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Article • Rev. Bras. Educ. 31 • 2026 • https://doi.org/10.1590/S1413-24782026310083 linkcopy

Open-access Preparation of a professional learning task for teaching linear systems in the degree in mathematics

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ABSTRACT

This study aimed to describe and explain the process of preparing a professional learning task (TAP) for the mathematics degree. The study collaborators are a mathematics teacher trainer and a researcher. This is a qualitative-interpretative study. The results indicate that the participation of a teacher trainer in the development of a TAP enables an interaction between theory and practice; however, there were challenges to maintaining a didactic proposal that provided a learning environment. From this perspective, the participation of mathematics teacher trainers can provide an understanding of needs in the initial training of future mathematics teachers, the expansion of professional knowledge to teach and the use of formative tasks, becoming a possibility to explore connections between mathematics school and academic and for didactic and mathematical discussions.

Keywords:
Professional Learning Tasks; Systems of Linear Equations; Mathematics teacher trainer; Professional development

RESUMO

Este estudo objetiva descrever e explicar o processo de elaboração de uma tarefa de aprendizagem profissional (TAP) voltada para a licenciatura em matemática. Os colaboradores do estudo são um formador de professores de matemática e uma pesquisadora. Trata-se de um estudo qualitativo-interpretativo. Os resultados mostram que a participação de um formador de professores na elaboração de uma TAP possibilita a interação entre a teoria e a prática, no entanto houve desafios à manutenção de uma proposta didática que viabilizasse um ambiente de aprendizagem. Nessa perspectiva, a participação de formadores de professores de matemática pode proporcionar a compreensão das necessidades na formação inicial do futuro professor de matemática e a ampliação de conhecimentos profissionais para ensinar. O uso de tarefas formativas torna-se uma possibilidade para explorar conexões entre a matemática escolar e a acadêmica e para realizar discussões didáticas e matemáticas.

Palavras-chave:
Tarefas de Aprendizagem Profissional; Sistemas de Equações Lineares; Formador de professores de Matemática; Desenvolvimento Profissional

RESUMEN

Este estudio tiene como objetivo describir y explicar el proceso de elaboración de una Tarea de Aprendizaje Profesional (TAP) dirigida a la Licenciatura en Matemáticas. Los colaboradores del estudio son un formador de profesores de Matemáticas y un investigador. Se trata de un estudio cualitativo-interpretativo. Los resultados indican que la participación de un formador de docentes en el desarrollo de un TAP permite la interacción entre teoría y práctica, sin embargo, hubo desafíos para mantener una propuesta didáctica que brindara un ambiente de aprendizaje. Desde esta perspectiva, la participación de los formadores de profesores de Matemáticas puede proporcionar una comprensión de las necesidades en la formación inicial de los futuros profesores de Matemáticas, la ampliación del conocimiento profesional para enseñar y el uso de tareas formativas, convirtiéndose en una posibilidad para explorar conexiones entre la escuela de Matemática y lo académico. y para discusiones didácticas y matemáticas.

Palabras clave:
Tareas de Aprendizaje Profesional; Sistemas de ecuaciones lineales; Formador de profesores de matemáticas; Desarrollo profesional

INTRODUCTION

Researchers within the field of mathematics education have conducted investigations into teacher education and have seen growing interest in studies on mathematics teacher educators (Loughran, 2014), particularly those focusing on their professional development (Coura and Passos, 2017; 2021). In this case, the mathematics teacher educator, a university professor, is the subject of studies that highlight aspects of their knowledge and the relationship between how they conduct their courses and their focus on teaching pre-service teachers (PSTs) (Li and Superfine, 2016; Coura and Passos, 2017).

Regarding the professional learning processes of teacher educators, Ping, Schellings and Beijaard (2018) highlight the importance of these professionals’ identity and activities involving individual and collaborative reflection. This allows for the sharing of personal reflections and/or interactions related to the exchange of experiences among peers (other professionals, more experienced educators, or even students) regarding a critical incident (Ping, Schellings and Beijaard, 2018; Silva, Albrecht and Pina Neves, 2023; 2024).

The study by Ping, Schellings, and Beijaard (2018), which is also related to the professional learning of teacher educators, further discusses the professional activity involving the different roles and responsibilities of these individuals. They argue that these roles and responsibilities vary depending on the context, highlighting what (content), how (activity), and why (reason/motivation) teacher educators learn throughout their professional trajectories (Ping, Schellings and Beijaard, 2018).

From this perspective, Silva, Albrecht and Pina Neves (2023) discuss the need for these individuals to be involved in professional development training, particularly for teacher educators working in licentiate degree programs in mathematics. Considering this study so far, working in initial teacher education requires that the teacher educator promote a set of actions that demand a complex understanding of teaching and learning, regarding preparation for teaching mathematics (Freitas and Fiorentini, 2008; Silva, Albrecht and Pina Neves, 2023). This implies that they play a fundamental role in professional development, focused "on the pre-service teacher's mathematical knowledge and on establishing clearer and more consistent relationships between the content that the student teacher learns and that which they will teach in the future" (Almeida and Cristovão, 2017, p. 531).

In this context, the purpose of this article is to describe and explain the process of developing a professional learning task (PLT) designed for the licentiate degree in mathematics, focusing on the relationship between academic mathematics and school mathematics in the teaching of systems of linear equations with two variables (2x2). Given the above, we intend to answer the questions that guide this study: How did the process of developing a PLT for the licentiate degree in mathematics unfold, particularly regarding the relationship between academic mathematics and school mathematics for teaching 2x2 systems of linear equations? What relationships were considered in the development of the PLT regarding the pre-service teaching practice of prospective teachers for the instruction of 2x2 systems of linear equations?

LITERATURE REVIEW

PROFESSIONAL LEARNING TASKS AND EXPLORATORY TEACHING

PLTs are formative tasks designed to provide teachers and/or PSTs with opportunities to engage in instructional activities related to mathematical tasks, as well as to influence the way they select and describe these tasks (Ball and Cohen, 1999; Silver et al., 2007).

According to Smith (2001), one way to design a PLT is to consider the set of actions involved in teaching work. Under this definition, to compose a PLT, open-ended mathematical tasks with a high cognitive level and practice artifacts (records of practice) are generally used. These artifacts of real-world practice are authentic records, such as lesson plans, classroom planning done by the teacher, excerpts from curriculum proposals, student activities and problem-solving protocols, images, audio, videos, dialogues between students and teachers in the classroom, and the teacher's class notes (Smith, 2001; Ball, Ben-Peretz and Cohen, 2014; Ribeiro and Ponte, 2020).

According to Li and Superfine (2016), these practice records facilitate the integration of PSTs into the school setting of basic education. On this point, Smith (2001) argues that, in order to utilize real and authentic practice artifacts, they must be organized. The author adds that this organization must be practice-oriented and must be structured as specified in order to design a PLT, which can then be incorporated into a teacher education curriculum.

From this perspective, Ribeiro, Aguiar and Trevisan (2020) highlight the organization and composition of the PLT, which are guided by four elements, namely:

  • it utilizes a mathematical task (MT) of high cognitive level;

  • it begins with a didactic approach to exploratory teaching;

  • practice records (PRs)1 are incorporated into the PLT;

  • the PLT proposes to discuss, in an integrated manner, mathematical knowledge and pedagogical knowledge.

Furthermore, the organization of the PLT in this study considered its development in an exploratory teaching environment (Canavarro, 2011).

Considering that a PLT should be initiated by a didactic approach to exploratory teaching implies that this perspective contrasts with traditional (teacher-centered) mathematics instruction in higher education, as it presents multiple possibilities for resolution and meaning, which can facilitate progress toward student-centered teaching.

According to Canavarro (2011), for the development of an exploratory teaching environment to occur, it is necessary to consider the lesson planning, generally through an exploratory task, and the orchestration of discussions based on the five practices of Stein et al. (2008). These five practices by Stein et al. (2008) involve:

  • anticipating students’ responses to the chosen exploratory task;

  • monitoring students’ responses;

  • selecting students’ responses for discussion;

  • sequencing responses that establish connections;

  • establishing connections with students’ mathematical learning, related to the collective development of mathematical ideas.

The PLT framework aims to bring classroom practice into the realm of teacher education (Ribeiro, Aguiar and Trevisan, 2020), and to promote didactic-pedagogical and mathematical discussions, in order to facilitate and/or provide opportunities for learning in specific situations and contexts (Trevisan et al., 2023).

LINEAR ALGEBRA TEACHING

Some studies on the teaching of linear algebra have focused on identifying new approaches and/or elements of instructional design and teaching practices for its instruction (Bianchini, Lima and Gomes, 2019). However, some authors consider the teaching and learning of linear algebra content to be complex and difficult (Aydin, 2009). In this regard, Silva, Barros, and Fernandes (2019) point out that the difficulties encountered in the teaching and learning of linear algebra constitute a dilemma, which creates an opportunity for research that presents teaching strategies focused on learning.

In the analysis conducted by Bianchini, Lima and Gomes (2019), the need to structure the linear algebra course with the aim of preparing pre-service basic education teachers is highlighted. Regarding the understanding of linear algebra instruction in the mathematics teacher education program, Prado's (2016) research described the similarities and discrepancies among mathematics teacher education programs, which allowed for the organization of four observations, two of which we highlight here: the question of the teacher educator's responsibility for student learning and their openness to seeking strategies that promote learning and improve classroom practice; and the linear algebra curriculum in mathematics teacher education programs, which relates to the necessary training for graduates, given that the curriculum lacks reflections and alternatives for teaching in basic education.

This occurs due to the formalism employed by teacher educators in the classroom, which leads to difficulties in learning the formal proofs and elements of academic mathematics necessary to interpret the formal language used in addressing concepts, to conceptual difficulties associated with the objects of linear algebra, and to cognitive difficulties, which involve the thinking required to study this discipline (Prado, 2016).

Regarding the teaching of systems of linear equations, research highlights decontextualized approaches in which students fail to master elementary skills, exhibiting difficulties in learning the concepts and skills related to solving equations (Bertolazi and Savioli, 2018). According to Bertolazi and Savioli (2018), this occurs because this reality is found in various basic education school textbooks, which address aspects of the concept of systems of linear equations through a traditional teaching approach, with learning focused on a technical and rote memorization of the concept.

MATHEMATICAL DISCOURSE

Moschkovich (2003, p. 3) explains that mathematical discourse (MD) arises from a situated and sociocultural perspective, which "includes not only ways of speaking, acting, interacting, thinking, believing, reading, and writing, but also mathematical values, beliefs, and points of view," varying across different communities ranging from mathematical researchers to teacher educators and teachers in basic education.

Silveira (2010, p. 82) reflects that the interaction between student and teacher is subject to the language used, "for it is through dialogue that communication can take place." The author explores that, in the process of teaching and learning mathematics, it is necessary to share the "same discursive universe," especially so that the teacher can understand the student's potential based on their access to the mathematical domain. This understanding can be complemented by Trevisan et al. (2023), who consider that the MD includes the communication of definitions and verbalized communication regarding problem-solving approaches.

On this point, Freitas and Fiorentini (2008) demonstrate that discourse in the mathematics classroom can take various forms, such as expressing thoughts and ideas through written, oral, symbolic, pictorial, or technological language.

The authors Heyd-Metzuyanim, Tabach, and Nachlieli (2016), on the other hand, examined how certain assumptions underlying mathematical learning are reflected in the discourse of a teacher educator during mediated discussions in the development of a mathematical task, mobilized in two opposing fields: ritual mathematical discourse (RMD), which establishes rigid routines for solving mathematical tasks based on the teacher's authority and focused on procedures and the correct result, and exploratory mathematical discourse (EMD), which establishes flexible routines, with narratives constructed from emerging logical connections and a combination of internal authority and mathematical rules.

The results showed that some algebraic generalizations were identified in classroom conversations, including frequent statements made by the teacher educator, particularly in her discourse, which was not aligned with that of the PSTs. This manifested itself in the prevailing cultural norms in her actions, which involved a RMD, with guided discussion and attempts to minimize the opportunities for error in the education of PSTs (Heyd-Metzuyanim, Tabach and Nachlieli, 2016).

METHODOLOGY

This article is part of a doctoral study,2 based on a qualitative-interpretive research design, as the meaning of the research findings is fundamentally interpretive, drawing on the participants’ lived experiences and the knowledge and meanings they construct during the research process (Creswell, 2014; Scheiner, 2019).

CONTEXT AND PARTICIPANTS

The study focuses on the context of teacher professional development based on collaborative work (Silva, Albrecht and Pina Neves, 2023). The participants include a mathematics teacher educator, whom we will call João, who teaches the course linear algebra in a licentiate degree program in mathematics at a public higher education institution (HEI) in the Center-West region of Brazil, and a researcher who is the first author. In addition, the advisor, the co-advisor, and the mathematics pre-service teacher in initial education participated indirectly in the linear algebra course taught by the teacher educator.

The formative process involved: a theoretical study; the creation/development of a mathematical task and a formative task (referred to as professional learning tasks — PLTs); the planning of a higher education class, the delivery of this class, and reflection on it (Silva, Albrecht and Pina Neves, 2023). The planning meetings occurred remotely.3 During the first session, an episodic interview (Silva, Albrecht and Pina Neves, 2024) was conducted between teacher educator João and the researcher.

The research data were collected in 2022 and 2023. The initial conversations began in April and May 2022, and the formative process happened from June 2022 through August 2023, through a theoretical-practical study. From November 2022 to January 2023, a didactic proposal for initial education was organized, involving 19 remote meetings, most of which took place weekly, lasting 1 to 1.5 hours each.

Chart 1 presents some of the sessions that will be analyzed in this study. These correspond to the stages of developing the PLT and planning a lesson for initial teacher education (from the 9th to the 13th session). Initially, the organization was intentional in developing the formative activities, and throughout the sessions, collaborative work was established (Silva, Albrecht and Pina Neves, 2023). Accordingly, the selection of the teacher educator was intentional, considering their specific profile, professional identity (Silva, Albrecht and Pina Neves, 2024), and their willingness to participate in a formative process focused on classroom practice and conducting research.

Thumbnail
Chart 1
Activities in the formative process: a theoretical and practical study.

STUDY DESIGN

The formative process was based on collaborative work between the teacher educator and the researcher (Silva, Albrecht and Pina Neves, 2023). During this process, there were two moments of reflection, which were facilitated through semi-structured interviews (Rosenthal, 2014). This collaborative work, illustrated in Figure 1, also involved the teacher educator's own teaching practice and was used as a resource for the researcher to facilitate moments of reflection.

Figure 1
Collaborative work development.

The organization involved meetings among collaborators, following the guidelines proposed by Fiorentini (2004), which were considered essential, as shown in Chart 2.

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Chart 2
Aspects of the organization of meetings aimed at professional development.

Throughout the collaborative work process among the participants, theoretical and practical studies were conducted (Silva, Albrecht and Pina Neves, 2023), a lesson was planned for the linear algebra course (covering linear systems), and opportunities for reflection were provided. All stages of the collaborative work were recorded on video and audio by the researcher, followed by the selection of excerpts for reflective analysis. The planning begins after the completion of a mathematical task (MT) on systems of linear equations with two unknowns (2x2), as shown in Figure 2. The lesson planning for a linear algebra I class was based on the concept of exploratory teaching (Canavarro, 2011), in which the teacher educator adopted a PLT (Silver et al., 2007).

Figure 2
Mathematical Task – System of Linear Equation.

This article focuses on the section related to collaborative work planning (Figure 1), with the activities of the meetings detailed in Chart 1.

The development of the PLT included:

  • a MT (Figure 2) on the content of linear systems, developed by the mathematics teacher educator and the researcher (Silva, Albrecht and Pina Neves, 2023);

  • use of the exploratory teaching didactic approach;

  • PRs (Ball, Ben-Peretz and Cohen, 2014) related to the work of basic education students and vignettes from a class taught by a basic education teacher (which is not the class in which the MT was developed);

  • an integrated discussion linking mathematical knowledge and pedagogical knowledge, addressing didactic-pedagogical issues involving connections between mathematical, linguistic, and pedagogical themes.

DATA COLLECTION

Data collection was carried out using the following strategies:

  • documentary collection of textual sources (Bauer and Aarts, 2017), which included transcribing audio and video recordings of the meetings, as well as pre- and post-meeting communications via messaging apps (WhatsApp), email exchanges, and the use of shared folders on cloud storage;

  • unstructured participant observation (Fiorentini and Lorenzato, 2006);

  • keeping a logbook throughout the course of the research.

DATA ANALYSIS

The data analysis was organized based on the work conducted using inductive and deductive methods (Benedicto et al., 2012). To provide a clearer understanding, we structured the results based on a description of the planning stage of the collaborative work (Figure 1). Based on the selected data, we sought to conduct an in-depth analysis of the planning process, first through an inductive analysis of the data, which allowed us to identify certain markers, followed by a deductive analysis, as illustrated in Figure 3, which shows the organization of the data analysis process.

Figure 3
Order of data analysis.

However, while conducting the deductive analysis of the data, we delved deeper into the observations and characteristics identified in the actions that occurred during the selected phase, in line with the theoretical framework. The themes identified from the data were:

  • PLT and exploratory teaching;

  • difficulties in teaching linear algebra;

  • MD.

Following this process, further reflections led to the decision to conduct an additional inductive analysis of the data, focusing on the theoretical framework presented earlier.

RESULTS

To conduct the data analysis, specific time points were established for meetings dedicated to planning collaborative work. This section examines the steps involved in organizing the PLT, which include developing the PLT design, selecting practice records, and planning a lesson.

ORGANIZATION OF THE PROFESSIONAL LEARNING TASK'S DESIGN,4 SELECTION OF PRACTICE RECORDS, AND LESSON PLANNING

The concise reports to be presented refer to sessions 9 through 13, as detailed in Chart 1. The presentation of these sessions follows the structure planned for the meetings, although it does not always adopt a fixed order in the choices that compose the parts of the PLT. During these sessions, the teacher educator and the researcher explored the development, organization, and design of the four stages of the PLT, namely: the first is called "The mathematical task, the content, and the instructional sequence;" the second deals with "The students and teaching;" the third relates to "Teaching practice;" and, finally, the issues discussed in the fourth stage of the PLT are known as "Plenary."

All stages of the PLT occurred after the organization and selection of concepts related to linear systems, which were explored in previous meetings (Silva, Albrecht and Pina Neves, 2023). The first stage involved the development of guiding questions designed to facilitate discussions and reflections by mathematically exploring the PR, as can be seen in Figure 4, which presents the selection of questions regarding reflective action on TM (Figure 2).

Figure 4
Questions from Part 1 of the professional learning task (PLT) for the pre-service teachers.

It had been agreed in advance that the researcher would present some options related to classroom practice. Accordingly, during the week prior to the meeting, a video was shared on YouTube that allowed the teacher educator to view a possible video-based role-play to serve as the third stage of the PLT program's teaching component. After watching the video, the teacher educator questioned the reasoning behind its selection.

João: The idea here is that my student who is at university will look at this record of my teaching and, in a way, analyze it. This could influence their own teaching practice. […] I don't know if my students will notice what I observe there. The teacher begins by saying that solving the system means finding "x" and "y" that satisfy the equations at the same time. However, when he represents the answer to the exercise, he uses ordered pair notation, with the exception of the first example… I found it curious that, in the first example, "x" and "y" were left out. In the other examples, he wanted to present the solution set correctly… So, these are already the different languages of the same class, and as a teacher, I already ask, what is one thing or the other? Because, in fact, it's not about finding "x" and "y." In fact, it isn't, but it's very important for them to know that it's ordered; the order there is what's important.

In this regard, it is noted that João highlights the importance of language in teaching practice, as the teacher in the video relates his knowledge of linear systems by saying "find x and y," as well as the use of different languages within the same lesson. It is therefore evident that João intends to encourage the correct and appropriate use of mathematical language, as demonstrated by the teaching practice in question. However, it is crucial to adapt this approach to the level of instruction, as the goal is to enable mathematics teachers to recognize the importance of mathematical communication, since this skill is essential for teaching, involving the articulation of different languages, from the mother tongue or natural language to mathematical language (the symbolic system that follows specific rules).

João: It will allow them to reflect as pre-service teachers, to draw conclusions about the instructional sequence of a math task; we work on both the mathematical content and the pedagogy. The focus of academic mathematics is the student's strict academic perspective, and later they will also have to consider the perspective of school mathematics. When they're in these courses, they start to acquire this practice, but they aren't guided to look at a teaching context; they haven't had that experience yet, which technically should begin in the third semester. Since the linear algebra I course is offered in the first year of the undergraduate program at the institution, João states that this will likely be students’ first exposure to an educational opportunity that integrates mathematical and pedagogical knowledge, using potential teaching scenarios to explore the content of linear systems.

At the 10th meeting, the researcher began by sharing her impressions of the previous meetings before continuing with the organization of the PLT. After watching the video selected as the PR again, the researcher added some of her impressions from the previous week, during which João had observed other aspects of the lead teacher's speech in the video: "Regarding pre-service teaching practice, regardless of the setting or the students being taught, it is essential to recognize the importance of mathematical communication, as it provided an opportunity to integrate different languages" (researcher). This influenced the choices of the basic education students’ PR and the reflections for organizing the questions for the second and third parts of the PLT. At this meeting, the questions for the second part of the PLT were also finalized (Figure 5), which aimed to facilitate the PST's reflection during interactions with the teacher educator.

Figure 5
Questions from Part 2 of the professional learning task for the pre-service teachers.

Between the week spanning the 10th and 11th meetings, João presented the edits, which resulted in the decision to use five vignettes and to obtain permission to use the images for research purposes. In addition, during that meeting, the number of vignettes was discussed, and the researcher mentioned that the PLT was becoming too extensive to be implemented in a higher education classroom. However, João considered that the selection of the five vignettes was relevant and necessary for observing pre-service teaching practice and for the class's learning process.

Continuing the discussion, the researcher discussed the matter with João, considering the institution itself with its curriculum, syllabus, and the context of the course. She emphasized that: "listening and socializing was our choice; your perspective as a professional is important, as it is a practice that involves school mathematics and your view on how it relates to academic mathematics, based on your teaching practice" (researcher). At that moment, João felt comfortable and shared his reflections on the language, teaching approaches, and mathematical concepts used by the prep course teacher in the video.

João: The professor begins by discussing a system and defining its elements, but he doesn't define all the elements right away. For example, to be very clear, he mentions the unknowns at step zero, but he doesn't mention the coefficients. However, at some point, in order to continue his explanation, he does mention the coefficients, but in my opinion, he should have mentioned them when he first introduced the system. The system consists of its coefficients and its unknowns, and the goal is to find a set of solutions. Another criticism concerns the set of solutions. He says "solution" but does not use the term "set of solutions"; only at a later stage, in the second example, does he use "bracket notation," but at no point does he use the word "set." Mathematical notation is very important, along with mathematical language, because I felt a disconnect between the two. At various points, he shares his opinions; of course, opinion is part of the teaching and learning process, but in my opinion, he makes value judgments, and this can act as a pedagogical barrier for the student. In his speech, he uses the term "a bit of a pain" in relation to fractions, which intimidates the student, especially since we don't live in a world of whole numbers. So, the way he frames things could be reconsidered — for example, "Look, dealing with fractions isn't as natural as dealing with whole numbers" — but the way he puts it creates fear of working with fractions.

In addition, João asked himself: "Based on that observation, I wonder, ‘Have I, João, ever made that kind of statement?’ So, it served as a moment of reflection for me; I'm not here to judge the professor — it's truly a reflection stemming from the stage I'm at in this collaborative project." It is evident that this was a moment in which João reflected on his own practice, recognizing that, as a university professor, he is also prone to errors and misunderstandings. He then supplemented his observations with specific points regarding how the professor in the video presents equivalence in the examples of linear systems.

João: When it comes to two linear systems being equivalent, he uses another way of putting it — which I've noticed — that I really emphasize with my students: "identical things" and "equivalent things." In math, you have to be very careful, so he talks about identical systems and equivalent systems. As for identical systems, I really don't know what they would be. Now, equivalent systems are those that have the same set of solutions; the presentation of the systems can be completely different, but they have the same set of solutions, so they are equivalent.

It is noted that João reflects on errors in the use of notation in the explanations provided in the video, he even appoints them when selecting the vignettes to compose the PLT. In this case, we see the importance of initial education in academic mathematics, but also in knowing how to correctly relate it to the mathematical language and MD necessary for teaching school mathematics. We also identify João's concern with exploring the definitions, notations, theorems, and proofs present in the practice records, always involving academic mathematics.

The YouTube channel is geared toward a preparatory course; the teacher uses language and MD to make the content accessible to everyone, without worrying about potential misunderstandings in the communication of mathematical concepts, since the words in the language of mathematics can be challenging, including terms such as "coefficient," "unknown," and "equivalent." At this point, both the researcher and the teacher educator are dealing with some considerations that will be necessary for lesson planning.

João: Equivalent systems must be written equivalently… You know, for example, look at this with me: the definition of a rational number is an equivalence class defined by an equivalence relation that we studied in the Cartesian set of integers, consisting of integers other than zero. So, the knowledge that the teacher has — when he uses this kind of expression — of course I don't expect him to use this language with his students, but it struck me: could it be that during his higher education, he wasn't challenged in this way — the way I challenge my students — to simply avoid certain types of language?

The educator adds that it is important to clarify, at the outset, why "x" is a number, emphasizing the importance of ordered pairs — which, interestingly, the teacher does not mention in the video when presenting the solution in isolation and only later does he introduce brackets without explaining the concept of a set. It is observed that João notes that language and MD also function as a strategy for developing teaching skills, reflecting the care taken to consider academic mathematical knowledge as the foundation for teaching school mathematics — a fact considered important, that contributed to completing the questions in the third part of the PLT (Figure 6).

Figure 6
Questions from Part 3 of the professional learning task for the pre-service teachers.

The 12th meeting focused on organizing the questions for the plenary session (the fourth part of the PLT), designing the materials (the PLT and the educator's guide), and making decisions regarding the final plenary session of the PLT, reaching a consensus, as shown in Figure 7.

Figure 7
Questions from Part 4 of the professional learning task for the pre-service teachers.

The 13th planning meeting involved selecting the learning outcomes for basic education, which had already been shared in advance with the teacher educator. The organization of the PLT to be implemented in the linear algebra I course was also finalized. The researcher again raised concerns about the length of the PLT, but João believes it addresses the previously identified objectives and sees no need to shorten it, noting that the time assigned will be sufficient for development. Therefore, the researcher accepted João's response. After this, they returned to the educator's file and finalized the anticipatory section, which also forms part of the organized lesson plan.

At this point, the researcher emphasized the difficulties that PSTs may face when solving the linear system.5 João recalls the initial sessions in which PSTs are likely to behave like high school students when solving the linear system. However, he discusses with the researcher — who has already taught the content on linear systems — that the pre-service teachers will likely struggle to choose a strategy for solving the linear system, stating: "It's a simple 2×2 system, which can use the addition or substitution method. But, since I think they are in a different process, there is the possibility of creating the matrix version, performing row reduction, based on everything they have learned from me" (João).

It is evident, then, that both are attentive to the five practices of Stein et al. (2008), as they organize some anticipations, which involve waiting for the teacher's reflection and observing, on the day, the perceptions regarding the similarities in teaching possibilities and the mathematical content covered in the MT, therefore an anticipation of resolution based on high school teaching strategies recalled by João in the initial linear algebra classes. By incorporating possible anticipations regarding the resolution of the MT into the planning process, reflections are facilitated on the use of academic mathematics in order to understand school mathematics.

RESULTS DISCUSSION

The meetings alternated between moments of feedback and reflection; it is noted that, at each collaborative work session, there was an initial reflection, shared work, brief final reflections on the work in progress, and discussions for the next meeting regarding both parties’ responsibilities for weekly sharing (WhatsApp messages, email, or Google Drive) until the following meeting (Fiorentini, 2004; Darling-Hammond, Hyler and Gardner, 2017).

The collaboration considered the professional field of teacher educator João and the subject under his responsibility. Accordingly, during the PLT development phase, it fell to him to select the necessary vignettes that captured moments where the content presented challenging mathematical and pedagogical potential, as well as problematic aspects. This can be observed in João's narratives regarding the language and communication used by the classroom teacher in the video, highlighting the importance of attention to mathematical notation and MD (Moschkovich, 2003; Freitas and Fiorentini, 2008; Silveira, 2010; Heyd-Metzuyanim, Tabach and Nachiele, 2016).

João focuses on formal and axiomatic language, which have played a fundamental role in the teaching and learning processes of linear algebra concepts as a discipline (Aydin, 2009; Silveira, 2010; Prado, 2016). This contributes to studies on the difficulties in teaching linear algebra, as the PST does not understand the disciplinary language adopted by the teacher educator and remains on the sidelines of the class, the concepts, and the discussion, since the language holds no meaning for them (Heyd-Metzuyanim, Tabach and Nachlieli, 2016). In many cases, they do not interact and do not ask questions due to a lack of common ground with the classroom discourse (Prado, 2016; Bertolazi and Savioli, 2018; Silva, Barros and Fernandes, 2019).

During the planning stage, João took advantage of video-based professional reflection to question and reflect on his own practice, which allowed for the reframing of his professional knowledge and the continuation of his learning throughout his professional career (Fiorentini, 2004; Mizukami, 2005; Loughran, 2014; Darling-Hammond, Hyler and Gardner, 2017). This facilitated reciprocity in decision-making, allowing for reflection on the selection of vignettes and the sequence in which they follow one another, considering the target audience and the objectives related to the context of initial teacher education, which involved the organization of teaching situations for linear systems in basic education. It is observed that there were connections to professional development, as João's participation in the meetings provided new perspectives on learning due to the meaningful reflections established between school mathematics and academic mathematics and on how to teach through collaborative work with the researcher (Mizukami, 2005; Darling-Hammond, Hyler and Gardner, 2017).

Regarding academic mathematics, the previously selected content on linear systems (Silva, Albrecht and Pina Neves, 2023) constituted the common field of knowledge, present in the linear algebra course and in high school within basic education. This choice made it possible to draw on mathematical, didactic, and pedagogical reflections from academic mathematics to develop the PLT as a tool for use by teacher educators, in order to discuss pre-service teaching practice and bridge the gap between academic and school mathematics. Furthermore, it sought to provide pre-service teachers in initial education with experience in a practice different from those they had experienced as students in basic education (Silva, Albrecht and Pina Neves, 2023).

As shown in Figure 1, teacher educator João engaged in the experience of relating the content of systems of linear equations to practice, seeking to establish curricular coherence between knowledge and practice. In this process, he came to understand teaching and learning from a different perspective in order to instruct the pre-service teachers (Loughran, 2014). We can infer that this helps to bridge the gap and reduce the disconnect often found in academia between theory and practice (Freitas and Fiorentini, 2008; Loughran, 2014).

In the context of learning to teach, teacher educator João and the researcher reflected on the conflicting demands related to academic mathematics and school mathematics, as well as on theory and practice. This reflection required a broader view of content knowledge and pedagogical knowledge, promoting a shift in focus in the education of pre-service teachers and in the enhancement of the role of the teacher educator (Loughran, 2014).

It is observed that there is common content that spans both basic education and higher education, specifically linear systems. The concern is to develop this content not only from the perspective of academic mathematics,6 but also with a focus on the teaching practice of PST who will teach it in basic education, based on school mathematics (Moreira and David, 2003; Freitas and Fiorentini, 2008). To achieve this objective, records of high school students’ practices and written work were used, along with a video illustrating teaching practice. This material highlights pedagogical situations based on school mathematics, emphasizing the fundamental ideas of academic mathematics related to the shared knowledge of linear systems.

This approach enabled the creation of a teaching resource (PLT) designed to discuss the teaching of school mathematics content from an academic mathematics perspective, aimed at initial teacher education (Ball and Cohen, 1999; Silver et al., 2007; Ball, Ben-Peretz and Cohen, 2014). This initiative focused on offering a student-centered development opportunity, going beyond a merely technical and rote-learning approach, as is often found in studies on linear algebra (Bertolazi and Savioli, 2018; Bianchini, Lima and Gomes, 2019; Silva, Barros and Fernandes, 2019).

In this regard, we organized some PLT relations, as shown in Chart 3.

Thumbnail
Chart 3
Relations based on the explanation of the professional learning task development process.

Chart 3 illustrates the descriptive and explanatory process underlying the development of the PLT for this study, as the collaborative reflections on the design of instructional materials for use in the initial education of pre-service mathematics teachers highlight aspects related to teaching practice, future classroom instruction, and opportunities for collective discussion that directly relate to the nine elements providing formative opportunities within the developed PLT.

In this regard, the discussions highlighted the reflective efforts aimed at creating, through the PLT, opportunities for PSTs to:

  • engage with real classroom experiences, using student writing samples (written work by high school students and a video of a basic education class);

  • reflect on their actions in situations related to pre-service teaching practice by using vignettes to foster meaningful discussions based on the proposed questions;

  • represent an opportunity for a licentiate degree program in mathematics to provide a space where knowledge of mathematics and knowledge of teaching mathematics in basic education can be integrated;

  • contribute to bridge the gap and reduce the disconnect that are common in academia regarding theory and practice.

The interactions, which occur throughout the formation and development of the PLT, provide moments of reflection that can lead to the reevaluation, disruption, and/or continuation of actions interwoven into teaching practices, as well as their subsequent adaptation/modification (Coura and Passos, 2019; 2022). Presenting itself as a cyclical process that results in the elaboration (refinement) and/or construction of (new) knowledge, this process can promote opportunities for the professional development of teacher educator João and the improvement of teaching practice (Coura and Passos, 2019). Furthermore, it is important to note that participation in collaborative work contributed significantly to the improvement of PLT and, consequently, to the professional development of the teachers involved in this study.

FINAL CONSIDERATIONS

This formative process, which began as a collaborative effort, explored the development of a lesson plan for a higher education class from the perspective of exploratory teaching. This process took as its starting point a local reality in which mathematics teacher educators frequently face difficulties in establishing an effective connection between academic mathematics and school mathematics (Moreira and David, 2003; 2005). The formative process involved the organization of intentional activities, which, throughout the sessions, were accompanied by constant deliberations and reflections.

The accounts described how the design unfolded, involving the inclusion of MT, the PR, and the formulation of guiding questions, in order to develop a lesson through exploratory teaching based on the set of five practices by Stein et al. (2008). It is observed that teacher educator João gradually comes to understand the process of constructing the PLT by establishing connections between content, modeled instruction, and professional practices. However, the teacher educator himself is concerned with making connections in a systematic and planned manner, particularly regarding academic mathematics, likely because this professional practice requires time to organize, within the course, a formative process that facilitates connections with school mathematics (Moreira and David, 2003).

By experiencing the proposed process in the organization of a lesson from the perspective of exploratory teaching, the teacher educator can identify anticipated misconceptions among basic education students and represent concepts in different ways, in order to sequence the topics taught (Stein et al., 2008; Canavarro, 2011). By offering a formative process that explores the connections between academic mathematics and the teaching of school mathematics through practice, the aim is to break away from traditional models in the teaching of linear algebra, and contribute to the improvement of the teaching and learning of this content (Smith, 2003; Coura and Passos, 2017; Almeida and Cristovão, 2017).

This teaching plan for the linear algebra course serves as a sample tool for initial teacher education. It explores connections between the content of linear systems and school mathematics, as well as opportunities for PSTs to experience a classroom setting and an educational approach that differs from the model through which their own mathematics teacher educators were educated (Ping, Schellings and Beijaard, 2018). In this way, it also enables PSTs to experience this new approach (Coura and Passos, 2017; 2019; 2022). Given this, the use of resources, such as formative tasks, becomes an opportunity to explore connections between school and academic mathematics and to foster didactic and mathematical discussions in the licentiate program in mathematics.

It is observed that although teacher educator João was not the focus of this study, based on his ability to conceptualize, he conducted his own learning regarding the methods necessary for the careful planning of the PLT's organization and for planning with deliberate actions (Darling-Hammond, Hyler and Gardner, 2017).

A gap in this research7 is the need to elucidate the didactic weaknesses and possibilities of the designed PLT and to evaluate its role as a didactic proposal in consolidating the learning of the PSTs. Therefore, this relates to future actions to present the results of lesson development based on the scenario created using this PLT. This will involve observing the teaching and learning approaches advocated in the curricula of mathematics teacher education programs and the extent to which they are modeled by mathematics teacher educators in their own practice.

It is hoped that the linear algebra course in mathematics teacher education programs will provide a space where knowledge of mathematics and knowledge of mathematics teaching can be integrated to address content related to the teaching and learning of school mathematics. Therefore, we believe that the participation of mathematics teacher educators can foster an understanding of the needs in the initial education of pre-service mathematics teachers, the expansion of their professional knowledge for teaching, as well as the need for studies that address the professional development and learning of mathematics teacher educators.

From this perspective, it is emphasized that studies involving mathematics teacher educators and the organization of instructional materials can provide an opportunity to understand the needs in the initial education of pre-service mathematics teachers. We hope, therefore, that such research will contribute to the quality of initial teacher education by facilitating mathematics educators’ understanding of school mathematics and by encouraging reflection on the ways in which teachers and these educators are/were educated.

  • 1
    Practice records (PR) are authentic artifacts related to teaching practice and the classroom, such as: lesson plans; lesson planning developed by teachers or teacher educators; excerpts from curriculum proposals; activities; records of student work; images; audio recordings; videos; dialogues between students and teachers in the classroom; and teacher's lesson notes (Smith, 2001; Ball, Ben-Peretz and Cohen, 2014). PRs enable the integration of PST into the school setting of basic education. On this point, Smith (2001) argues that, to utilize real and authentic PRs, they must be organized and practice-oriented, enabling the design of a PTL, and that this can be incorporated into a curriculum for teacher training.
  • 2
    This study adopted the format of a research report known as a "multipaper," with one principal investigator and one co-investigator. It has also received approval from the Research Ethics Committees (CEP) of the following HEIs: the proposing institution, the Federal University of ABC (UFABC) (Certificate of Presentation for Ethical Appreciation — CAAE 56236822.4.0000.5594, Opinion No. 5.400.645/2022) and the co-participating institution (CAAE 56236822.4.3001.5083, Opinion No. 5.478.447/2022).
  • 3
    Meetings held via videoconference using Zoom Meetings.
  • 4
    First version of the PLT organized and developed during the course: https://docs.google.com/document/d/1oNuK9qlVKky_qEb1VOi2apt85imDOqEe/edit?usp=sharing&ouid=108131784422861479326&rtpof=true&sd=true. Accessed on: May 21, 2026.
  • 5
    A MT file developed in a second-year high school class at the Instituto Federal de Brasília (IFB), in which the students’ practice records constituted the second stage of the TAP. Files for the first and second parts of the PLT for the class on January 25, 2023, and the third and fourth parts of the PLT for the class on January 27, 2023, to be developed with students in the linear algebra I course taught by teacher educator João.
  • 6
    "Mathematics education in the licentiate's program is guided by the conceptual and aesthetic values of scientific mathematics, thereby ensuring, in theory, a status of theoretical-scientific education. The integration of the undergraduate training process with school practice is thus conceived as a task to be carried out from outside the realm of mathematics education" (Moreira and David, 2005, p. 59, emphasis added).
  • 7
    This is because this stage of presenting the research results did not focus on the PST data generated from the development of classes in higher education.
  • Funding:
    This work was carried out with the support of the Coordination for the Improvement of Higher Education Personnel - Brazil (CAPES) - Funding Code 001.

Data availability statement:

Research data are only available upon request.

ACKNOWLEDGEMENTS

To the Study and Research Group on Education in Science, Mathematics, and Sexuality (GECIAMS/UFABC); to the Mathematics Education Research Group at UnB (GIEM/MAT/UnB); to the Graduate Program in the Teaching and History of Science and Mathematics (PEHCM); and to the Federal Agency for Support and Evaluation of Graduate Education – Brazil (CAPES) – Funding Code 001.

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Edited by

Publication Dates

  • Publication in this collection
    07 Aug 2026
  • Date of issue
    2026

History

  • Received
    03 Sept 2024
  • Reviewed
    29 May 2025
  • Accepted
    18 June 2025
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