Levels of Complexity in Mathematics Teachers’ Knowledge Connections: An Approach Based on MTSK and Piaget’s Schemas

Abstract

This paper presents a study whose aim was to formulate a conceptual framework that allows us to understand the degree of complexity of the connections between the knowledge that a mathematics teacher can possess, based on the model of the mathematics teacher’s specialized knowledge model and the Piaget’s schemas. To this end, a literature review was carried out to analyze how these connections are made in the minds of mathematics teachers and how complex they are, identifying a deficit. Therefore, the schema structure proposed by Piaget, with its three stages of intra-, inter-, and trans-development, was considered. Based on an instrumental case study with two prospective teachers, an analysis was performed on semi-structured interviews conducted while the teachers separately discussed a lesson plan based on the understanding of multiplication and division of natural numbers, aimed at third-grade students. The results show that three types of knowledge schemas emerged among the participating teachers, with the respective types of connections between the knowledge. This shows that the connections between the knowledge of the model in teachers’ minds can vary in complexity.

1. Introduction

In order to investigate mathematics teachers’ professional knowledge, several analytical models have been developed in recent decades, among other approaches that have identified specific elements that compose it (e.g., Ball et al., 2008; Carrillo et al., 2022). In particular, the theoretical and analytical model referred to as Mathematics Teachers’ Specialized Knowledge (MTSK) has focused on examining this notion in depth through three domains: mathematical knowledge, didactic knowledge about the content, and beliefs about mathematics, and about its teaching and learning (Gamboa et al., 2022).

According to Scheiner et al. (2019), the knowledge of the mathematics teacher flees from a reductionist approach in which knowledge is interpreted as independent of the knower and tends more toward a perspective that conceives it as an evolutionary process in which the interaction between knowledge, the knower, and the context is emphasized. In the processes of knowing and learning, they establish the teachers’ vision of unraveling the students’ understandings and making their ways of mathematical thinking visible; that is, they agree with the students’ vision by constructing mathematical ideas that are used as a starting point in the teaching complex (Scheiner et al., 2019).

In this way, knowledge is viewed as a process rather than an object, where different facets or types of knowledge are in constant dialogue with each other, informing each other and interacting dynamically to form emergent structures (Scheiner et al., 2019), in line with the complexity paradigm, where the whole is greater than the sum of its parts (Grinberg, 2002).

Current empirical research on this topic, especially that based on the MTSK model, is in the phase of identifying the connections that may exist between the different types of knowledge of mathematics teachers (e.g., Flores & Carrillo, 2014; Otero-Valega et al., 2023; Pacheco-Muñoz et al., 2023; Tascón & Juárez, 2024; Zakaryan et al., 2018). In this paper, we discuss how these connections can be understood in the teacher’s mind and propose a conceptual framework for their identification and understanding based on the MTSK model and the epistemological proposal of Piaget’s (1975) schemas. Thus, the research question guiding this work is: How can we formulate a conceptual framework that allows us to understand the degree of complexity of the connections between the knowledge in the MTSK model that a mathematics teacher can possess? The aim of the research is to formulate a conceptual framework that allows us to understand the degree of complexity of the connections between the knowledge that a mathematics teacher can possess, based on the MTSK model and the Piaget’s schemas.

The justification for this work is the need to investigate not only the knowledge or connections between the knowledge of the MTSK model that the mathematics teacher can build in their mind, but also to deepen and characterize the complexity of these connections in stages. This is in order to identify possible avenues for teacher education “to empower teachers to impact practice in a timely, relevant and meaningful way” (Chapman et al., 2022, p. 1).

2. Conceptual Framework

This section sets out the proposed conceptual framework for the research and supports the analysis of the information by providing meaning and interpretation. It is divided into four subsections: a description of the MTSK model, the connections between knowledge as established in the literature, the connections as defined in this study, and Piaget’s schemas.

2.1. MTSK Description

The MTSK model is a theoretical-analytical proposal that makes it possible to investigate, analyze, characterize, and deepen the knowledge of mathematics teachers, which is not a matter of isolated elements but a complex network of relationships (Carrillo-Yañez et al., 2018). It consists of three domains: Mathematical Knowledge (MK), Pedagogical Content Knowledge (PCK), and beliefs and conceptions about mathematics and its teaching and learning. This third area is not the subject of the present study.

The first two domains consist of subdomains of knowledge. The MK domain consists of Knowledge of Topics, Knowledge of the Structure of Mathematics, and Knowledge of Practices in Mathematics. PCK, on the other hand, is composed of Knowledge of Mathematics Teaching, Knowledge of Features of Learning Mathematics, and Knowledge of Mathematics Learning Standards (Flores-Medrano et al., 2022). These subdomains are studied through categories that allow for refinement to identify knowledge descriptors. These are presented in detail below.

• Knowledge of the Topics (KoT): This subdomain refers to what and how mathematics teachers know about the different topics they teach. This includes knowledge of how mathematical content is structured (intraconceptual connections, i.e., connections with elements within the same topic), such as concepts, procedures, properties, rules, foundations, theorems, and their meanings. It also includes knowledge of the phenomena that give meaning to mathematical content (in terms of origin) and its applications (in mathematics and other fields), as well as knowledge of the registers used to represent content (Carrillo-Yañez et al., 2018).
• Knowledge of mathematical structure (KSM): Carrillo-Yañez et al. (2018) mentioned that this subdomain considers the connections between mathematical elements that may be associated with an increase in complexity or simplification. For example, the complexification connection that exists between linear equations of the first degree and systems of two equations with two unknowns. It can also be interconceptual connections, such as auxiliary connections, i.e., when the contribution of a mathematical element is necessary for more comprehensive processes. For example, the use of equations as an auxiliary element in the calculation of the roots of a quadratic function (Carrillo-Yañez et al., 2018). Finally, transversal connections occur when different mathematical elements have common properties that allow them to be linked together. For example, the concepts of limit and continuity are connected by the notion of infinity (Carrillo-Yañez et al., 2018).
• Knowledge of Practices in Mathematics (KPM): Refers to the knowledge related to the functioning of different aspects of mathematics (Zakaryan & Sosa, 2021), to the construction, validation, and communication of mathematical knowledge in an educational context. In terms of construction, there is the category of definitional practice (e.g., constructing, formulating, or specifying properties that characterize a mathematical object). In validation, there is the practice of proof (e.g., testing the truth value of a theorem, methods of proof, or distinguishing between a necessary and a sufficient condition) (Zakaryan & Sosa, 2021), and the practice of problem solving (e.g., using heuristics such as dividing the problem into cases or Polya’s four steps to solve a difficult or non-routine problem). Mathematical language plays a role in communication (e.g., the use of mathematical symbols to convey mathematical ideas in an abbreviated form and with precision with respect to the strict meaning of the terms) (Zakaryan & Sosa, 2021).
• Knowledge of Features of Learning Mathematics (KFLM): This subdomain refers to knowledge of mathematical content as objects of learning, i.e., the phenomena that occur when someone learns mathematical content. This knowledge comprises categories of learning theories that explain the learning of mathematical objects and give them meaning. The strengths and difficulties that students show when learning mathematical content. The forms of interaction with mathematical content, i.e., the strategies and processes students use, their language, and vocabulary to address the content. The interests and expectations of students influence how they approach content (Zakaryan et al., 2018).
• Knowledge of Mathematics Teaching (KMT): The categories of didactic resources (physical or digital) for teaching mathematical subjects are considered. The teaching theories that the teacher knows (institutionalized theories) or constructs (personal theories). The knowledge of when and how to help students, i.e., the different strategies, techniques, tasks, and examples to teach mathematical content (Zakaryan et al., 2018).
• Knowledge of Mathematics Learning Standards (KMLS): Carrillo-Yañez et al. (2018) mentioned that this subdomain considers the knowledge of the content to be taught at a certain level. Knowledge of the sequence of topics. The knowledge of what students should and can achieve at a certain level, considering what the student should know from previous topics and the requirements for later levels.

Carrillo et al. (2022) mention that the MTSK allows us to understand what knowledge underlies the different actions of teachers. Thus, given a professional action, such as planning a lesson, helping students, or selecting materials, there is a variety of knowledge that makes it possible to explain this action through the mobilization of knowledge structured in the system of categories. In this sense, although the categories allow us to isolate knowledge, the MTSK allows us to understand that it is interconnected, and one of the goals of the model is to explore how this knowledge interacts and what the most common relationships between them are (e.g., Otero-Valega et al., 2023; Pacheco-Muñoz et al., 2023; Tascón & Juárez, 2024) (see

Table 1. Domains, subdomains, and categories of the MTSK model.

Domains	Subdomains	Categories
Mathematical knowledge, MK	Knowledge of Topics, KoT	Procedures Definitions, properties, and their foundations Registers of representation Phenomenology and applications
	Knowledge of the Structure of Mathematics, KSM	Connections to complexity Connections to simplification Helpful connections Transversal connections
	Knowledge of Practices in Mathematics, KPM	Practice defining Practice demonstrating Practice solving problems The language of mathematics
Pedagogical content knowledge, PCK	Knowledge of mathematics teaching, KMT	Theories for teaching mathematics Physical or virtual resources Strategies, techniques, tasks, and examples
	Knowledge of features of learning mathematics (KFLM)	Theories on learning mathematics Strengths and difficulties in learning mathematics Forms of interaction with mathematical content Emotional aspects of learning mathematics
	Knowledge of mathematics learning standards (KMLS)	Expectations of learning Expected level of conceptual or procedural development Sequence of topics

, based on Delgado-Rebolledo & Espinoza-Vásquez, 2021).

2.2. Connections Between the Knowledge Within the MTSK

A large body of research within the MTSK model has attempted to connect mathematics teacher knowledge (e.g., Aguilar-González et al., 2018b; Delgado-Rebolledo & Zakaryan, 2020; Escudero-Ávila et al., 2017; Flores & Carrillo, 2014). This is a topic of interest due to its potential to see the teacher’s knowledge as an integrated entity, while identifying specific aspects of these complex relationships within this model.

Two major trends can be identified in the works analyzed, depending on the interests involved. On the one hand, research has reported that it only investigates relationships between mathematical knowledge and didactic knowledge of content in various mathematical topics (Delgado-Rebolledo & Espinoza-Vásquez, 2021; Delgado-Rebolledo & Zakaryan, 2020; Pacheco-Muñoz et al., 2023; Paternina-Borja & Juárez-Ruiz, 2023; Otero-Valega et al., 2023; Tascón & Juárez, 2024; Zakaryan et al., 2018). However,, some research links teachers’ conceptions of mathematics and their teaching and learning with mathematical knowledge or mathematics didactics (Aguilar-González et al., 2018a, 2018b; Flores & Carrillo, 2014).

Regarding the conceptualization of the construct relationship between knowledge within the MTSK model, in the work of Delgado-Rebolledo and Zakaryan (2020), the units of analysis (text fragment) in which a subdomain is identified are established as evidence of the relationship between knowledge that supports or conditions the appearance of one or more subdomains.

This conceptualization is taken up in the work of Pacheco-Muñoz et al. (2023), who propose the construct of the directional relationship, stating that “the subdomain that occupies the point of departure has a conditioning role and the subdomain as the point of arrival assumes the role of the mobilized” (p. 63). It is also clearly stated that “the role of conditioning can be taken by one category and the role of mobilized can be taken by one or more categories in a given argument” (p. 63).

In the latter, they specify that a directional relationship is established between the categories of knowledge subdomains of the MTSK model. Moreover, they confirm that this type of relationship begins with a category of knowledge that conditions the presence of one or more categories, i.e., it must not be a one-to-one relationship. It is also clearer in the sense that the relationship must take place at a specific moment in which the teacher’s thinking establishes the connection of directionality between the categories of knowledge, which manifests itself in their verbal expression in an interview or in a class, in a series of juxtaposed sentences (specific argument), but not at different moments separated in time.

Finally, Delgado-Rebolledo and Espinoza-Vásquez (2021) provide a different way to classify the relationships between the different knowledge subdomains of the MTSK model. Specifically, they refer to the relationships between knowledge in the same subdomain as intra-subdomain, within a domain as intra-domain, and between different domains as inter-domain. The authors did not elaborate on the reason for the prefixes intra or inter. It is assumed that they were used because of their general meaning: the prefix intra stands for within or inside, while the prefix inter stands for between or in the middle. In this article, we will give these prefixes a different meaning, in the sense of Piaget’s (1975) schemas.

2.3. Conceptualization of the Constructs Specific to the Connection Between the Knowledge of the MTSK Model

From now on, they will be referred to as connections rather than relations (with the exception of quotations from other works where they are so called), because the term “relations” might refer more to a structured association between variables and quantitative studies, whereas the term “connections” suggests a more flexible link that emphasizes how ideas, concepts, or knowledge are integrated or interact in the mind of the subject.

In order to establish the various kinds of connections between the knowledge of the MTSK model of mathematics teachers considered in this paper, we first consider the nature of this knowledge. We adhere to the proposal of Scheiner et al. (2019), who present it as intrinsic knowledge that renounces external reference points and explains its specialization as a process of becoming rather than a state of being. In this way, we understand mathematics teachers’ knowledge as a continuous process of internal development. In this sense, Scheiner et al. (2019) state that mathematics teachers’ knowledge should not be treated as static and general properties, but as adaptive and evolutionary actions.

Furthermore, Carrillo et al. (2013) consider this knowledge inextricably linked to context, noting that knowledge is born and developed in context. Scheiner et al. (2019) define context as that which consists of situations and activities embedded in the learning-teaching complex in the immediate moment. Thus, the connections between mathematics teachers’ knowledge are inextricably linked to the context; they are not static and are constantly evolving.

In this paper, we explain, from the perspective of the MTSK model, what we mean by a non-connection, a directed connection, and an interactive connection between the mathematics teacher’s knowledge. In doing so, we draw on the work of Spencer et al. (2014), who defined some types of relationships between phenomena. In this paper, the phenomena under investigation are the mathematics teacher’s knowledge.

If the subject does not make a conscious connection between the knowledge, the knowledge appears isolated, without any kind of link. A single piece of knowledge may appear or be separated by an intervening discourse or appear sequentially, describing an event without any conscious connection. Spencer et al. (2014) called this phenomenon a sequential relationship, but in this paper, it is considered as no connection between the knowledge of the MTSK model. One possibility of what we consider a non-connection is the description of an event that occurred in time or was given by another person or entity. In this case, the events are described sequentially because they occurred or were given, but not because the subject made an intentional connection between them.

An example of a non-connection from which only the knowledge The language of mathematics of the subdomain KPM emerges, is the question to the teacher Eva: “What is the role of symbols in mathematics?”. She answers:

Well, mathematical symbols are used to reduce language and connect mathematical concepts and distinguish them from natural language, in this case… They identify mathematical concepts so as not to confuse them and to distinguish them. In the case of similarity, for example, the symbol for triangles does not stand for another element; the use of the similarity symbol does not mean something else.

(Zakaryan & Sosa, 2021, p. 80)

Another relationship between the phenomena considered by Spencer et al. (2014) is the structural relationship, where one type of knowledge promotes another. This type of relationship exists in one direction but not in the opposite direction. This type of relationship is identified by the directional relationship established by Pacheco-Muñoz et al. (2023).

In this paper, we adhere to Pacheco-Muñoz et al. ’s (2023) definition of a directional connection between mathematics teacher knowledge of the MTSK model, adding that this connection is associated with a specific context. An example of this type of connection can be seen in the interview fragment shown in

Table 2. Example of a directional connection.

Researcher:	You’re somewhat against these standards because they somehow limit the teacher’s work.
Teacher:	Indeed, this is precisely what happened with the 2011 curriculum. They asked us for results; they gave us, for example, tests based on these standards, on the expected learning outcomes by core areas. So we were going to focus more on that aspect of the results, rather than on understanding, and we were going to focus on the student’s process. (Mary, Interview excerpt, 17 July 2024)	Expectations of learning, KMLS Strategies, techniques, tasks and examples, KMT

, where it is observed that the teacher’s knowledge about the subdomain KMLS, specifically the Learning expectations, conditioned the emergence of her knowledge of KMT), specifically, a teaching strategy based on drills rather than understanding.

A more complex relationship, which Spencer et al. (2014) define as an interactional relationship, occurs when two phenomena interact, i.e., when the first promotes the second, in a reciprocal manner. In this paper, we consider an interactional connection between two or more types of mathematics teachers’ knowledge when they influence each other and emerge in a particular context. The direction of the interaction is mutual. In this type of connection, the teacher’s knowledge is interwoven into a continuous fragment of the teacher’s discourse. The subdomains of knowledge at stake, which are revealed in particular by their categories, interact in a back-and-forth manner, i.e., the first may emerge, then the second, the first may emerge again, a third may emerge, the first or second may emerge again, and so on. It is important that this emergence takes place in a continuous fragment of the teacher’s discourse, because it is at this moment that the teacher connects this knowledge in their mind. Several examples of this type of connection are presented in the Results Section of this paper.

2.4. Schemas and Their Stages of Development

Piaget (1975) established the construct of schema to explain the mental processes that occur in a subject, which can be differentiated into three stages according to their level of development, essentially according to the level of relationships that exist between their elements. By examining how knowledge develops in people’s minds, we can use the MTSK model to examine the knowledge of the mathematics teacher. This is done in order to understand the various knowledge schemas that the teacher can build, their level of abstraction or complexity, and most importantly, to help us develop mechanisms for their identification and development through teacher education programs. These concepts are further developed below.

According to Piaget and García (1989), the development of knowledge in humans is not a linear process in which each stage replaces the previous one, maintaining some connection with the latter but no relationship with the former. In reality, they affirm that the successive stages of the construction of different forms of knowledge are sequential, each new stage beginning with a reorganization of the main achievements of the previous stages at a different level.

Piaget and García (1989) state that cognitive development occurs through reflective abstraction, which consists of the repetition of the same mechanism, constantly renewed and expanded through the alternation between the addition of new content and the elaboration of new forms or structures. This process originates in the subject’s actions and operations and unfolds through two interrelated processes: (1) a reflected projection (or mirroring) onto a higher level of what has been extracted from a lower level and (2) a reflection that reconstructs, reorganizes, and extends what was transferred through the initial projection.

More specifically, assimilation is the part of the process proposed by Piaget that involves absorbing new experiences or information into the schemas already existing in the subject, where the information is not adopted “as is” but is modified (or interpreted) in some way to fit in with the schemas already present in the subject (Boyd & Bee, 2014). Similarly, accommodation is the part of the process where a person modifies their existing schemas based on new experiences (Boyd & Bee, 2014).

Piaget and García (1989) showed that cognitive development requires the reconstruction of what was acquired at previous levels each time a new level is reached. Therefore, it is a question of reorganizing knowledge in light of new perspectives and reinterpreting basic concepts. Piaget and García called the stages involved in the development of a schema intra, inter, and trans, and explained that “these stages occur in a compelling sequence from intra to inter and from there to trans” (p. 36).

The first, the intra-stage of these beginnings of knowledge, is characterized by a focus on the individual components of a schema (Arnon et al., 2014). The intra-schema leads to the discovery of a set of properties in objects or events, but without other explanations that are not local and particular (Piaget & García, 1989). “In this phase, the analysis of individual cases that are not related to each other takes place” (Piaget & García, 1989, p. 171). In our case, the objects or events are the descriptors of the mathematics teacher’s knowledge within the MTSK model, e.g., knowledge of how to solve a system of equations (mathematical knowledge) or knowledge of the theory of authentic situations for designing word problems (didactic knowledge). Thus, in this paper, we will refer to an intra-schema in MTSK as one in which the knowledge components of the schema are not linked, as there are no connections between them yet. In this stage, the teacher concentrates on specific knowledge and provides local and specific explanations. There are no connections in this stage.

To stay with Piaget’s explanation, the new schemas thus constructed in this way cannot remain isolated. Sooner or later, the assimilation process will lead to certain assimilations, and the demands of equilibrium will impose more or less stable forms of coordination and connection on the schemas or subsystems thus linked. This is where the inter-character of the second stage becomes apparent (Piaget & García, 1989). As knowledge develops, the understanding of local relationships begins to play a more fundamental role, and access to the necessary connections and the reasoning behind them begins to develop (Arnon et al., 2014).

In this paper, we identify an inter-schema in MTSK as a schema in which connections between knowledge are observed, which can be directional or interactional. These connections are observed in a continuous fragment of the teacher’s discourse. For example, if the teacher explains that her teaching strategy is to propose an a-didactic situation involving a word problem linked to concrete materials and also describes how she thinks the students will interact with the mathematical content, one can observe that her knowledge of strategies, techniques, tasks and examples, theories of mathematics teaching, and teaching resources from the KMT subdomain and ways of interacting with mathematical content from the KFLM subdomain emerge.

Once discovered, these connections require the establishment of links between them (Piaget & García, 1989). Connections, in turn, require a motivation that explains them, and the structures that respond to this new requirement are characteristic of the trans-stage, for “a total system of relations produces new relations and provides the reasons for their composition as a whole” (Piaget & García, 1989, p. 159). At this stage of the schema, it is necessary to determine the reasons for the local connections that make up the inter-stage of the schema. Specifically, the individual begins to see the schema as a whole, and through synthesis, a structure is built that can explain its composition as a whole (Arnon et al., 2014). The structure is now coherent, and the individual can determine whether it is applicable to a particular situation. This is referred to as the trans-schema in APOS theory (Arnon et al., 2014). Participants can explain and justify these relationships (Trigueros et al., 2024).

In this paper, a teacher is considered to have a trans-schema in their MTSK if they can justify the reason for the connections they have made in the inter-stage of the schema and see the schema as a coherent whole. You can determine whether such connections are applicable to a particular situation in your profession.

This conceptual framework allows us to understand what is going on in the mathematics teacher’s mind at a given moment by sketching the content and stage of development of his knowledge schema in one of these three stages. We speak of a sketch because we assume that knowledge schemas can generally be more complex than what the teacher can provide. The formation of these schemas also depends on the activity that the teacher performs, whether in class with their students, in a group discussion, or in an interview, to name just a few examples.

3. Methodological Design

Starting from an interpretative paradigm, this qualitative study is based on an instrumental case study, which Stake (1995) defines as a method where there is “a need for general understanding, and a feeling that we may get insight into the question by studying a particular case” (p. 3). The focus is not on the case itself but on the phenomenon under investigation. Through the information from the informants that make up the case, we can delve deeper into the phenomenon being investigated. In this paper, the phenomenon under investigation is the specific knowledge schema of the math teacher, especially its recognition, and the case consists of the teachers who participate in the research.

The case involved two mathematics teachers. Participant Mary (pseudonym) is a Mexican teacher with a bachelor’s degree in basic education, which she obtained in 2017. She has been working at this level since 2017 (seven years of teaching experience) in a rural community in the Mexican state of Puebla, where the Nahuatl language is still spoken. At the time of the interview, he was teaching second grade in a secondary school (students aged 13–14).

The second participant, Laura (pseudonym), was a Colombian teacher with a degree in mathematics. She has three years of teaching experience in primary, secondary, and high schools, one year at each level. At the time of the interview, both were studying for a master’s degree in Mathematics Education.

The data collection instrument consisted of a lesson plan previously carried out by a Colombian math teacher to teach elementary school students the multiplicative structure of natural numbers through problem solving. This planning is based on the Colombian Ministry of Education (Ministerio de Educación Nacional, 1998), which is grounded in competency-based learning. It was obtained from another study by one of the authors of this paper. This lesson plan was used ethically and carefully, without revealing the identity of the developer.

The data collection technique consisted of a semi-structured interview conducted separately with the informants via videoconference at different times. The interviews were conducted until data saturation was reached, which in our case meant identifying the three schemas of the mathematics teacher in order to verify that the proposed conceptual framework was coherent and sufficient. The first interview was conducted with each participant and lasted approximately one hour for Laura and one and a half hours for Mary. The initial data analysis revealed intra- and inter-schemas for both participants, but the trans-schema did not emerge. This schema is more difficult to identify as it relates to a previous inter-schema held by the teacher. Participant Mary had two strong inter-schemas that indicated the possible presence of trans-schemas in this teacher. Therefore, a second interview was conducted, which lasted for 28 min. The analysis of this interview revealed two trans-schemas in this teacher, one of which is presented in the Results section.

For data collection, the informants were first invited to participate in the study, as they were highly regarded as experts and graduate students. Once they agreed, they were informed about the care and confidentiality of their data, and a consent form was signed by both parties.

They were then given a lesson plan and asked to analyze it before the interview. After a few days, the interviews were scheduled for each participant individually and conducted via videoconference through the institutional Google Meet platform, which was videotaped. The aim of the interview was to analyze lesson planning. To do this, the teacher was shown the lesson plan and asked whether she agreed or disagreed with the aspects shown, or what she would change and why. Some questions were related to the knowledge subdomains of the MTSK model, but it was agreed that they should not be asked in a biased way, but in a general way. An example of the type of question asked is as follows: I would like to know what you think about the standards for the core competencies. To explore your knowledge of the Mathematics Learning Standards (KMLS). It was also agreed that the questions should follow the flow of the conversation so that it could develop smoothly. In the second interview, participant Mary was shown parts of the lesson plan and the transcript of her first interview (especially the fragments where we had discovered inter-schemas) and asked about the reasons for her comments and statements.

The information was analyzed in several phases. First, evidence of specialized knowledge was identified in each interview, supported by the categories of the MTSK model (deductive analysis). Next, the fragments of the interviews in which more than one piece of knowledge was identified. In the third phase, we analyzed whether there was evidence of a non-connection, directional connection, or interactional connection. This information was used to determine the type of schema that emerged: intra- or inter-schema. In the final phase, we analyzed the existence of trans-schemas grounded in their inter-schemas. These analyses were performed separately by the researchers to verify that the information was interpreted correctly.

4. Results

The results of the analysis are presented below. First, the fragment of the class planning that was analyzed at the time is presented, followed by the fragment of the interview that refers to this part of the planning. The excerpts from the interviews were arranged in three-column tables. The first column indicates the participating teachers or researchers. The second column contains the interview fragment, and the third column contains the identified knowledge. A color code has been used for each subdomain: turquoise for KoT, olive for KSM, lime green for KPM, gray for KMT, pink for KFLM, yellow for KMLS, and blue for statements that justify a proposal as a whole. In some sentences, which are later referred to as S01, S02, etc., codes have also been placed in parenthesis. The text in the figures has been faithfully translated from the original Spanish.

We begin with the results by showing an intra-schema demonstration by Laura. The researcher was shown the lesson planning objectives (see Figure 1).

Therefore, the teacher was asked the following questions in

Table 3. First interview excerpt with teacher Laura.

Researcher:	Here, in the next part, the goals, I would like to ask you again what you think about this. In one part, he tells us: establish strategies for solving multiplicative problems. I would like to know if you know of any strategies that you have seen children use to solve these kinds of problems.
Laura:	Well, some strategies I’ve seen students use are to make marks like chopsticks and counting their chopsticks to do the multiplication. Also his fingers (S01). These are the strategies that are most commonly used in this case. Like I said, I feel like they’re most often scratching, trying to count with certain strokes, sticks, balls, figures that they make and then counting (S02). When it comes to big numbers, this’s what the students usually do (Laura, interview excerpt, 12 July 2024).	S01 and S02. Forms of interaction with mathematical knowledge, KFLM

:

It can be observed that the teacher explains that students mark sticks or balls or even use their fingers to perform multiplications (S01) and (S02), which indicates knowledge of KFLM, more specifically the category Forms of student interaction with mathematical content. This is an intra-schema, as only this knowledge appears.

At another point in the interview, teacher Laura showed an inter-schema. The researcher showed the expected learning process (see Figure 2) of planning, which states that students should solve everyday situations involving multiplication and division, and it is suggested that the analysis they make of the situations and the resolution strategies they propose should be assessed.

This is what the teacher was asked during the interview (

Table 4. Second excerpt from the interview with teacher Laura.

Researcher:	Would you use this method to introduce multiplication and division with everyday situations right from the start in your lessons?
Laura:	Yes, I would do that and, as I said, recognize very well the context in which the students relate to each other and try to adapt it to that. Yes, I would do that as an introductory activity (S03). Suggest a not very complex situation (S04) where the students tend to multiply, and that they don’t get so caught up in trying to solve the problem as they might with addition, (S05) but get them to see if they can do it with multiplication. Because that can also happen that a situation arises and the student uses repeated addition instead of multiplication (S06) (Laura, interview excerpt, 12 July 2024).	S03. Strategies, techniques, tasks and examples, KMT S04. Phenomenology and examples, KoT S05 and S06. Forms of interaction with mathematical knowledge, KFLM

):

In this excerpt from the interview, the teacher shows an interactional connection between her Teaching strategy (KMT), the proposal of an introductory activity (S03), a real contextualized situation (S04) (Phenomenology and applications, KoT), and the instruction to solve it by multiplication, as she affirms that it may happen that they perform repeated additions (S06) (Forms of interaction with mathematical knowledge, KFLM). Therefore, it is assumed that teacher Laura has an intra-schema.

In the case of teacher Mary, an intra-schema is shown for continuation, in which several pieces of knowledge appear in a non-connection. The context in which this occurred is described below. The researcher was shown the basic competency standards established by the Colombian Ministry of Education (Ministerio de Educación Nacional, 1998) and the basic learning rights for planning (see Figure 3).

Teacher Mary, based on her experience with the methodology proposed by the Mexican Ministry of Education in the 2016 reform, expressed that she did not agree with the way it had been introduced, whereupon the researcher asked (

Table 5. First fragment of the interview with Mary.

Researcher:	You feel that setting certain standards restricts the learning process a little. So how would you design or plan a lesson to help students understand the basics of multiplication and division without setting goals as specific as the ones seen here, standardized, for example.
Mary:	Well, you see, what I really liked were the books of the 2016 reform plan (S07). In that [reform], a theme […] was developed in several sessions. The sessions are the classes, so there were topics that had three sessions, for example. In the first session, we analyzed everything that had to do with prior learning or prior knowledge, for example, when it came to multiplication, we talked about addition, we talked about subtraction, we maybe talked about place value, etc. (S08). But in the second session, we gave the students problems where they had to search using the tools that they already had, for example, the shortcut of addition, the shortcut of subtraction, etc., which in this case gives us multiplication or distribution [respectively] (S09). And at the end we talked about how to apply this in everyday life or extend the knowledge further. Expand. Looking for other sources, looking for a case, I don’t know, that had to do with situations that caught their attention, even if they weren’t in context (S10). (Mary, interview excerpt, 17 July 2024)	S07. Teaching resources, KMT S08. Definitions, properties and their foundations, KoT S09. The practice of problem solving, KPM S10. Phenomenology and applications, KoT

):

In this interview excerpt, various insights can be observed in chronological order. They appear one after the other, but there is no conscious connection between their knowledge; rather, they appear as a description of a curriculum created by this institution. The teacher provides no connecting words to link them together. The conclusion is that this is evidence of a non-connection between knowledge and, thus, an intra-schema.

Below is another example of inter-schema from the same teacher, where an interactional and a directional relationship can be observed. The context of this interview extract is that the researcher was shown the initiation phase of planning, as illustrated in Figure 4. The activity consisted of a video showing the story of a person buying 30 loaves of bread to distribute to his family (six people, including himself) and also having to travel on three buses that cost a certain ticket. They asked themselves two questions: how many loaves are there for each family member? How much did they pay for transportation?

In the interview, the teacher was asked about how the topic was presented and whether it was appropriate to present it in this way. The teacher opted for the answer of how she would work on the topic in class.

Table 6. Second excerpt from the interview with Mary.

Researcher:	I would like to know how this phase is presented here in the plan, for example, and whether you consider it appropriate to present it in this way.
Mary:	Ok, well, I don’t know if it’s appropriate, because I haven’t put it into practice this way. I can’t say if it works, but from my experience, for example, I would first start from the context of the students (S11), from [a] situation or give them a problem, a task where they have to find a way to maybe, I don’t know, distribute 10 candies to five kids (S12) And then they see that “ah! I need to add this many piles of candy to make it enough for everyone” and so on, and then the addition and subtraction will happen naturally. But realize that the path traveled can have a shortcut called multiplication and division (S13). (Mary, fragment of interview, 17 July 2024).	S11. Teaching Strategy, KMT S12. Phenomenology and applications, KoT S13. Forms of interaction with mathematical knowledge, KFLM

shows the interview fragments and the knowledge shown by the teacher.

In this fragment of the interview, an interactional connection between a Teaching strategy (KMT) (S11) and contextualized situations (Phenomenology and applications, KoT) (S12) can be observed. It is noted that this is an interactional connection, as the teaching strategy and knowledge of the context seems to be inseparable. The directional connection is then given as follows: When the teacher talks about a distributive situation (S12) (Phenomenology and examples, KoT), her knowledge of the way the students will interact with that specific situation (KFLM) (S13) arises from saying the dialogue that the students would have in mind (note the word “then” connecting the sentences). The specific situation of the distribution makes her think about how the students will interact with that mathematical content. In this way, the teacher proves an inter-schema.

In relation to Figure 4, the teacher was asked (

Table 7. Third excerpt from Mary’s interview.

Researcher:	If you were presented with a problem of this nature, what steps would you take to solve it so that students would understand it? So how would you solve it?
Mary:	Well, here you tell me that there are six people in the family, so it would be teams of six, and then I would distribute (S14) maybe it’s not loaves, but, I don’t know, seeds, maybe it’s bones, maybe whatever I have on hand, and I would distribute 30, or you draw your own loaves. There will be 30 in total (S15). Maybe there’s a student who likes to draw and does 10, and others are too lazy or can’t do it and only do one or two, but together they add up to 30 (S12), and we also work on collaboration. From there I ask them to cut or divide these loaves, divide them among these people and give them the same amount, because here we have not specified that they get the same amount (S16). Then they will see, oh, well, if I put this many pieces together, I have one loaf, and this many pieces, then I have this many loaves, I have five loaves, that’s it (S17). Well, how did you do it? What problems did they encounter? So that we don’t just focus on one outcome, but also think about that part of the process (S18). Then talk to them about how this person [the problem], well, travels, travels too much and takes three busses to get there and three busses to get back. And if each bus costs a total of $1,750, how are we supposed to know how much he’s going to spend? Well, here you tell me that there are six people in the family, so it would be teams of six, and then I would distribute (S14) maybe it’s not loaves, but, I don’t know, seeds, maybe it’s bones, maybe whatever I have on hand, and I would distribute 30, or you draw your own loaves. There will be 30 in total (S15). Maybe there’s a student who likes to draw and does 10, and others are too lazy or can’t do it and only do one or two, but together they add up to 30 (S16), and we also work on collaboration. From there I ask them to cut or divide these loaves, divide them among these people and give them the same amount, because here we have not specified that they get the same amount (S17). Then they will see, oh, well, if I put these many pieces together, I have one loaf, and these many pieces, then I have these many loaves, I have five loaves, that’s it (S18). Well, how did you do it? What problems did they encounter? So that we don’t just focus on one outcome, but also think about that part of the process (p15) (S19). Then talk to them about how this person [the problem], well, travels, travels too much and takes three busses to get there and three busses to get back. And if each bus costs a total of $1750, how are we supposed to know how much he’s going to spend? (S20). (Mary, excerpt from interview, 17 July 2024).	S14, S17 and S19. Strategies, techniques, tasks and examples, KMT S15 and S20. Phenomenology and applications, KoT S16 and S18. Forms of interaction with mathematical knowledge, KFLM

) what steps she would take to solve this task so that the students could understand it.

In her response, the teacher shows an inter-schema with an interactional connection between her knowledge of strategies, techniques, tasks, and examples (KMT) (S14), (S17), and (S19), phenomenology and applications (KoT) (S15) and (S20), and forms of interaction with mathematical knowledge (KFLM) (S16) and (S18). It is interesting to observe how the teacher describes in detail what she would do in the classroom at each stage. She explains that her teaching strategy would be to form teams of six and ask them to share the 30 loaves of bread (S14). She mentions that she could replace the sandwiches with other materials (S15) or even have them draw the sandwiches. She describes how she thinks the students would work with the math content (S16 and S18) until they solve the problem.

Another example of an inter-schema with an interactional connection was shown by teacher Mary. In the interview, the suggested planning performance was discussed (Figure 5), which states that students will be able to propose strategies to solve problem situations involving multiplication and division.

Therefore, the researcher asked the following question in

Table 8. Fourth excerpt from interview with Mary.

Researcher:	In this part of the performances, he says that the student proposes strategies to find solutions to problems involving multiplication and division. I would like to ask you what situations or strategies you use to help students focus better on certain topics or pay more attention to what is being explained
Mary:	Well, basically my strategy is to structure the knowledge so that it grabs their attention [S21], to show them situations that appeal to them [S22]. In the project method, I look for a problem that worries them so that they get involved in solving it [S23], I motivate them, show videos and so on. I look at them as situations, yes, they are resources that serve the students’ learning channels [S24] (Mary, interview excerpt, 17 July 2024).	S21 and S22. Emotional aspects of learning mathematics, KFLM S23. Strategies, techniques, tasks and examples, KMT S23. Phenomenology and applications, KoT S24. Teaching resources, KMT

:

In this excerpt from the interview, teacher Mary shows in statements (S21–S24) an interactional connection between her Teaching strategy (KMT) (project-based learning) (S23), the Emotional aspects of learning mathematics (KFLM) (S21, S22), Phenomenology and applications (KoT) (S23), and Teaching resources (KMT) (videos) (S24), as this knowledge seems to interact with each other in a fragment of her discourse. It should be noted that their knowledge of Phenomenology and applications (KoT) is enunciative rather than descriptive, as it does not specify a concrete situation.

In the second interview with teacher Mary, she demonstrates a trans-schema. The context is that the researcher was shown Figure 5 and the excerpt from the interview in Table 8, without highlighting, so that the teacher would remember this part of the interview conducted three months earlier, and she was asked the following (

Table 9. Excerpt from the interview with teacher Mary.

Researcher:	Could you explain the reasons for your proposal? Like a synthesis or something. That you explain what it is for you, is that how you do it or is it something very particular?
Mary:	In my experience, I can say that this way of working is more effective, it is more entertaining for the students and it is more efficient when it comes to putting knowledge into practice (S25). I work project-based because that is the method that is used in the New Mexican School and it is very interesting when we are not limited to one format or one time, so to speak (S26). So yeah, I’ve worked on that and the heart of the projects is definitely motivation and context (S27). In any subject, especially in science, it’s too appealing to discover things, to find concepts, to make sense of them (S28) and not just, for example, pose a series of problems that don’t affect them as such. More than for cognitive activity, it is interesting to see them, to experience them, to put them into practice, to test them (S29). […] If we want the student to learn something, he or she must definitely experience it, he or she must live it (S30). Otherwise, there would be no point in giving a video more importance than an experiment or a real problem (Mary, interview excerpt, 10 October 2024)	S25. Reason for the connection between the project-based learning strategy, KMT and the emotional aspects of mathematics learning, KFLM S26. Why she uses the project-based learning strategy, KMT S27. Reason for her teaching strategy, KMT in relation to emotional aspects of learning, KFLM and phenomenology and applications, KoT S28. Rationale for the relationship between emotional aspects in mathematics learning, KFLM and the practice of defining, KPM S29 and S30. Justifications for the proposal in general.

):

In the phrase “this way of working,” it is evident that the teacher conceives of her schema as a whole, which is characteristic of trans-schemas, and she justifies the connection between the project-based learning Strategy (KMT) and the Emotional aspects of learning mathematics (KFLM): “is more effective, it is more entertaining for the students and it is more efficient” (excerpt from the interview in Table 9). Next, in (S27), she argues when mentioning project work, which is the methodology of the New Mexican School. In this statement, she provides a rationale that explains the interactional relationship between her Teaching strategy of project work (KMT), the Emotional aspects of learning mathematics (KFLM), and Phenomenology and applications (KoT). She refers to her inter-schema, which was analyzed in the interview fragment in Table 6.

Next, in fragment (S28), she establishes an interactional connection between the categories of Emotional aspects of learning mathematics (KFLM) and the Practice of defining (KPM). In this way, the teacher establishes a connection between knowledge that did not appear in the interview fragment shown to her. Thus, we can show that there are further connections between knowledge in her inter-schema that she did not demonstrate in the first interview.

Finally, fragments (S29) and (S30) provide two general justifications that show that the teacher perceives this knowledge as a coherent whole by justifying the connections she has made and explaining its applicability in the classroom. From this, it can be concluded that she has a trans-schema connected to the inter-schema she was shown.

5. Discussion and Conclusions

This study contributes to the theoretical development of the MTSK model by explaining the connections between its knowledge elements. However, its scope extends beyond the model itself and may be useful for similar phenomena in other theoretical frameworks. Schemas have been extensively studied in APOS theory, a cognitive theory that describes and predicts the types of mental structures that students need to build in order to learn abstract concepts (Arnon et al., 2014). Arnon et al. (2014) describe an individual’s schema for a mathematical topic as their total knowledge associated (explicitly or implicitly) with that topic. It is a framework in an individual’s mind that can be applied in a problem situation involving that concept. In our study, schemas were proposed to understand the degree of deepening of the connections between the knowledge in the mathematics teacher’s mind that occurs when he reflects on his profession, analyzes, teaches mathematics, or, more generally, performs activities.

We consider it important to emphasize that the teacher must show their knowledge in a continuous fragment of their discourse in order to be able to speak of the emergence of connections or non-connections. The results of this work show the existence of three types of schemas that show a congruence between the constructs of non-connection, directional connection, and interactional connection and the definitions of intra-, inter-, and trans-schema proposed by Piaget and García (1989).

This study also contributes to the understanding of the simplicity or complexity of schemas. In the first case, two examples of intra-schemas are shown: one in which only a single piece of knowledge is evidenced, and another in which several pieces of knowledge are evidenced in a temporal sequence that turned out to be non-connected after analysis, in part because the linking words that connected them could not be identified. In the second case, the participating teachers’ inter-schemas presented both types of connections, i.e., directional and interactional. In one of these cases, a trans-schema was established in which the teacher justified the connections created in her inter-schema. In addition, an interactional connection between the KFLM and KPM subdomains, which had not previously appeared in her inter-schema, was shown. This shows the deep complexity and stability of the schemas that a teacher can develop in their mind.

In Professor Laura’s intra-schema shown in Table 3, it can be seen that the teacher knows how students usually make marks with sticks, lines, spheres, or figures when solving multiplication and division problems with natural numbers (forms of Interaction with Mathematical Knowledge, KFLM). In this way, it is possible for the teacher to know that the students are in the concrete operations stage (Jun & Kan, 2024), in which children can only solve problems that relate to concrete events or objects. This intra-schema is linked to her inter-schema, as shown in Table 4. There, the teacher suggests working with contextual situations involving multiplication and division (Phenomenology and Applications, KoT), in which students tend to multiply and do not try to solve the problem through repeated addition (forms of Interaction with Mathematical Knowledge, KFLM). In this way, it can be observed that the teacher understands the importance of proposing contextualized situations with concrete materials to students in order to promote their understanding of mathematics, as found in some studies (see, for example, Prosser & Bismarck, 2023).

The study also sheds light on how teachers use knowledge connections based on the MTSK model in their teaching and classroom activities. Mary’s trans-schema (Table 9) shows how she develops her lessons to promote motivation and interest through project-based learning. In doing so, she presents contextualized situations that interest students so that they feel engaged in their solutions and develop mathematical concepts that give them meaning. This teacher’s approach to teaching is consistent with the findings of various studies that have examined project-based learning. It enables elementary students to develop a deeper understanding of mathematical concepts, helps develop critical thinking skills, and makes learning more interesting and relevant for students, which increases their engagement and motivation (see, e.g., Arrieta-Cohen et al., 2024; Lzic et al., 2021). In this way, the teacher demonstrates a trans-schema that is consistent with the research on this topic. Both teachers agree that it is necessary to work with contextualized situations and concrete materials to help primary school students understand multiplication and division with whole numbers.

The fact that the lesson planning analyzed by the informing teachers was not done by them allowed for some disagreement with what was written in it, motivating reflection, analysis, and reasoning about why they were in discrepancy with each other and how they would do it, which favored the emergence of their knowledge schemas by deepening their connections. This can contribute to the methodological design of future research.

While the study conducted by Tascón (2024), in which lesson planning was obtained, focuses on analyzing the specialized knowledge of two teachers and determining how this knowledge from the subdomains of MTSK is related, the study presented here focuses on the interpretation and analysis of a planning and shows how different teachers can perceive and connect the mathematical and didactic knowledge present in the same lesson planning. The differences in knowledge schemas (intra-, inter-, or trans-) and the ways in which they are arrived at show the diversity of understandings and the richness of teachers’ expertise and highlight the importance of reflection and in-depth analysis of their knowledge.

The conceptual framework proposed by the constructs of non-connection, directional connection, and interactional connection, as well as intra-, inter-, and trans-schemas, allows a deeper understanding of the level of complexity of the knowledge connections that a mathematics teacher can build in his mind within the MTSK model. Therefore, this work contributes to the theoretical development of MTSK. However, we are aware that mathematics teachers’ knowledge is extremely complex, dynamic, and constantly evolving, so this paper only opens a door to understanding it from the perspective of the MTSK model.