled in a primary education mathematics course. The MDPm is a collective discussion involving PSTs and their facilitator about the prior teaching experience based on a Mathematical Discussion (MD) model. In this article we try to give a more detailed definition of MDPm by analyzing different examples drawn from a collection of data gathered in the frame of different teacher training experiments. The results indicate that engaging PSTs in both a direct MD and a subsequent reflective meta-discussion leads to enhanced awareness of the teacher’s role and a more robust conceptualization of the MD model. These experiments aimed to help future teachers recognize the characteristics of their instructional actions within a pedagogical model, engaging them actively in their training, in and from practice. In this setting, PSTs assumed a dual role as both students and future teachers. The research has been conducted in a technological environment, which assumed a fundamental role in supporting the processes involved in the teaching experiment. The examples are analyzed through a specific lens aimed at identifying key elements supporting and refining the MDPm, with the aim of more precisely characterizing it.

1. Introduction

A fundamental research topic in the field of Mathematics Education is teachers’ training. During the last few years, several research projects have been conducted in this area and several theories have emerged by highlighting the complex articulation of different contents and their relationships. It remains difficult to know how to intervene in practice.

In this contribution, we propose a new training approach designed to introduce pre-service teachers (PSTs) to the pedagogical model of Mathematical Discussion (MD), both from a theoretical and practical perspective. This study, part of deeper research involving different experimentations, focuses on pedagogical model and its implementation in the classroom, aiming to provide new perspectives on the training of PSTs. In our study, we propose that PSTs directly experience the pedagogical model by assuming the role of students in different kinds of tasks (geometric and arithmetic). They are then engaged in a reflective situation guided by the facilitator, consisting of a collective discussion centered on the role of the teacher previously played by the facilitator.

Our hypothesis is that, through the combination of a direct experience in which PSTs, as students, participate in an MD and a reflective experience, where PSTs engage in a second reflective discussion, orchestrated by their facilitator, to analyze the MD they previously participated in, the PSTs become aware and construct the characteristics of the MD pedagogical model. We refer to this reflective discussion as the “Meta-Discussion on a Pedagogical Model” (MDPm) (Montone et al., 2023).

Our research involved 400 PSTs enrolled in the mathematics education course for primary education. Data analysis showed that the training activity—where future teachers experienced Mathematical Discussion (MD) by taking on the role of students—facilitated their conceptualization of both the theory itself and the teacher’s actions within this theoretical framework. Additionally, it fostered awareness of the teaching profession.

According to this hypothesis, a research project was developed, and it is still in progress: teaching experiments have been carried out at different university courses aimed at clarifying the characterization of MDPm.

This paper reports on some results of this project, focusing on better articulating the original idea of MDPm.

In the following sections, we outline the conceptual framework, present the experimental design and research methodology, and analyze selected data through the specific lens aimed at identifying key elements supporting our idea of MDPm.

2. Conceptual Framework

In order to situate our intervention within the broader literature on teacher professional development, we now elaborate on several theoretical approaches, from cognitive and situated learning perspectives (Putnam & Borko, 2000) to sociocultural and linguistic frameworks (Vygotsky, 1978), that support reflective practices in education. For example, classroom-based analyses have often employed interpretive frameworks (Schön, 1983) and task analysis methods that prompt teachers to reflect metacognitively on their instructional goals through video-based lesson reviews (van Es & Sherin, 2008).

In the field of teacher professional development, numerous studies and theories have emerged in recent years. According to Shulman (1986), teacher education in a specific discipline requires both subject-specific knowledge and pedagogical content knowledge. Consequently, pedagogical content knowledge should not be understood as a disconnected set of disciplinary knowledge but rather as an integrated body of knowledge tailored to specific subject content. This highlights the need for training that fosters an integrated vision of disciplinary content, and the challenges associated with its teaching.

In the context of mathematics education, several models of specialized mathematical knowledge for teaching have been proposed, including the Mathematical Knowledge for Teaching (MKT) model by Ball et al. (2008) and the more recent Mathematics Teacher’s Specialized Knowledge (MKTS) model by Carrillo-Yañez et al. (2018).

Ball and Even (2009) suggest focusing on teacher education that, beyond emphasizing the learning of theoretical and pedagogical concepts, is grounded in and from practice, reinforcing the effectiveness of this approach. This perspective underscores the importance of providing systematic and long-term opportunities to learn from practice and develop teaching effectiveness by addressing the complex tasks inherent in the profession.

Building on the proposal by Ball and Even, our research is based on the hypothesis that a teacher training model can be structured around two key components: a direct teaching experience, in which PSTs take on the role of students within a pedagogical model; and a reflective experience, in which PSTs engage in a discussion—facilitated by their facilitator—to analyze the teacher’s actions during the activity of the MD.

Building on these diverse perspectives, we argue that the meta-discussion phase (MDPm) constitutes an effective learning tool. By engaging PSTs in a structured reflective dialogue about an experienced MD, our approach encourages them to recognize the characteristics of mediator and moderator role, by the analysis of the teacher’s actions. These roles are central to effective classroom management and instructional design. This dual perspective supports not only the internalization of the pedagogical model (PM) but also the development of an awareness of how theoretical principles come to life in practice.

This approach combines the lived experience of a pedagogical model (PM) with a reflective discussion on it, leading to the Meta-Discussion on a Pedagogical Model (MDPm).

3. The Mathematical Discussion: An Overview

The Mathematical Discussion (MD) model, originally developed by Bartolini Bussi (1998), has been defined as a “polyphony of voices articulated around a mathematical object (concept, problem, procedure, etc.), which serves as the driving force of the teaching-learning activity.” In this model, the teacher’s role is to “orchestrate the polyphony” of students’ voices while coordinating them with the voice of mathematical culture, represented by the teacher (Bartolini Bussi & Mariotti, 2008, p. 763).

Mathematical discussions differ based on their aims. Two main types can be distinguished: balance discussion (MDb), a collective process aimed at sharing, analyzing, and evaluating individual solutions to a given problem; and conceptualization discussion (MDc), a collective process for constructing mathematical concepts by establishing meaningful connections between past experiences and specific mathematical terms (Bartolini Bussi et al., 1995, pp. 11–12). At times, an MDb naturally transitions into an MDc, as observed in our experiment. For clarity, we will use the general term MD to refer to the combination of these two types of discussions.

The teacher “orchestrates the polyphony” assuming the role of mediator and moderator. To fulfil these roles, the teacher implements several key actions, including: “Returning to the task” to reconstruct the context and facilitate the reemergence of meanings and processes; “Focusing” to highlight aspects consistent with the didactic objective; “Requesting synthesis” to support students in generalizing concepts; and “Offering synthesis” to introduce targeted terminology, validating the acceptability and mathematical status of a specific meaning.

4. The Meta Discussion on a Pedagogical Model: A Characterization

In this article, we deepen the characterization of a new theoretical pedagogical construct. We believe that the combination of a direct experience in which PSTs participate in an MD, along with a reflective experience where PSTs engage in a second discussion to analyze the previous one, can be considered an effective training model.

In our model, PSTs take part in the first discussion (MD), led by the facilitator, to conceptualize a specific mathematical topic. Subsequently, they participate in a second discussion aimed at reflecting on the prior experience and identifying the key theoretical aspects that characterize the previous MD model. Both discussions are orchestrated by the facilitator, and the aim of the second discussion is to focus on the teacher’s actions within the MD.

The second discussion, that we named as the Meta-Discussion on a Pedagogical Model (MDPm), consists of a collective discussion involving PSTs and their facilitator about the prior teaching experience of solving a mathematical problem within the specific pedagogical model of Mathematical Discussion (MD). This reflective experience allows participants to identify the key aspects that characterize the studied pedagogical model.

To further distinguish the MDPm from the initial MD, we now underscore that MDPm makes the PSTs able to analyze the teachers’ actions, not solely the mathematical content. In doing so, the PST transitions from being only a participant to assuming an emerging observer role that informs their future practice as educators.

In this article, we aim to define the concept of MDPm in relation to MD, while also demonstrating that this definition is suitable for characterizing other pedagogical models. For instance, it can be applied to the analysis of the semiotic potential of an artifact within the framework of the Theory of Semiotic Mediation (Bartolini Bussi & Mariotti, 2008), as ongoing research—soon to be published—has been demonstrating.

In this new perspective, the teacher’s actions in the MD framework are redefined with new objectives within the MDPm construct:

• “Back to the task” becomes “Back to the experience”, aiming to reconstruct the context and highlight the interactions between students and the teacher in relation to the pedagogical model (PM).
• “Focusing” becomes “Meta-focusing”, making explicit the key elements that characterize the MD within the lived experience.
• “Requesting synthesis” now supports PSTs in decontextualizing and generalizing the PM, fostering awareness of its practice through a process of distancing.
• “Offering synthesis” now aims to introduce the defining characteristics of the PM, validating its structure and the teacher’s role within a given PM.

The action “back to the task” is reconceptualized as “back to the experience” to emphasize the dynamic, lived interaction during the MD, as opposed to the more static nature of task-based (MDc) discussions. In this model, the pedagogical model (PM) refers explicitly to the integrated processes and teacher actions that characterize the Mathematical Discussion. Specifically, in the shift from “back to the task” to “back to the experience”, PSTs are prompted to reflect not merely on the problem-solving steps but on the interaction dynamics that facilitate learning. Similarly, when the role of the teacher evolves from delivering content to acting as a discussion mediator, PSTs are encouraged to observe and later replicate these behaviors in their own practice.

Within this complex conceptual framework, our specific hypothesis can be summarized as follows:

Through the combined experience of MD and MDPm, PSTs can be introduced to the pedagogical model of MD and develop both the theoretical knowledge and practical experience of this model, with a focus on the teacher’s actions.

In this combined experience, PSTs transition from active participants in an MD as students to observers recognizing the defining characteristics of the PM, particularly the teacher’s role and actions. Our study aims to answer the following research questions:

• Can the direct experience of implementing a mathematical discussion help PSTs become aware of how this teaching methodology functions?
• How can the combined experience of engaging in both an MD and an MDPm help introduce PSTs to the PM and enable them to implement it in practice?
• How can technology support the professional development of PSTs?

In line with this hypothesis and these research questions, various teaching experiments (Montone et al., 2023) were designed and conducted to further explore and validate these ideas.

5. Technological Environment’s Support

The development of activities in our experimental research was supported by the technological environment. Indeed, the technological environment enabled the activation of collaborative processes among PSTs, encouraging them to participate actively. Each PST identified some solutions and solved the problem individually. Subsequently, through group division in virtual rooms, the PSTs shared and discussed their solutions, identifying a common one. This process fostered listening, reflection, and the participation of all students, both in small and larger groups.

Therefore, technology played a crucial role in reducing students’ uneasiness in speaking up, making materials available, and allowing for video and audio recordings of all discussions. The role of technology proved useful during the design phases of all activities; indeed, it facilitated the easy sharing and visualization of materials, the processing and analysis of collected data, and created an environment where PSTs could interact, work individually or in groups on a task, or explore mathematical/scientific content (Albano et al., 2020; Perry et al., 2021). To ensure effective participation of PSTs in solving the mathematical problem, which is not always solved quickly, our study found that the structured digital environment effectively facilitated the identification of solutions in adequate time and consequently promoted careful reflection and equal participation among PSTs. In line with the above-mentioned research, the technological tools provided flexibility and reduced the anxiety often associated with in-person discussions, thereby empowering pre-service teachers to assume greater agency in the learning process.

6. Research Methodology and Experimental Design

This study focuses on experimental research within a broader project. The aim is to foster the professional development of pre-service teachers (PSTs) by directly engaging them in practice as students and subsequently combining their reflection on this practical experience with theoretical knowledge about a specific pedagogical model. By doing so, PSTs build an understanding of the teacher’s role and actions in the classroom.

To achieve this goal, we designed and implemented a training method involving PSTs in two types of activities, developed at two different levels. At the practical level, PSTs take on the role of students, while the facilitator applies the pedagogical model under examination. At the reflective level, within the framework of the Meta Discussion on a Pedagogical Model (MDPm), PSTs reflect on their experience under the teacher’s guidance, analyzing the teacher’s role and actions.

From the vast amount of data collected during the experiments, we analyzed transcripts of the MDPm, referred to different MDs following the solution of different types of mathematical problems (geometric and arithmetic problems). The excerpts presented in the following sections were selected because they provide evidence of the training method’s effectiveness, particularly in constructing the characteristics of the MD. Following the criteria of credibility, reliability, transferability, and confirmability (Guba, 1981), we conducted a qualitative analysis of the transcripts to ensure the reliability of our findings.

The study involved 180 PSTs in the first experimental research and 220 PSTs in the second one, fourth-year students in a primary education degree program specializing in mathematics education. The training method, including both the Mathematical Discussion (MD) and the MDPm, was conducted synchronously via the Microsoft Teams platform. The discussions involved the entire class group, and were recorded and transcribed for analysis.

6.1. Structure of the Experimental Training Method

The training method follows a sequence of phases based on our hypothesis regarding the educational relationship between direct experience in an MD and participation in an MDPm reflecting on that experience.

The first phase introduces the theoretical model of MD. The facilitator presents the pedagogical model through a lecture, providing the theoretical elements that characterize MD, particularly the teacher’s actions necessary to orchestrate the discussion.

In the second phase, an open problem is introduced to activate an MD in an online workshop. The first experiment was about a geometrical problem (Montone et al., 2023). The second experiment, instead, was about an arithmetic problem. Specific mathematical topics were chosen for their inherent openness and their curricular relevance. This selection supports a discussion that is both accessible to all PSTs and rich enough to foster multiple solution perspectives, thereby facilitating deeper reflection on teaching methods. In addition, the proposed problems must be sufficiently elementary for all students to understand and be included in school curricula. The use of an open problem (Pehkonen, 1997) is crucial in fostering a rich and effective mathematical discussion, as it cannot be solved mechanically and requires different solution strategies. It must be complex enough to engage students both cognitively and emotionally.

The selected arithmetic problem focused on arithmetic relationships in a 100-table, consisting of 10 rows and 10 columns with numbers from 1 to 100. The task given to students, to be solved in online groups via the Teams platform, was as follows:

“Build a table of 100 (each row contains 10 numbers, the first goes from 1 to 10). Consider any square of side 2 squares extrapolated from the table of 100. What can be observed when adding the numbers on the diagonals? And when multiplying these diagonal numbers together? Argue your answers, justifying why.”

In both experiments, following the proposed solutions, a Mathematical Discussion (MD) was developed. In the arithmetic experiment, the goal was to uncover various resolution strategies, reach a common solution, and clarify the underlying relationships involved in constructing the 100-table. The facilitator leading the discussion adhered to the prescribed actions of the pedagogical model, allowing PSTs to engage in the MD as students while also observing the teacher’s role and actions. The technological tool was fundamental in helping students reflect on the progression of the activity, develop their responses at their own speed, listen to their peers, and ensure equal participation from all members of the discussion.

In the third phase, PSTs participated in an online workshop reflecting on the previous MD in an MDPm. The goal was to identify the teacher’s actions and the characteristics of the pedagogical model, in this case, MD. Here, the focus shifted from mathematical content to analyzing the teacher’s actions and pedagogical framework. In this phase, PSTs changed their perspective on that of future teachers. Indeed, the arithmetic is not the principal topic of the discussion, but PSTs are required to recognize the teacher’s actions during the previously MD. Through observing and analyzing the facilitator’s work, PSTs internalized their future professional role, gradually distancing themselves from the student’s perspective.

The technology environment permitted the PSTs to share observations on the teacher’s actions during the MD. In this environment PSTs developed a greater awareness of the pedagogical model, generalizing its characteristics and distancing themselves from their previous experience.

Our training method distinguishes itself from traditional professional development models—where teachers typically reflect on recorded lessons or engage in peer discussions—by integrating an immediate experiential component (the MD) with a guided meta-reflection (the MDPm) in a single, dynamic session. This format, supported by technology, is designed to accelerate the recognition and internalization of effective teaching practices.

6.2. Data Analysis

The MDPm sessions were audio/video-recorded and transcribed. A qualitative coding procedure was employed, drawing from grounded theory principles. Two independent researchers coded the transcripts using pre-defined categories—such as “back to the experience”, “meta-focusing”, “requesting synthesis”, and “offering synthesis”—which correspond to the teacher’s actions in the MD. Episodes were selected based on their clarity in illustrating shifts between the PSTs’ experiential and reflective stages. The final sample presented in this paper comprises two illustrative episodes: one centered on a geometric problem and the other on an arithmetic problem.

7. MDPm Characterization in Two Episodes

In this section, we present the analysis of two episodes drawn from the MDPm, orchestrated by the facilitator after the two experimentations described above about geometric and arithmetic problems. The common goal of each of the two MDPm was that of fostering the recognition of the teacher’s actions emerging during a MD. The analysis of the MDPm’s transcriptions confirms our research hypothesis concerning the awareness of the MD’s characteristics by PSTs, the role of MDPm in helping to introduce PSTs to the PM after having implemented it in practice, and finally the useful role of technological environment.

7.1. Episode 1

In this first episode, PSTs were involved in the MDPm following the MD concerning the geometrical problem. They explicitly recognize the characteristics of the MD pedagogical model thanks to the MDPm.

The following episode occurred after the MD that was held with the PSTs after sharing solutions of the geometrical problem in the second phase.

1.

(0:00) Facilitator: What did we do in the last lesson [referring to the lesson where the MD on the open mathematical problem related to geometry was conducted]? Why did I conduct this lesson with you? What was the teacher’s objective?

2.

(0:35) Maria: We analyzed the solutions to the geometry problem, and at the same time, we reviewed the stages of a mathematical discussion, […] thus the role of the teacher, who must act as a mediator but also as a moderator, the sharing of our solutions, […], it makes me think of an analysis of mathematical discussion as a pedagogical model…

3.

(1:15) Tiziana: In my opinion, we had a mathematical discussion, and I think so because we were divided into groups to solve a problem, […] it was not just an exercise but a real problem with different solutions, we discussed within the groups among ourselves and then with the teacher, […] we sought a shared solution under the guidance of the teacher in the role of discussion mediator.

The facilitator initiates the MDPm with the intent of bringing out the characteristics of MD by performing the action of “back to experience.” Indeed, they propose discussing the last lesson in which the PSTs participated, aiming to retrace the discussion experience to recognize MD as a pedagogical model. The facilitator’s question was formulated without referring to the problem to be solved (“Why did I conduct this lesson with you?”) to highlight MD characteristics and, in particular, to focus attention on the teacher’s actions. The answers from Maria and Tiziana demonstrate the effectiveness of the facilitator’s intervention. Indeed, Maria and Tiziana recognize and share with others the pedagogical model of MD in practice. Following the question that induces and promotes shifting the discussion’s attention from mathematical content to the pedagogical model, Tiziana’s response shows recognition of the MDPm objective of engaging PSTs in a discussion about MD. Through Tiziana and Maria’s answers to the facilitator’s questions, we highlight their recognition of MD as a pedagogical model.

These answers reveal that the facilitator’s question prompted a shift in the discussion focus from mathematical content to the pedagogical model. Additionally, the PSTs recognized the teacher’s specific role as a moderator and mediator in an MD.

The MDPm continues with Matteo’s intervention:

4.

(5:01) Matteo: your interventions and guidance made me think that… in a way, mathematics can be discussed, thanks to the different ideas that emerged, and the various hypotheses proposed to solve the problem…

5.

(6:00) Oscar: I agree with Matteo. In the last lesson, we conducted an MD, and I also had the impression that… a problem can have many different solutions, that mathematics can be discussed, precisely because through reasoning guided by you facilitators… various ideas emerged…

Matteo recognizes the characteristic of an MD that allows sharing “various ideas” and solutions. We believe that it was easier for Matteo, Maria, and Tiziana to recognize the characteristics of the pedagogical model because they had previously experienced MD in the mathematical problem session. Oscar acknowledges that the activity in the previous lesson was an MD; he also asserts that a discussion can help evolve a mathematical meaning, which is a distinctive feature of MD.

A clearer articulation of MD characteristics is shown in the following excerpt.

6.

(6:16) Facilitator: Matteo and Oscar said that we discussed about mathematics, but can you tell me which elements allowed you to recognize this activity as a discussion?

7.

(6:47) Alessia: I noticed that… you never interrupted our discussion or provided the solution to reach a conclusion, and even when we reached uncertainty… you moderated our discussion… in fact, you never said, “this is the correct solution,” or “this solution is wrong” … you repeated some of our words exactly, especially those we could reflect on again. Maybe also because we were connected online, we felt free to express our solution without the fear of being judged…

Here, the facilitator performs the action of “meta-focalization,” which can be considered a key action for the MDPm aimed at making explicit the characterizing elements that enabled the recognition of an MD in the experienced activity. Alessia refers to the experience linked to the concept of a moderator as a characteristic of the teacher’s role in an MD. From Alessia’s words, it emerges that when the student presents their solution, the teacher intervenes to allow many to participate during the MD and never states whether the solution is correct or not. When Alessia says, “you repeated some of our words exactly,” she highlights a personal indicator recognizing the teacher’s action of repeating students’ words, which we can classify as the “mirroring” action, a characteristic action of the teacher in an MD. Moreover, Alessia brings out the importance of the online context, which allowed all participants to express their proposed solution, respecting each person’s time and eliminating the embarrassment of intervening.

8.

(14:30) Facilitator: So… to summarize… Oscar said we discussed about Mathematics, Tiziana said the teacher’s role is seen as that of a mediator, Alessia added that it is also that of a moderator… but can you tell me which elements allowed you to recognize this activity as a discussion?

9.

(14:59) Sabrina: Well, while Oscar was speaking and outlining his solution to the geometric problem, [referring to the MD] you explicitly asked why… you tried to rephrase that concept multiple times, attempting to ensure that the entire class understood, not just Oscar. You wanted to make sure to guide everyone along that path.

Here, the instructor performs the action of “offering a synthesis” explaining what emerged from previous interventions. In particular, referring to Tiziana’s words, they recall the teacher’s role as a mediator, another characteristic of the MD model. In the first part of her intervention, Sabrina explicitly refers to her experience during the MD, highlighting specific teacher actions as a mediator in continuously asking why. In the second part, Sabrina recognizes another mediator action related to the teacher’s concern for involving all students. Reflecting on the summarizing intervention and Sabrina’s response, we can observe an evolution from direct reference to the experienced situation—what was done and said—to reference to the pedagogical model.

7.2. Episode 2

In this second episode, PSTs were involved in the MDPm following the MD concerning the arithmetic problem. They let the significant role of technology emerge. Furthermore, it is possible to emphasize the importance of having previously experienced MD directly to make PSTs aware of their future professional perspectives.

10.

(40:23) Facilitator: During the MD, what was the role of technology, in your opinion?

11.

(40:45) Matteo: Through online discussions, I had time to reflect, listen to others, and compare my colleagues’ proposals with my own. In this way, my ideas evolved for the better. The online discussion gave me the opportunity to understand how important it is to listen to others’ ideas and to take the time to rethink my own. Listening is an activity that requires respect for others. Listening does not mean staying silent, but understanding, leaving space to offer what the other person is communicating while also keeping space for oneself to connect with them. Online work facilitates this process because everyone is in their own space, and no one feels rushed. Perhaps, in person, we would not have had the same amount of time to reflect and listen to as we did while working remotely.

12.

(42:17) Oscar: During today’s workshop, it seemed that each of us had the opportunity to think about our own solution without being influenced by others’ ideas. Later, by listening to others’ solutions, I was also able to come up with a new solution that no one had thought of, and I realized that we all have completely different ways of thinking. Being online, without the teacher physically present and waiting for an immediate response, made us feel free to express our solutions without fear of being judged. I was able to observe the teacher’s actions as they guided us toward a shared solution. I recognized the act of focusing when the teacher repeated my statement and asked for clarification. For the first time, I felt like I was truly “experiencing” a theory and seeing its effects in a discussion.

13.

(45:33) Sabrina: I would like to add to what already said Oscar and Matteo that today, during the MDPm, I reflected a lot. Once the MD was over, I felt a sense of calm that had been present throughout the discussion. I didn’t expect everything to be so well-organized remotely, and despite the fact that most of us had never participated in a collective discussion before, it seemed like we were structured in a collaborative way, recognizing the importance of exchanging ideas.

14.

(48:10) Tiziana: During the MD where we were all together on Teams, I noticed that we used a different kind of language compared to when we are all physically in the classroom. In the breakout rooms, the discussion was among peers, so the language was less formal. When we are connected remotely, we feel freer to express our thoughts. It seems like there is more respect for others, and I had the opportunity to write down others’ reflections and think about them. The language I used became increasingly specialized, and above all, I wasn’t afraid of making mistakes.

From these reflections the value of using technology for this activity seems to emerge. Matteo highlights the potential offered by the technological environment to listen to others and understand their opinions, “leaving space” for reflection in order to share ideas. As Matteo states, this was possible because everyone worked remotely in their “own space.”

Oscar emphasizes the opportunity to take time to develop a solution and express it freely without fear of being judged and without the possibility of being influenced by others. Additionally, Sabrina reflects on the value of remote work, stating that she perceived a calm and well-organized working environment where exchanging opinions was essential for making collective decisions. By stepping back from her personal experience, Sabrina also observes an unexpectedly positive working atmosphere from a teacher’s perspective.

Finally, Tiziana underscores the importance of peer discussion conducted online, without the presence of a teacher, where the language used was informal, typical of a discussion environment among students. Tiziana also notes that this language becomes more formal in the presence of a teacher, marking a shift from the student perspective to that of a future teacher.

8. Preliminary Results

From the analysis, it seems that our main hypothesis, implemented in the training method tested, has been confirmed. The new theoretical construct MDPm, applied to the MD pedagogical model, enabled PSTs to reflect on their previous experiences, identifying the key theoretical aspects that characterize the MD pedagogical model.

It should be noted that the results reported herein are based on a preliminary analysis. Further, longitudinal studies are required to assess the sustained impact of the MD and MDPm intervention on teacher professional development.

In this research study, the implementation of direct experience provided the opportunity to engage in an experience that must be put into practice in their future profession (activities designed according to the pedagogical model). Moreover, it was possible to directly “experience” the way an MD could be implemented. The specific training method presented here offers a particular interpretation of the suggestive expression “in and from practice”: through MDPm, PSTs, after personally experiencing the teacher’s actions as students, recognize them according to the specific pedagogical model.

As emerged from the analysis of the two episodes, thanks to the teacher’s mediation activity, PSTs implement the recognition of theoretical aspects from the lived experience. In particular, PSTs transition from the role of student to that of a future teacher by identifying the teacher’s actions, recognized through a process of detachment from the same situation, which was also facilitated by the online learning mode. In fact, Matteo, in his intervention (05:01), recognizes MD as an appropriate pedagogical model that is actively constructed. Specifically, the student identifies and internalizes the characteristics of the model, acknowledges its effectiveness in relation to its function, and finally distinguishes two specific functions, solving the problem and constructing the mathematical concept, recognizing the role of the instructor’s mediation in the construction of new concepts.

9. Discussion

This study contributes to the literature on teacher professional development by demonstrating that a dual-phase intervention, combining a direct experience (the Mathematical Discussion) with a subsequent meta-discussion (the MDPm), can foster both practical and theoretical understandings of effective pedagogy. The structured reflective process observed among PSTs aligns with both cognitive learning theories and situated learning frameworks, suggesting that real-time discussion about teaching practices significantly enhances the assimilation of complex pedagogical constructs.

In comparison with more traditional video-based or peer review approaches, the MDPm model appears to provide additional support by creating a live forum for metacognitive reflection, thus better preparing PSTs for the multifaceted nature of classroom instruction.

10. Conclusions

The findings from our analysis indicate that the integration of direct participation in a Mathematical Discussion with a subsequent reflective meta-discussion (MDPm) offers considerable promise for PSTs development. The dual-role approach supports PSTs in developing a more nuanced understanding of the teacher’s role, ultimately facilitating stronger instructional decision-making.

It is also interesting to note the discovery of the nature of mathematics as something open to discussion.

Furthermore, the data analysis suggests that engaging PSTs in training activities could foster both the theoretical conceptualization and practical implementation of a pedagogical model. This occurs because PSTs can personally experience and reflect on their learning, thereby contributing to their professional development.

Finally, technology also played a fundamental role, as it allowed students to collaborate in a stimulating environment, sometimes even without the perceived judgment of the teacher. The online platform tools proved to be very useful for mutual understanding and self-improvement, thanks to peer contributions during both MD and MDPm.

Future work will explore long-term outcomes of this intervention.