Linear algebra in engineering: a study of specialized knowledge of Chilean and Brazilian teachers

Abstract

The objective of this paper is to establish comparisons between the Linear Algebra (LA) approach practiced in Engineering courses offered by Chilean and Brazilian Higher Education Institutions and, for that, teachers from both countries were interviewed. From a methodological point of view, the research consists of two case studies. For the analysis of the interviews, we used the precepts of Content Analysis. The data were qualitatively analyzed considering the MTSK theoretical model. In this analysis, we sought to identify what specialized knowledge these teachers have regarding AL and about its teaching and learning. The results have shown that, in relation to the Mathematical Knowledge (MK) domain, which refers to content knowledge, the Knowledge of Topics subdomain (KoT), linked to the way that teachers know the topics they teach, is the most present among interviewees. In the Pedagogical Content Knowledge (PCK) domain, covering specific pedagogical knowledge linked to Mathematics, the Knowledge of Mathematics Teaching (KMT), regarding strategies, teaching theories, resources and materials, and Knowledge of Mathematics Learning Standards (KMLS) subdomains, concerning the curricular specifications and the moments in which the student must or can learn certain content and with what level of depth prevail.

1 Introduction

This article is the first product of a research project carried out jointly by Brazilian and Chilean teachers within the scope of the PEPG Excellence Program (PEPG-Ex) of the Pontifical Catholic University of São Paulo (PUC-SP) and ‘The Teaching of Mathematics in the Initial Years of Engineering Courses via the Didactic Model of Mathematics in Context’ project linked to the Mauá Institute of Technology. The objective of this project is to establish comparisons between the Linear Algebra (LA) approach practiced in Engineering courses offered by Chilean and Brazilian Higher Education Institutions and, for that, teachers from both countries were interviewed.

But what is the relevance, for the area of Mathematics Education, of carrying out comparative studies on the teaching of Mathematics in different countries? Are the results of such studies of interest only to researchers from the countries involved or can they also contribute to expanding knowledge in the area at a global level?

According to Tchoshanov et al. (2017), this type of study is not only globally relevant but has increased in the last decade. These authors point out that the potential of this research modality is related to the opportunities provided to ‘compare, share and learn about issues in an international context, which, in turn, helps researchers to understand their own context, their own teaching practices, teachers’ knowledge and students’ learning’ (Tchoshanov et al., 2017, p. 1658).

The benefits of knowing comparisons about similarities and differences in teaching and learning processes in different countries are also highlighted by Andrews (2009), according to which, such knowledge makes us ‘more aware of our own implicit assumptions about mathematics learning’ (Andrews, 2009, p. 98). In many cases, researchers from a given country ‘cannot appreciate their academic strengths until they compare them with those of other nations’ (Vistro-Yu, 2013, p. 146).

Shimizu & Kaur (2013) complement this point of view by stating that: ‘by looking at similarities and differences in educational policies and practices across cultures and contexts, international comparative studies offer opportunities to verify typical dichotomies such as “high performance” versus “low performance”, “teacher-centered” versus “student-centered” [...] often found in recent educational debates’ (Shimizu & Kaur, 2013, p.1). Tchoshanov et al. (2017) point out that, although the number of international comparative studies has increased in the last decade, there are still few investigations that focus on teachers’ knowledge.

Bianchini et al. (2019) reported that ‘the relevance of Linear Algebra (LA) is well known for different fields of knowledge and also the difficulties faced by engineering students when studying this content’ (Bianchini et al., 2019, p. 1097). In the same publication, the authors emphasize the importance of researchers highlighting the knowledge of teachers who teach LA in Engineering courses, especially specialized knowledge, in order to provide them with initial or continuing training to prepare them for the development of ‘an interdisciplinary work that requires the establishment of dialogues between LA teachers and those from specific areas, in addition to the planning of didactic-pedagogical organizations that go beyond the so-called traditional classes, based on content exposition’ (p. 1108).

In summary, what we have mentioned in the previous paragraphs highlights: the relevance of carrying out comparative studies regarding the teaching of Mathematics in different countries; the need for a greater number of investigations of this nature focusing on teachers’ knowledge and, finally, the importance of identifying the knowledge of teachers who teach Mathematics—and particularly LA—in Engineering.

In the research that originated this article, a study on the specialized knowledge of some teachers who teach LA in Engineering courses in relation to this area of Mathematics, its teaching and learning, was carried out. To achieve this objective, four Chilean teachers, who teach LA at the same university, and six Brazilian teachers from six different institutions, who teach the same subject, were interviewed.

The option to compare Brazil with Chile is justified by the fact that there are more cultural and educational similarities between two Latin American countries than, for example, between Brazil and a country on the European continent. In addition, in a perspective that has been called Decolonial, we understand how important it is, in Mathematics Education, to reflect on teaching practices and knowledge made explicit in different geographic locations. The objective of these Decolonial studies is to contribute to the deconstruction of Eurocentric narratives that, in general, have long time has determined knowledge, ways of being in the world and teacher training processes (Fernandes, Giraldo & Matos, 2002).

From the theoretical point of view, some assumptions of the MTSK model—Mathematics Teacher’s Specialized Knowledge were adopted as approach.

In the following, the mentioned theoretical framework, the methodological aspects of the research and the characterization of teachers interviewed in Chile and Brazil are explained. Then, we focus on the analysis of answers to some specific questions related to the main difficulties faced by students when studying LA; if they, playing the role of teachers, resort, during classes, to some digital technological resource; if there are dialogues between LA teachers and those who teach specific Engineering disciplines so that the approach is linked to the course objectives; and, finally, if contextualized problems in the Engineering area are addressed. Our objective, through this analysis, is to make inferences about the specialized knowledge of interviewed teachers about LA and the teaching and learning of this area of Mathematics in Engineering in the light of the reference framework presented below.

2 The MTSK model

The Mathematics Teacher’s Specialized Knowledge—MTSK model, which from now on is referred to as MTSK, was idealized, according to Carrillo et al. (2013a) and Carrillo et al. (2013b), in theoretical and practical agreement to a specialty of the knowledge manifested by Mathematics teachers in their teaching performance. It is a continuation of studies carried out by Lee Shulman on Pedagogical Content Knowledge (PCK) and Deborah Ball and collaborators concerning Knowledge of Mathematics Teaching (KMT), which brings as main contribution the idea that the knowledge necessary for the Mathematics teacher to perform tasks (planning, teaching and reflecting on classes, exchanging ideas with colleagues, etc.) must always be analyzed in view of its specialized nature.

In this sense, Carrillo et al. (2018) advocate that the specificity of teacher knowledge affects both mathematical knowledge (identified in the model by MK) and pedagogical content knowledge (which, in the model, is denoted by PCK), which implies that it is not adequate to consider specialized knowledge as a MK or PCK subdomain. The objective pursued in such a model is not to assess the specialized knowledge of mathematics teachers, but to understand and interpret them (Carrillo et al., 2018).

In our study, in view of our agreement with the specialized nature of the knowledge of teachers who teach LA in Engineering, we chose to resort to a theoretical model that takes this aspect into account and not a model that understands the specialization of knowledge only as a subdomain of teaching knowledge in a broader view. This is the justification for choosing the MTSK and not the MKT model, authored by Ball and colleagues to subsidize our analyses.

As mentioned in the previous paragraph, in the MTSK model, two knowledge domains are distinguished: MK (mathematical knowledge) and PCK (pedagogical content knowledge), which contain six subdomains, three of the MK domain—knowledge of topics (KoT), knowledge of the structure of mathematics (KSM) and the knowledge of practices in mathematics (KPM)—and three others in the PCK domain—Knowledge of Features of Learning Mathematics (KFLM), Knowledge of Mathematics Teaching (KMT) and Knowledge of Mathematics Learning Standards (KMLS). The model structure is schematized as shown in Fig. 1.

Regarding Mathematical Knowledge (MK), Carrillo et al. (2018) emphasize that they understand.

[...] Mathematics as a systemic knowledge network structured according to its own rules. Having a good understanding of this network—nodes and connections between them—the rules and characteristics related to the process of developing mathematical knowledge allows teachers to teach the contents in a connected way and validate their own mathematical conjectures and those of their students. Thus, we divided the mathematical knowledge into three subdomains: the mathematical content itself (Knowledge of Themes), the systems that interconnect the different themes (Knowledge of the Structure of Mathematics) and how mathematics works (Knowledge of Practices in Mathematics) (Carrillo et al., 2018, p. 6).

In turn, in relation to Pedagogical Content Knowledge (PCK), the authors state that:

It is the area of knowledge that most closely concerns to classroom practice. However, we consider that PCK represents only part of the set of knowledge for teaching, and needs to be complemented by MK. Working together, they subsidize and guide the decisions and actions of teachers in the course of their practice. The specific focus of PCK is related to Mathematics itself. More than being the intersection between Mathematics knowledge and General Pedagogy, it is a specific type of Pedagogy knowledge linked to Mathematics. Therefore, we do not include in this subdomain the General Pedagogy knowledge applied to mathematical contexts, but only that knowledge in which the mathematical content determines the teaching and learning practices. It is in this domain that the literature in Mathematics Education plays an essential role as a source of knowledge for teachers (Carrillo et al., 2018, p. 11).

We then proceed to present details about each of the MK and PCK subdomains.

In relation to the KoT, the authors state that it ‘describes what and how the Mathematics teacher knows the topics, which implies a deep knowledge of the mathematical content (e.g., concepts, procedures, facts, rules and theorems) and their meanings’ (Carrillo et al., 2018, p. 7). Through Table 1, we explain the different elements that make up this knowledge subdomain.

KSM includes knowledge of the main ideas and structures of Mathematics, such as knowledge of properties and notions relating to specific items being worked on, knowledge of connections between topics currently being treated, as well as precedent and subsequent ones. Regarding connections, Carrillo et al. (2018) point out that in KSM, only inter-conceptual ones are considered. Intra-conceptual connections and those with the contents of other disciplines are considered in KoT. In summary, it could be said that the essence of KSM is the knowledge of connections established between mathematical objects (Contreras et al., 2017). In Table 1, we explicitly present the elements constituents of the Knowledge of the Structure of Mathematics.

Finally, KPM is related to the ways of proceeding, creating or producing in Mathematics (syntactic knowledge); it also includes aspects of mathematical communication; reasoning; verification and explanation of counterexamples about some statement. It also includes knowing how to define, use definitions and understand the characteristics of definitions and their roles in the construction of mathematical knowledge; establish relationships (between concepts, properties, etc.), correspondence and equivalences; select representations; to argue; generalize and explore. According to Flores-Medrano et al. (2014, p. 8), the elements of KPM are relevant for teachers both to ‘give solidity to their own knowledge, and to know how to manage the mathematical reasoning used by students when accepting, refuting or refining them’.

We now explain the main characteristics of each of the Pedagogical Content Knowledge (PCK) subdomains.

KFLM is related to the teacher's knowledge of how students think when faced with a mathematical activity or task, what difficulties they may find when studying a specific mathematical topic. It also includes knowledge about theories and models related to learning Mathematics. In KFLM, ‘the main focus of the learning process to be analyzed turns to the mathematical content as a learning object, with the interest of understanding the knowledge related to the learning characteristics resulting from the student's interaction with the mathematical content, and not in the characteristics of the former’ (Lima, 2018, p. 91). For Carrillo et al. (2018, p. 11), ‘the main sources of knowledge of teachers within this subdomain tend to be their own experience accumulated over time together with the results of research in Mathematics Education’. The authors also state that the KFLM subdomain also includes knowledge about the main mistakes and misconceptions that, in general, students make when studying a certain mathematical content, in which areas they have greater obstacles and in which they present greater ease, knowledge about the different learning styles, about different ways of perceiving the inherent characteristics of certain contents, etc. Table 2 shows the elements that constitute KFLM.

KMT is related to the knowledge of teaching strategies and theories, resources and materials. It is the type of knowledge that allows the teacher to choose a specific material or textbook to help students learn a mathematical concept or procedure and to select examples that can help students understand the meaning of a mathematical object. KMT integrates mathematical knowledge and knowledge related to the teaching of mathematics, not encompassing aspects of general pedagogical knowledge, since, according to Flores-Medrano et al. (2014), this subdomain includes knowledge intrinsically dependent on mathematical themes themselves. ‘It involves awareness of the potential of activities, strategies and techniques for teaching specific mathematical content, along with any potential limitations and obstacles’ (Carrillo et al., 2018, p. 12). The constituent elements of KMT are presented in Table 2.

According to Lima (2018, p. 91), from the MTSK perspective, a learning standard ‘is understood as an “instrument” that indicates the level of skills assigned to students at a given school moment so that they can understand, build and know mathematics’. Assuming this idea as a premise, it is emphasized that the KMLS is related to the knowledge of curriculum specifications, regulations of different associations of Mathematics and Mathematics Education about what, at any given moment, the student should or could learn, both in terms of content and in terms of depth levels. This subdomain also includes the knowledge of results of investigations in the area of Mathematics teaching, as well as the opinions of expert teachers. In addition, this category of knowledge includes the minimum curricula required by professional associations and external organizations responsible, for example, for large-scale assessments. Table 2 summarizes the constituent elements of KMLS.

The MTSK model also includes the teachers’ beliefs both in relation to Mathematics and in relation to the teaching and learning of this science. According to Flores-Medrano et al. (2014, p. 88), these are placed ‘in the center of the scheme and with dotted lines (once) about the teachers’ beliefs in relation to Mathematics, its teaching and learning permeate the knowledge they have in each of its subdomains’. In this article, as this is not the objective, we did not perform an in-depth analysis of the issue of beliefs.

In the next section, we present a brief overview of investigations already carried out by other authors about the teaching and learning of LA, using the MTSK model as theoretical framework.

3 Overview of LA-related investigations supported by the MTSK model

In this section, we seek to outline a small overview of LA-related investigations supported by the MTSK model related to LA in Engineering courses. Obviously, there are investigations using the aforementioned model to analyze the specialized knowledge of teachers who work at other levels of education or in university courses other than Engineering and who turn their attention to teachers who teach topics other than LA. However, in this article, as our focus of study is the knowledge of teachers who teach LA in the training of future engineers, we restrict our literature review to this specific context.

As will be evidenced from data presented, the mathematical objects usually focused on in such research are matrix, determinants and systems of linear equations. We did not find investigations related to other LA themes.

The MTSK model has been used in several studies on issues related to LA, its teaching and learning. Vasco-Mora (2015) analyzes, through a case study, the specialized knowledge of two LA teachers, who work in Engineering, through the analysis of their educational practices when approaching the contents matrix, determinants and systems of linear equations. One of the limitations of this study, pointed out by the author herself, is the fact that data were obtained only through non-participant observations in classes and through interviews. In her view, analyses could have been richer if she had proposed a series of activities and problems to teachers, which should develop in classroom, favoring greater interaction with students. The main contribution of this thesis to our research concerns the validation carried out by the author of the MTSK model as an analysis tool to characterize the specialized knowledge of a Mathematics teacher at university level and, especially, of LA.

Sosa et al. (2015), with the objective of advancing in the characterization of one of the subdomains of pedagogical content knowledge (PCK), namely the KFLM, developed a case study with two LA teachers in order to identify, know and understand indicators for KFLM. The authors obtained ten indicators and grouped them into three categories: (a) language and processes with which students interact with the content; (b) errors and difficulties associated with learning; and (c) ideas related to theories and ways of learning. As the main limitation of their research, Sosa et al. (2015) indicate the fact that the study was carried out only for a particular case of KFLM. Reading this work inspired us to organize the analysis of our data into categories related to indicators of teachers' specialized knowledge, inferred from the answers given by them during interviews.

The same authors, in Sosa Guerrero et al. (2016), once again using a case study, analyzed the knowledge of teachers, focusing particularly on one of the MTSK subdomains: KMT. The results of the research showed which knowledge teachers manifest about the potential and didactic use of examples and about the different ways to help students in the process of knowledge construction. The limitation of the study, in the authors' point of view, lies in the fact that they explored only KMT, despite being aware of the existence of interrelationships between the different model subdomains. The study of this article inspired us, when analyzing the transcripts of interviews carried out, to look for examples that teachers use in classroom, an inspiration that gave rise to one of the analysis categories of this research, namely the concern of interviewees in contextualizing the contents to be taught.

Vasco-Mora et al. (2015), from the observation, in a case study, of the practice of a university LA teacher when developing the topic of matrixes and determinants, analyzed his specialized knowledge. Knowledge related to the KoT subdomain was mainly identified, which is particularly useful, in the authors' view, for the identification of the teacher's specialized knowledge. In this article, data were organized for analysis following the principles of Content Analysis according to the conception of the French researcher Laurence Bardin. The authors, resorting to what is advocated in this methodology, selected in the teachers' statements elements that, in their views, signaled the mobilization of knowledge inserted in some of the MTSK subdomains. Inspired by this work, we then decided to follow the same procedure for the organization and categorization of data produced from interviews with Brazilian and Chilean teachers in the research carried out. As a possible limitation of the study that they developed, Vasco-Mora et al. (2015) point out that the instruments used for data collection (classroom observation and interviews) may have limited the scope of knowledge revealed by research subjects. In their perceptions ‘other subdomains and categories of specialized knowledge may be revealed through the use of other data collection tools’ (Vasco-Mora et al., 2015, p. 3288).

The same authors, in Vasco-Mora et al. (2016), analyzed, having the case study as methodology, the specialized knowledge evidenced by a university LA teacher when teaching a class on matrix multiplication and concluded that this teacher seems to explain the origins of the aspects that are most present in the mathematical work proposed by such teacher. As the authors point out in this work, we understand that it is relevant to understand the knowledge of teachers who teach LA classes in Engineering courses because such knowledge may explain the type of approach given to this area of Mathematics in the training of future engineers. The reading of this work contributed, in our research, to ratify our understanding. The limitations of the study developed by Vasco-Mora et al. (2016) are, according to the authors, those linked to the very limitations of models that supported the investigation, the MTSK in conjunction with the idea of the Professor’s Appropriate Mathematical Workspace which, according to the authors, refers to ‘an environment designed and organized to facilitate mathematical work, in which the teacher uses strategies and promotes tasks that articulate the components of this environment’ (Vasco-Mora et al., 2016, p. 223).

In Vasco-Mora & Climent-Rodrígues (2017), the authors, with the objective of understanding which knowledge and beliefs are used by two LA teachers as personal resources that guide their practices, with regard to contents matrix, determinants and systems of linear equations, collected, through a case study, data through videos and semi-structured interviews and concluded that: elements related to MK, PCK and evidenced beliefs seem to be consistent with each other and their relationships made it possible to identify key issues in their practices, such as the use of examples and error. The authors point out that the preliminary results of the study could be a starting point for the structuring of teacher training and development, in which the role of examples to help students overcome difficulties in LA was considered. Not specifically as a research limitation, but as an unexplored aspect that may be the subject of further investigations, the authors emphasize the importance of broadening the view of the relationship between teachers’ specialized knowledge and their conceptions about teaching and learning of Mathematics and, in addition, to investigate how these can influence, among other aspects, the teacher’s beliefs in the construction of specialized knowledge. Vasco-Mora et al. (2015) served as an inspiration for our research on how to use the Content Analysis methodology, articulated to the MTSK model to, based on data collected through interviews with LA teachers, establish categories of analysis and highlight specialized teaching knowledge.

Vasco-Mora & Climent-Rodríguez (2018), through a case study, analyzed classes of two university LA teachers during their approaches on topics matrixes, determinants and systems of linear equations in order to identify which knowledge subsidizes their practices. The authors observed a ‘conceptual and procedural emphasis with evidence of KoT, referring to procedures, phenomenology and applications, representation records, definitions and properties’ (p. 129). They also identified elements related to ‘Knowledge of Features of Learning Mathematics (KFLM), regarding students’ difficulties; and KMT, in relation to teaching examples’ (p. 129). The main contribution of this work to our study was to show the potentialities of the analysis of the specialized knowledge of LA teachers to understand the teaching of this content, from another perspective, other than those destined to the actions carried out in this teaching and to, from the explanation of using this knowledge, develop interventions aimed at contributing to the professional development process of teachers who teach LA in Engineering. In turn, the limitation of the work, according to the authors, lies in the fact that they did not investigate the teachers' knowledge in situations other than those analyzed in this study. For example, they did not investigate the possible knowledge of teachers about extramathematical applications of other LA contents and what importance they attribute to them in teaching. Thus, according to the authors, we cannot ‘say that there is a bi-univocal correspondence between teachers’ specialized knowledge in relation to this content and the teaching of this topic’ (Vasco-Mora & Climent-Rodríguez, 2018, p. 144).

Gutiérrez-Garay (2019) aimed to identify, resorting to the case study methodological strategy, the specialized knowledge of two LA teachers of the Engineering course, about systems of linear equations. The authors concluded that the research subjects have the content knowledge necessary to solve the tasks proposed in the texts of the level they teach; however, in relation to the specialized knowledge of such content, that is, what is desirable for a teacher, in some aspects, it is limited or non-existent. The contribution of the work of Gutiérrez-Garay (2019) to our investigation concerns the fact that the author emphasizes the relevance of studies of this nature, especially involving LA teachers in Engineering, in the context of Latin American countries, since, since according to the author, these are still relatively scarce. The main limitation of the work, according to Gutiérrez-Garay (2019) is linked to the fact that the teachers (subjects of their investigation) did not consent to the recording of their classes, which, consequently, led to a restriction in data collection.

In a work carried out in 2020, Vasco-Mora and Climent-Rodriguéz, through a case study, investigated the knowledge of a university teacher of students’ errors in relation to matrixes and determinants. ‘The results show the teacher’s knowledge of common errors, which may have different origins, in the learning of the aforementioned contents, as well as their use in teaching, aiming to overcome such errors’ (Vasco-Mora & Climent-Rodríguez, 2020, p. 97). This work, in addition to being quite recent, attracted our attention for the fact that it presents a favorable argument for carrying out studies such as the one we had carried out, with which we are fully in agreement, with regard to a latent concern in our research: the explanation of the specialized knowledge of teachers who work in higher education institutions can be a means for elaborating teacher training programs that contribute to the implementation of university practices cantered on the student, in order to overcome the paradigm of predominance of lectures, in which the professor is the protagonist. Not as a limitation, but as an opportunity to unfold the study, the possibility of investigating how university teachers who have experienced some specific training on the mistakes commonly committed by LA students use such knowledge in their teaching activities is evident, since, as emphasized by the authors, the study they carried out had as subjects, teachers who had not received any special training in this regard.

Finally, Vasco-Mora et al. (2021), with the objective of deepening the understanding of the practice of a LA teacher, analyzed, based on a case study, in the work with the topic matrixes, the connections made by the teacher between his content knowledge (MK) and pedagogical knowledge (PCK). Data collection took place through video recordings of classes and semi-structured interviews with the teacher and data allowed establishing connections between the teacher’s knowledge about KoT, KFLM and KMT subdomains. Such connections became evident in the teacher’s practice, especially in the use of varied examples to introduce new content and helping students about possible errors and difficulties to be faced in the study of the topic. The study by Vasco-Mora et al. (2021) attracted our attention, especially for emphasizing the relevance of studying LA teaching and learning at the university not from the perspective of difficulties faced by students, an option made in most of the investigations, but from the point of view of the knowledge mobilized by teachers. Once again not as a limitation, but as a path to be followed in future research, the authors suggest carrying out studies on teachers’ knowledge regarding the interconnections between KSM, KPM, the categories of KMT and those related to the KFLM.

In view of the scenario outlined from investigations presented in this section, it was observed that there are some gaps to be filled and, in the research reported in this article, we sought to take a first step towards filling one of them, namely: the indicators of specialized knowledge of teachers who teach this discipline in Engineering courses, that is, evidence, from the answers to questions proposed to interviewees, of knowledge linked to the different MTSK subdomains. For that, we limited to analyze the specialized knowledge of Chilean and Brazilian teachers related both to LA and to the teaching and learning of this discipline in Engineering, assuming for this analysis the categorization proposed in the MTSK model.

In the following section, we will present the methodological procedures that allowed us to operationalize the MTSK model, placing it in the specialized knowledge of a Mathematics teacher who teaches LA in Engineering courses.

4 Methodological procedures and characterization of the interviewed teachers

This investigation, from the methodological point of view, is qualitative in nature and is characterized as a field research, according to Marconi & Lakatos (2003). It was carried out through two case studies, considered in the view of Stake (2000), for whom the option for this type of investigation is linked to the observation of a group of individuals who share a certain similar experience or the same specific environment. In the case of this research, the specificities of subjects are that they are all Mathematics teachers who teach LA in Engineering courses. It should be noted that the option for the case study is explicit in all investigations mentioned in our literature review. In addition, in these previous studies, reduced groups of teachers were considered; therefore, there are no intentions of broad generalizations of results obtained, nor of representativeness, in statistical terms, of selected subjects. As expected in a qualitative research, from the participation of volunteers interested in contributing to the investigation, reflections on specific cases are sought from which other researchers can carry out similar studies, compare the results obtained, expanding the scope of the analyses and, consequently, building a knowledge base about the theme under analysis. According to Van Zanten (2004), for the generalization of a study, a fundamental element is the comparison. The author states that: ‘if we have a local situation and other local situations that have already been studied, we can resort to comparisons to show that there is a process that is possible, or not, to generalize, as well as to mark the limits of generalization. [...] the comparative work can be within the research and there are comparative works outside it [...]. Comparison allows building the generalization of internal processes and establishing comparisons in relation to external ones’ (Van Zanten, 2004, p. 39–40).

In this article, we seek, at the end of the analyses, to establish a comparison between the results we obtained and those resulting from the investigations that made up the literature review of this work.

As mentioned in the introduction, this article was produced within the scope of an internationalization project developed between institutions to which the authors of this article are linked. Due to the interest of its participants in LA and the teaching of Mathematics in Engineering, due to the fact that one of the Brazilian authors has long been one of the main researchers on LA teaching and learning in the country, and since the Chilean author is one of the great exponents of the subject, we chose to start collective investigations seeking to identify and compare, still in an exploratory way through two case studies, the knowledge of some Chilean and Brazilian teachers who teach LA.

A general characterization of teachers, subjects of our investigation, is presented in Table 3, which shows aspects related to the (fictitious) name, the country in which they teach, academic education and teaching time. Given the previously mentioned nature of the research—case study with qualitative approach—subjects were not selected in order to compose a representative sample of their institutions. Teachers were selected based on indications and for making themselves available to voluntarily participate in the research.

As evidenced by data presented in Table 3, in relation to the academic education of teachers, it was observed that among Brazilians, only one does not have a doctorate degree, while among Chilean interviewees, the maximum title is a master’s degree and half of them have only degree in mathematics. In the case of Brazilian teachers, the predominant graduate degree is in the area of Mathematics Education, while among Chileans, there is one teacher with master’s degree in this area and another in Mathematics. Regarding teaching time, in Brazil, there is predominance, among subjects, of teachers with 10 to 20 years of professional experience and professors with more than 30 years of experience. In Chile, in turn, teachers with 10–20 years of experience predominate among interviewees. Considering the ten interviewees, we highlight that only two (one Brazilian and one Chilean) who have less than 10 years of teaching experience. Among Chilean deponents, all work in private Higher Education Institutions (HEIs), while among Brazilians, there are four teachers from private HEIs and two from public HEIs.

As for the methodology, from now on we call: Case 1: Brazilian Teachers and Case 2: Chilean Teachers. When studying these cases, we will proceed as summarized in Table 4 and will describe in more detail below.

For data collection and analysis, as mentioned in Table 4, the stages described below were adopted.

First Stage: We started work in Chile in November 2019 and, in this first meeting, we sought to know how Engineering courses are being developed at the institution where the research would be carried out. In this first stage, we aimed to understand how LA is taught in Engineering courses offered at the institution. The teacher who received us explained about how LA is taught in courses, which contents are worked on and explained that teachers who teach specific disciplines in Engineering point to the need for changes in the way in which basic disciplines, including LA, are taught in the initial years of these courses. We then decided that we would interview both Chilean teachers and teachers from different Brazilian institutions, and then, from data collected, explain, using the MTSK model, indicators of the specialized knowledge of these professors who teach LA in Engineering in these countries, both in relation to this subarea of Mathematics in terms of its teaching and learning.

Second Stage: As the first step of this stage, we interviewed the coordinator of the Mathematics area for Engineering at the Chilean institution with the aim of having a global view of the subject under study and if she could indicate teachers for future interviews.

Third Stage: The coordinator indicated a list of four professors with whom we should dialogue to collect data for our investigation. The coordinator scheduled our meetings with these teachers. We developed a semi-structured interview script (Laville & Dionne, 1999) to be used to obtain data from these teachers. Each of the interviews, carried out individually, lasted approximately one hour and was audio-recorded with the permission of interviewees.

Fourth Stage: Using the same script, we conducted interviews with six teachers from six different Brazilian HEIs. The Brazilian interviewees were selected based on previous knowledge of the authors of the article about teachers who teach LA to Engineering courses and who, therefore, could contribute to the investigation. In Chile, interviews were audio-recorded, with the permission of participants.

Fifth Stage: After data collection, all recorded audios were transcribed in full and later textualized, and in this action, we chose to exclude language vices from the transcribed text and fill in possible gaps, with the objective of making the reading of documents produced from testimonies more fluent (Garnica, 2003).

In this article, we focused on the analysis of data obtained from the teachers’ answers to some specific questions that make up the script, namely, questions number 4, 7, 8 and 9. When asking such questions to interviewees, our objective was to seek elements about aspects mentioned in the second column of Table 7. The answers obtained allowed us obtaining indicators related to MTSK subdomains, explained in the third column of Table 5.

For the organization of collected data, we resorted to the precepts of Content Analysis, in the conception of Bardin (2006), understood as one of the possible techniques to treat data in qualitative research. According to the author, Content Analysis can be defined as a set of communication analysis techniques aimed at obtaining, through systematic procedures and objective description of the content of messages, indicators (quantitative or not) that allow the inference of knowledge regarding the conditions of production/reception (inferred variables) of these messages. (Bardin, 2006, p. 44).

We then proceed to describe how this methodology was used. The corpus of documents with which we work is composed of textualizations of respondents’ answers to the four mentioned questions. After textualizations, we started what Bardin (2006, p. 122) calls fluctuating reading, which ‘consists of establishing contact with the documents to be analyzed and getting to know the text, allowing oneself to be invaded by impressions and guidelines’.

Due to the objective of our study and the procedures used, we can consider that the themes present in the four questions provided us, a priori, with the categories of analysis that, later, were related to the different types of specialized knowledge of teachers who teach LA in Engineering courses that are, from the MTSK model, presented and discussed in this article, namely: technological resources used in LA classes, difficulties faced by LA students, dialogues between LA teachers and those of specific Engineering disciplines, concern to contextualize LA contents in Engineering. As a result of this fluctuating reading, we obtained indices, which according to Bardin (2006, p. 126), can be explicit mentions of certain themes, which began to guide us in ‘operations of cutting the text into comparable units of categorization for thematic analysis’.

The next step was the treatment of the corpus. In this step, we carry out its codification, ‘an action that includes the cutting of the text into the so-called units of analysis, and the elaboration of the categories of analysis from the classification and aggregation of such cuttings’ (Bardin, 2006, p. 129). In this case study, cuttings were made by registration units, which are ‘units of meaning to be coded and correspond to the content segments to be considered as base units’ (Bardin, 2006, p. 130). The recording units sought in the textualizations that make up the corpus of analysis were the excerpts from the teachers’ answers to questions 4, 7, 8 and 9, in which they made reference to: teaching resources; didactic strategies; their practices; errors and difficulties with mathematical content; dialogue with the specific area of Engineering and its influence on the design of the LA discipline program, on the use of practical applications of contents in classroom and its potentialities for teaching; the purpose of the LA discipline; and relevance or not of knowing practical applications of contents and their potential for teaching.

Finally, we categorize data obtained from reading the selected texts, that is, we grouped them into previously established categories.

When discussing each of the categories, we sought to bring to the article some significant excerpts from the textualizations in order to ratify the reflections triggered by interviewees and, consequently, to explain indicators of their specialized knowledge both in relation to LA and its teaching and learning.

5 Inferences about the specialized knowledge of interviewed teachers

In this section, we present some inferences regarding the specialized knowledge of interviewed teachers, which could be made from the answers given by them to the four questions presented in Table 5, each one linked to one of the developed analysis categories.

(I) Inferences from the technological resources teachers use.

It was observed that, both among Chilean and Brazilian teachers, which were part of the case study, the use of Digital Information and Communication Technologies (DICT) is not part of the university routine of AL classes. Among the four Chilean teachers, two (Benito and Beatriz) mention them and among the six Brazilian teachers, four (Luís, Paula, Eva and Rui) make reference to such resources. However, in general, these reports are related to occasional uses. Benito has Instagram and YouTube channels and makes videos available to his followers, including many of his students. He states that, although these videos are not institutional, students watch them: ‘when I arrive in the classroom, I notice that some students already know the contents because they have already watched my videos’. However, he regrets that his engagement with technologies, as it is a private initiative, is not officially validated or valued through institutional assessments, which, according to him, do not even contemplate the mobilization of technological resources in their issues.

The most cited software is the GeoGebra. Among Chilean teachers, Benito uses it to approach systems of linear equations, eigenvalues and eigenvectors. Beatriz, in turn, uses it as an auxiliary tool for understanding demonstrations, the same use made by Brazilian teacher Eva. Another Brazilian teacher, Rui, uses the same software to represent mathematical objects in |${\mathbb{R}}^3$|⁠. Finally, we highlight the Brazilian teacher Paula, who uses, on some occasions, a technological resource not mentioned by the other teachers, which, to some extent, resembles what Benito uses, but in the case of Paula, institutionally: she develops a blog for the LA discipline through which she makes her videos available for students. In the classroom, she quickly resumes the content from what was covered in videos so that students can work with it.

Regarding the role played by DICTs in the contemporary world and especially in the daily lives of university young people, we expected that, even for this small group of teachers who were the subjects of our research, they would be more present as auxiliary resources in the day-to-day of classrooms, for teaching, learning and even for evaluations, but as evidenced by data presented in this section, this was not the observed scenario.

Based on the technological resources used by teachers, we can highlight indicators related to one of the of Pedagogical Content Knowledge (PCK) subdomains, namely: KMT, since the teachers’ considerations are mostly linked to different digital teaching resources and teaching strategies contemplating them. However, elements belonging to two Mathematical Knowledge (MK) subdomains were also identified. KoT is evidenced when one of the Brazilian teachers claims to use one of the software to represent mathematical objects in space, indicating knowledge of the importance of exploring different ways of representing a mathematical entity during teaching. Finally, aspects related to KPM are also revealed in terms of knowing how to explore and generate new knowledge in Mathematics, use heuristic strategies for problem solving and how to perform mathematical tasks. These elements, in our view, are related to the knowledge mobilized by a Chilean and a Brazilian teacher when using software to understand the ideas present in a mathematical demonstration.

(II) Inferences from difficulties that, according to interviewees, are faced by students.

The difficulties that, according to interviewees, are faced by Chilean engineering students for which they have already taught at institutions where they work, are linked, above all, to the study of vector spaces. For Catalina, the most problematic aspects are linked to the basic ideas of a finitely generated vector space, determination of a set of generators of a finite-dimensional vector space and the change of basis. According to the teacher, working with linear transformations becomes easier for students because they have already studied the real functions of real variables. Benito, in turn, understands the situation in another way. For him, working with vector spaces is indeed a source of obstacles, since students find it difficult, above all, to understand that apparently so different sets, such as the real matrixes and real polynomials of real coefficients (including the null polynomial) have, from the point of view of the vector space structure, the same behavior, for example, of the set of vectors of Analytical Geometry, of |${\mathbb{R}}^2$|and |${\mathbb{R}}^3$|⁠, of the set of complex numbers, etc. In addition, unlike some other contents under study (in the discipline that Benito teaches, matrixes and vector geometry are also covered), in most situations, it is not possible to approach the issue of vector spaces from the geometric point of view. Unlike Catalina, Benito also observes difficulties on the part of undergraduates when working with linear transformations: ‘it is very difficult to recognize whether or not it is linear transformation (and) to find kernel and image. They have difficulty understanding that the kernel is a subset of the domain (of the linear transformation) and that the image is a subset of the codomain’.

Beatriz also claims to perceive the students’ difficulties when working with notions related to linear transformations, especially in the diagonalization of operators which, according to her, ‘is a topic that covers everything that has been studied in LA; it is the moment in which the concepts are applied’.

Difficulties related to the study of vector spaces were also widely pointed out by Brazilian interviewees. They were mentioned by four of the six teachers (Sofia, Luís, Eva and Rui). According to Sofia, the obstacles faced by students in relation to themes vector space and subspace are, especially, when working with unusual vector spaces, directly related to the difficulties they have in mobilizing the symbolic algebraic language and due to deficiencies related to basic contents widely used in the study of these themes, such as sets, matrixes and complex numbers. Luís, in turn, mentions issues related to obtaining bases and dimensions for finitely generated vector spaces and changes between non-canonical bases. Eva points out that the main obstacles become explicit when students are faced with unusual vector spaces, such as that of integral functions, and need to perform demonstrations considering them. In addition, she reports that the technical procedures are easily mobilized by students, who, on the other hand, show difficulties when having to work with conceptual aspects and definitions. She exemplifies her statement citing students who easily verify, via a practical procedure for obtaining |${\mathbb{R}}^n$|bases, whether a given set of vectors is linearly dependent or linearly independent, but they cannot verify this fact using the definition of linear independence. Another point worth mentioning for the teacher is the fact that students have difficulties when working with spaces, vector subspaces and inner product, with unusual operations that are sometimes present in exercises involving these contexts.

Luís and Rui also mention students’ difficulties in working with linear transformations. According to Luís’, these are specifically more present in the determination of kernel and image sets.

Paula and Sofia highlighted an aspect that for us, who are also LA teachers, is in fact relevant and acts as a complicating factor in subjects related to this content: the student’s difficulty with the specific language of LA and with the abstraction of Mathematics that is so present in the approach to the concepts of this discipline. Sofia exemplifies this difficulty by pointing out that, when working with finitely generated vector spaces, students do not understand the difference in terms of notation, when using braces (to identify the set of generators) and square brackets (to indicate the space generated by a given set of vectors).

Another point mentioned by two teachers (Pedro and Eva) concerns the students’ difficulties with the matrix representation present in the most common LA approaches, where the focus is on the applications of the concepts covered, as is the case, obviously, of Engineering.

Sofia, who, as we have already highlighted, emphasized the lack of prior knowledge of students as a complicating factor when working with vector spaces and subspaces, once again refers to this aspect by stressing that, when studying eigenvalues ​​and eigenvectors, the revealed difficulty is not in relation to these LA-specific objects, but to factoring polynomials.

In summary, it was observed that the main obstacles highlighted by Brazilian interviewees are directly related to LA content, but in their interviews, they also made references to two more general issues: difficulties with basic mathematical content (pointed out by Rui and Sofia) and those related to demonstrations (Luís, Eva and Sofia). In relation to demonstrations specifically, Sofia reported that, in the face of so many difficulties faced by students, she no longer works them in the classroom ‘only for one or another interested student I send the material with the demonstrations’. She also reported that when asking students to verify whether a given set of vectors satisfies the properties that characterize a vector subspace, they often mistakenly conceive such a task as a demonstration, revealing, albeit indirectly, lack of knowledge about what is, in fact, demonstration in mathematics. This idea is reinforced by another statement by Sofia: that some students use the artifice of giving a counterexample (something correct, from the mathematical point of view, to demonstrate that a statement is not valid) to prove that something is valid.

While Chilean teachers only mentioned difficulties related to vector spaces and linear transformations, Brazilian teachers explained these obstacles, but also others related to different contents and some of a more general nature. Therefore, new studies should be carried out considering larger groups of professors and with students, aiming to analyze whether these difficulties reported by Brazilians are in fact characteristics of LA learning in the country and, on the other hand, a new, more in-depth study in relation to Chilean Engineering students to understand whether these other difficulties mentioned by Brazilians, but not by Chileans, when considering a broader group, are also relevantly present in LA learning among Engineering students in Chile.

The teachers’ statements about the difficulties faced by students when studying LA reveal specialized knowledge related to all Mathematical Knowledge (MK) subdomains. The KoT is explained by the interviewees, both Brazilians and Chileans, in moments when they highlight the topics that students have greater difficulties and the possible reasons for this. The KSM is inferred from statements made by teachers (from both countries) related to obstacles (or the absence of these) by students, linked to connections between different contents (for example, linear transformations and functions). In turn, the KPM is manifested when one of the Chilean teachers mentions the fact that, in Mathematics, a common structure is sought in sets that, at first glance, may seem to have nothing in common.

In relation to the Pedagogical Content Knowledge (PCK) domain, the present subdomains, among Brazilian and Chilean teachers, are: KFLM, since teachers indicate knowing how their students interact with the LA contents and what are their strengths and weaknesses in learning this subject; and KMLS, as teachers know the topics and prerequisites of the discipline.

(III) Inferences from the teachers’ perception about the dialogues between LA teachers and those of specific disciplines.

Through data obtained, it was observed that, considering this group of subjects, both in Chile and in Brazil, dialogues between LA teachers and those of specific Engineering disciplines, in general, are personal initiatives of teachers who seek to establish them as a strategy to improve their classes, and not institutional practices. Pedro (from Brazil), for example, highlights that in the institution where he works, the dialogue between Mathematics teachers (especially LA teachers) and those of specific disciplines took place at a specific moment that was the reformulation of the LA program, by personal initiative of the teacher who, at the time, was responsible for this curricular unit and who also works as a researcher in the area of ​​Mathematics Education. In these reflections on the necessary changes, Pedro, as an engineer, ‘made this bridge between Mathematics disciplines and those specific to my area (Control)’. But, in general, the teacher reported that “there is not a very open channel, the teacher, on his own initiative, is the one who should, if he deems it necessary, establish such a dialogue.

Both Chilean and Brazilian teachers emphasize the need to rethink the structure, in terms of content and approach strategies, of mathematical disciplines in engineering. Catalina (from Chile) states, for example, that in conversations with the teacher responsible for these subjects at the institution where she works, there is the idea that ‘teachers are teaching too much about mathematics and what engineers need is a more instrumental mathematics as a tool’. Pedro (from Brazil) states that Mathematics teachers need to keep in mind that, in Engineering, ‘teachers use ready-made things and students, using software such as Matlab, solve problems’. It is up to mathematicians to approach the concepts so that the indirect mobilizations that their students make are significant, that they understand that Mathematics is what makes them possible. For example; ‘teachers of specific disciplines use eigenvalues and eigenvectors, but students do not know what they are doing, they do not know the meaning’ and it is therefore up to Mathematics teachers to demonstrate them. Eva (from Brazil) also makes considerations about the need to reformulate mathematical disciplines in terms of what, in fact, future engineers will use. She points out that ‘the teachers of specific disciplines report that teachers need to teach what students will need in the basic cycle of the course [...]. For an engineering course, students need a little more practice.’

Personal initiatives are observed in the search for strategies aimed at helping to rethink mathematical disciplines in Engineering, in the sense of what has been discussed in the previous paragraph. Beatriz (from Chile), as an alternative to understand what future engineers, especially mechanical engineers in the reported situation, needed to learn in Mathematics and concerned about how to help students in this sense, chose, based on the invitation from a teacher in the specific area, for attending the classes he taught in order to understand ‘where should I put more emphasis on the subjects I was teaching. Thus, I was able to look at my discipline and analyze what was of high interest to engineers and what was not. As much as there is conversation with engineers, the best thing to do is to attend classes’.

Another strategy adopted is research and dialogue with other professors/researchers at events related to Engineering Education. Luís (Brazil), for example, reported that, in order to meet expectations for changes in the approach to Mathematics in Engineering, ‘in recent years, a group of teachers, of which I am a member, managed to publish an article in the Brazilian Congress of Engineering Education (organized by the Brazilian Association of Engineering Education) on robotics, which involves the theoretical part of LA’.

In addition, when, for any of these reasons, such dialogues occur, engineering teachers often show that for them the mobilization of LA concepts in specific disciplines is not explicit, on the contrary, for example, to what occurs with some contents of Differential and Integral Calculus. Pedro, from Brazil, emphasizes that ‘teachers in the technical area use LA, but they do not know that they are using it, it is not clear for them that it is LA’. There are engineering teachers who are not even able to clearly measure the importance of Mathematics for the course. This situation is reported by Chilean Catalina, who states: ‘I asked, on my own account, to an Engineering teacher if Mathematics was important for students of the course and he replied that he did not know’.

In general, there are no space for collaboration between Mathematics teachers, especially LA teachers, and those of other areas for a number of reasons, among them: the reception, by LA teachers and by the Mathematics area coordination, of programs that, in their interpretation, must be rigorously fulfilled, with the possibility of inserting topics, but never suppressing (aspect mentioned by Juan—Chile). In this regard, Brazilian teacher Eva points out that:

If I had total autonomy, I would give engineering topics; would make a thematic Linear Algebra. I would give in a month or so the theoretical content: basis, transformations (linear), eigenvalue and eigenvector. I would then propose three major themes and from them I would apply the LA contents previously worked on. That is, I would give the theory and then try to contextualize it. I believe it is not easy, but it would be ideal for an Engineering course.

In addition, when there are institutional attempts, some teachers do not attend the meetings scheduled with the aim of establishing these dialogues (reported by Sofia—Brazil). Rui (from Brazil) also makes reference to this aspect, mentioning that, in the institution where he teaches, often, due to time issues, he is the only Mathematics teacher who participates in meetings. On these occasions, he ‘is able to talk about LA with the electrical circuits and robotics teachers’. Another limitation is the lack of communication, which occurs in certain institutions, of the Mathematics department in relation to the others.

Finally, we highlight an aspect reported by Paula (Brazil), which, in her view, in the current context in which the curricular structures of Engineering courses are being rethought, gains importance. ‘The articulation between basic and specific disciplines adds other institutional ingredients that depend on the ways in which teachers are hired and work and the size of the institutions in terms of number of students’. Paula exemplifies two distinct situations: a first scenario is the insertion of more advanced mathematical topics, which were previously taught by professors in this area, in specific disciplines (for example, Fourier series and Laplace transform that, in some Electrical Engineering courses, are already covered in the specific disciplines of this qualification). A second scenario is the hiring, by Higher Education Institutions, which opt for a bolder curricular structure, of professors in the area of Mathematics, no longer to teach mathematical disciplines, but to work together, at times, with professors of specific disciplines.

The testimonies of some teachers, both Brazilian and Chilean, reveal, in terms of Mathematical Knowledge (MK), the presence of KoT, specifically in relation to phenomenology and applications, since they claim to seek to understand, through dialogues with teachers in specific areas, participation in Engineering Education congresses and even attending classes in specific disciplines, in which way Mathematics, and especially LA, is articulated with the particular themes of different Engineering qualifications.

We also identified specialized knowledge linked to the Pedagogical Content Knowledge (PCK) domain, especially some related to the KMT subdomain, perceived when the interviewees, from both countries, indicate mastering the sequencing of topics and strategies they consider to be the most suitable—even if unfeasible in the contexts of the institutions in which they work—for the approach of LA, and of the KMLS subdomain, since they seem to know what of Mathematics and, especially LA, must be learned by a future engineer, both from the curricular point of view defined by the institution, and by associations that regulate the training of engineers and from the point of view of teachers who are experts in the specific Engineering disciplines.

(IV) Inferences from the teachers’ concern in contextualizing contents to be taught.

The concern in contextualizing LA contents, within the scope of Engineering, was highlighted by only one of the four Chilean professors (Benito). In relation to the other three, two (Catalina and Beatriz) only answered that they do not contextualize and one (Juan) points out that ‘it is very difficult to contextualize Linear Algebra contents’. However, he claims to relate different contents in the context of Mathematics itself: ‘for example, I make the link between linear systems and matrixes, and particular cases, I generalize’.

Benito, who shows concern with contextualization, highlighted that, due to his background in Mathematics and not in Engineering, uses books to look for examples of applications and, later, take them to the classroom. He exemplifies by mentioning the case of problems related to cracks in beams, treated in Civil Engineering. In addition, he uses a resource, which is the dialogue with some of his former students who are already in more advanced stages of the course, to understand in which subjects and in which situations LA concepts are used. He also reported that when asked by students about the applications of LA in Engineering, recommends students to talk to an engineer, since his area of expertise is Mathematics.

The six Brazilian interviewees expressed some kind of knowledge about the applications of LA concepts. However, few are those who effectively work with contextualization, in engineering situations, of what they are teaching. Obstacles to implement this type of action were highlighted: the fact that students need, to understand applications, knowledge related to specific contents of Engineering that are discussed in subsequent disciplines (Sofia); applications presented in textbooks are not adequate (Sofia); the essentially abstract characteristic of LA (Rui); the fact that some applications involve subjects that are not the teacher's domain (Rui, mentions Quantum Physics, a context that could be used for the study of vector spaces, but which he claims not to use, as he does not have the required knowledge).

The application that is most present in classroom, cited by three (Sofia, Pedro, Rui) of the six teachers, are electrical circuits and Kirchhoff's laws, in the study of systems of linear equations and linear dependence. Other applications mentioned were dynamical systems (which Sofia claims to work with at least one example in the LA course after addressing the concepts of eigenvalues and eigenvectors); graphs (Peter); the idea of robot programming (in this regard, Luís mentions that in this type of situation, knowledge associated with eigenvalues, eigenvectors and base change are mobilized and Rui, in turn, mentions the possibility of working with translations and rotations in three dimensions when exploring robotics); and flight dynamics (which Eva mentions as a possibility to work with linear transformation application).

A final aspect to be mentioned in this category is that two of our interviewees expressed, albeit implicitly, the idea that it is not necessarily up to LA teachers to evidence the applications of concepts, that mobilizing what has been studied will be natural in specific disciplines and that, in this way, when studying curricular units that succeed LA, undergraduates will realize the importance of contents studied for Engineering. Sofia takes an explicit position in this regard, stating that ‘applications will be seen in specific disciplines’. Paula, in turn, points out that, although she was never able to actually use an application in classroom, she always ‘signaled students where the concepts would be used later, warning that a given teacher, in a given discipline, will need that content for applications that students were interested in’.

In relation to the specialized knowledge evidenced by teachers when reflecting on the concern in contextualizing the contents to be taught, we observed the presence, with regard to Mathematical Knowledge (MK), of the KoT subdomain, especially in the phenomenology and applications, such knowledge being more evident in the group of Brazilian teachers who also manifest KSM by highlighting that they seek to establish connections, within the scope of Mathematics itself, among contents covered. Regarding the Pedagogical Content Knowledge (PCK) domain, aspects of the KFLM subdomain are present in both groups, since teachers mention knowing the ways in which students interact with mathematical content. Elements of the KMT subdomain are mentioned by one of the Chilean teachers (who seeks to identify, together with former students (egress or at a more advanced stage of graduation and textbooks, LA applications that can be worked as examples in classroom) and by several Brazilian teachers (who show some applications with which they say they work). Finally, evidence of the KMLS is present in the testimony of the aforementioned Chilean teacher who seeks to identify, together with former students and textbooks, what the future engineer needs to know about LA.

6 Discussion of results

In order to synthesize and discuss the results obtained from interviews carried out in relation to the specialized knowledge of Brazilian and Chilean teachers, who composed our case studies, regarding the Mathematical Knowledge (MK) and Pedagogical Content Knowledge (PCK) domains and their respective subdomains, Table 6 shows which aspects of each one of them could be inferred in each of the countries, from the four established categories.

Analyzing data presented in Table 6, it could be observed that: with regard to the Mathematical Knowledge (MK) domain, KoT is the most frequent, both among Brazilian and Chilean professors, although in relation to technological resources, it is evident, among the ten interviewees, only among Brazilian teachers. The KPM was shown in categories technological resources (in both groups of interviewees) and difficulties faced by students (among Chilean teachers). Finally, the last MK subdomain revealed was KSM expressed by teachers from both countries in the category difficulties faced by students and by Brazilian respondents in the category concern in contextualizing the contents to be taught.

Regarding the Pedagogical Content Knowledge (PCK) domain, the prevailing subdomains are KMT—evidenced, among teachers from both countries, in the categories technological resources, dialogues between LA teachers and those of specific disciplines and concern in contextualizing contents to be taught—and KMLS—revealed among respondents from both countries in the categories difficulties faced by students and dialogues between LA teachers and those of specific disciplines and only by Chileans in category concern in contextualizing contents to be taught. Finally, the KFLM could be perceived by the statements of respondents from both countries in the categories difficulties faced by students and dialogues between LA teachers and concern in contextualizing contents to be taught.

In the same way as investigations contained in the literature review presented in one of the preceding sections of this article, we also used the case study as investigation methodology and the semi-structured interview as instrument for data collection. In our view, one of the contributions of this research, even though it was restricted to 10 Brazilian and Chilean professors, was to allow reflections on the specialized knowledge of LA teachers in Engineering courses.

Among studies included in our bibliographic review, seven refer to the same two Ecuadorian LA teachers, whose specialized knowledge was identified through the doctoral thesis of Vasco-Mora (2015) and analyzed in different aspects in the other six studies. The eighth study that composes the review also has two LA teachers in Engineering, being, in this case, Peruvian teachers.

Unlike Sosa et al. (2015), which aimed to explain the subjects’ knowledge about the KFLM subdomain (Knowledge of Features of Learning Mathematics), in our study, according to Gutiérrez-Garay (2019), we were not focused on highlighting elements related to a specific subdomain, but understanding which emerged from the responses given by teachers. However, Gutiérrez-Garay (2019) focused on systems of linear equations, while we did not focus on a specific mathematical LA content.

Sosa et al. (2015), focusing on KFLM, perceived the knowledge of subjects in their investigation regarding the language and processes with which students interact with contents, errors and difficulties associated with learning and personal theories associated with learning. In our research, in turn, a single aspect related to KFLM was evidenced: knowledge of interactions with the content.

As in studies by Vasco-Mora (2015), Vasco-Mora et al. (2015), Vasco-Mora et al. (2016), Vasco-Mora & Climent-Rodrígues (2017); Vasco-Mora & Climent-Rodríguez (2018, 2020), Vasco-Mora et al. (2021), the most evident knowledge subdomain was KoT especially in relation to elements, registrations of representation, phenomenology and applications. In relation to KMT, unlike the subjects interviewed by Vasco-Mora (2015), those of this survey, in addition to the issue of examples, also showed knowledge related to digital resources. Participants surveyed by Vasco-Mora (2015) and Vasco-Mora et al. (2015) did not express aspects related to KSM nor to KMLS, while the subjects of our investigation indicated knowing the connections that can be established between the different objects of LA (inherent in KSM) and what a future engineer must learn about LA (elements of KMLS).

7 Concluding remarks

A first aspect to be resumed and highlighted in these concluding remarks concerns the MTSK model not having the objective of judging or evaluating a teacher’s knowledge, in the sense of attributing to the non-explicitness of this knowledge a status of ignorance, of lack. The aim is, on the contrary, to understand possible knowledge that has not yet been sufficiently absorbed and, based on this understanding, to provide the teacher with opportunities, through training or other types of actions, for these to be developed or improved.

It should also be noted that the fact that some knowledge related to the different MTSK subdomains was not evidenced by the research subjects does not mean that they have not somehow developed it. The questions we prepared for interviews may, for some teachers, not have led to the emergence of such knowledge in their arguments.

One more point to be highlighted is that, based on the results obtained, we do not intend to generalize the specialized knowledge manifested by teachers as being representative of all those who teach LA in Engineering in Brazil or Chile. It is, as already pointed out, a case study, which, in addition to potentialities for the investigation, also brings some limitations, which are, in turn, characteristic of qualitative investigations.

Studies of this nature are relevant, as highlighted in the introduction to this article, for challenging readers from countries other than Chile or Brazil, ‘to look beyond the research and determine how these lessons can be appropriated in their own context with the desired goal of improving the teaching and learning of mathematics in schools’ (in particular, LA in Engineering) (Vistro-Yu, 2013, p. 146). In addition, the difficulties faced, according to respondents from both countries, by students when working with LA are coincident with those that have been discussed for a long time by researchers in the area, among which: Rogalski (1991), Dorier (1997, 1998), Maracci (2006), Wawro et al. (2011), Hannah et al. (2016) and Sandoval & Possani (2016). Based on these results, it can be indicated as a way for readers and researchers from different countries to analyze whether, in their contexts, as observed in this case study bringing together Chile and Brazil, these difficulties still remain despite all the progress in studies of issues about LA teaching and learning or whether they have already been partially minimized and what paths led to such minimization.

The specific data of the work we carried out show that both teachers from Brazil and Chile advocate reformulation in terms of structure, content and approach strategies, of the Mathematics curricular units in Engineering courses, in which LA is inserted. It is necessary for teachers to know that engineers, in most cases, will not resort, to solve specific problems, to mathematical concepts in the way they are usually worked on in the course, but rather to software whose procedures are based on Mathematics. It is up for teachers who will teach this science in Engineering to propose approaches that allow students to understand, in a significant way, these indirect mobilizations that they will make of mathematical contents.

In our view, in order for the proposition of approaches with this orientation to be in fact possible, it is essential for teachers who teach mathematical subjects or even other basic sciences in Engineering courses, to be trained, from the didactic and pedagogical point of view, to carry out that activity. Obviously, the ideal would be for the Higher Education Institution in which the teacher works to provide the opportunity to experience formative moments in an officially integrated way, but if this does not occur, we understand that the teacher himself must seek continuous self-training. The results of this article can contribute to this process in order to point out aspects that, perhaps, still need to be strengthened regarding the specialized mathematical knowledge.

As Mexican researcher Patricia Camarena Gallardo points out, who since the beginning of the 1980s has dedicated herself to reflections on the teaching and learning processes of Mathematics and other Basic Sciences in Engineering courses, the key point must be to differentiate a traditional program of updating and training for teachers who work in higher education institutions, from those specifically aimed at Basic Sciences, highlighting the concern with the link between these Sciences and Engineering qualifications in which they are inserted and the recognition of skills developed from the Basic Sciences of Engineering courses.

According to Camarena (2004), only by knowing in depth the Engineering qualifications for which they will teach, teachers of Mathematics and other Basic Sciences will be able to establish links between their areas of knowledge and Engineering, the roles played by them in the future performance of the professional being trained, the notations used in Engineering and the different applications of contents with which they will work. But how can a teacher whose area of training is Mathematics, Physics, Chemistry or other Basic Science be able to appropriate all these aspects? In our view, institutions should promote teacher training based on dialogue between teachers of Basic Sciences and those of specific Engineering disciplines, so that each of these groups can benefit, in a symbiotic and dialogic relationship, from the knowledge in which each of them is an expert. To develop this type of training, a first step is precisely to be clear about the specialized knowledge that teachers already have and which ones need to be built or deepened.

As future developments of this research and aiming to mitigate the limitations inherent to the study, we intend to follow some classes of subjects in order to identify if the knowledge that emerged in interviews is also translated into their practices, analyze the resources they use to prepare and develop their classes and the evaluation instruments; identify those of these teachers’ conceptions and establish relationships between them and the identified specialized knowledge, to expand the scope of interviewees both in Brazil and in Chile with the objective of understanding if, in fact, the results obtained can be effectively considered in a more general context, of teachers who teach LA in Engineering courses in these countries. In addition, similar investigations should be carried out in other countries.

References

Barbara Lutaif Bianchini: bachelor degree (1978) in mathematics at PUC-SP. Master in Mathematics Education at PUC-SP (1992) and PhD in Education (Psychology of Education) at PUC-SP (2001). Associated Professor of the Postgraduate Studies Program in Mathematics Education at PUC-SP, leader of the Algebraic Education Research Group and member of the research group Mathematics in Professional Formation. E-mail: [email protected]—https://orcid.org/0000-0003-0388-1985.

Eloiza Gomes bachelor degree (1978) in Mathematics at Mackenzie. Master in Mathematics Education (1992) and PhD in Mathematics Education (2015) at PUC-SP. Titular Professor at Instituto Mauá de Tecnologia, member of the Algebraic Education Research Group and of the research group Mathematics in Professional Formation and Leader of the Research Group Education in Engineering, Design and Business. E-mail: [email protected]—ORCDI: https://orcid.org/0000-0002-1217-9904.

Marcela Parraguez González bachelor degree and master degree in Mathematics at Pontificia Universidad Católica de Valparaíso (PUCV). PhD in Mathematics Education at National Polytechnic Institute of Mexico. Professor and Researcher at the Institute of Mathematics (IMA) of PUCV. E-mail: [email protected]—ORCID: https://orcid.org/0000-0002-6164-3056.

Gabriel Loureiro de Lima bachelor degree (2001) and master degree (2004) in Mathematics at UNICAMP. PhD in Mathematics Education at PUC-SP (2012). Professor of the Postgraduate Studies Program in Mathematics Education at PUC-SP. Leader of the research group Mathematics in Professional Formation and member of the Algebraic Education Research Group. E-mail: [email protected]—ORCID: https://orcid.org/0000-0002-5723-0582.