Specialised knowledge for teaching geometry in a primary education class: analysis from the knowledge mobilized by a teacher and the knowledge evoked in the researcher

mathematics Article Specialised Knowledge for Teaching Geometry in a Primary Education Class: Analysis from the Knowledge Mobilized by a Teacher and the Knowledge Evoked in the Researcher María del Mar Liñán-García1,2,*, Maria Cinta Muñoz-Catalán1, Luis Carlos Contreras 3and Víctor Javier Barrera-Castarnado 1,2   Citation: Liñán-García, M.d.M.; Muñoz-Catalán, M.C.; Contreras, L.C.; Barrera-Castarnado, V.J. Specialised Knowledge for Teaching Geometry in a Primary Education Class: Analysis from the Knowledge Mobilized by a Teacher and the Knowledge Evoked in the Researcher. Mathematics 2021,9, 2805. https:// doi.org/10.3390/math9212805 Academic Editor: Jay Jahangiri Received: 28 September 2021 Accepted: 1 November 2021 Published: 4 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Mathematics Education, School of Education, University of Seville, 41013 Seville, Spain; [email protected] (M.C.M.-C.); [email protected] (V.J.B.-C.) 2CEU Cardenal Spinola Centre for University Studies, University of Seville, 41930 Bormujos, Spain 3Research Centre COIDESO, University of Huelva, 20071 Huelva, Spain; [email protected] *Correspondence: [email protected]; Tel.: +34-616-495-622 Abstract: Deepening the specialised knowledge of mathematics teachers is necessary for teaching, and it is one of our concerns as researchers; accordingly, in this study we analyze the knowledge that a teacher mobilizes when giving a first lesson on geometry to a class of fifth year of primary education at a Spanish school. We provide this analysis with a complementary perspective that leads us to broaden our focus of attention to other aspects of classroom management, thus identifying the specialised knowledge evoked in the researcher by the opportunities observed in the classroom. We take opportunities to mean classroom situations that occur as a result of the teacher’s management for which the researcher considers that an alternative management of such situations would be supported by relevant specialised knowledge for the teaching of the given content. Considering the possible alternative management of specific classroom situations makes this specialised knowledge emerge, which, together with that mobilized by the teacher, generates specialised knowledge useful for guiding the content of the training of primary education teachers. Keywords: specialised knowledge evoked in the researcher by the opportunities; mathematics teachers’ specialised knowledge; geometry; primary education; case study 1. Introduction One of the lines of research in mathematics education is to deepen the knowledge of the mathematics teacher. Characterizing this knowledge from a teacher’s professional practice is of fundamental interest for the authors and the working group SIDM (Research Seminar in the Didactics of Mathematics, University of Huelva) in which this research is framed. We share the view of [ 1 ] (p. 58) that, “an essential object of investigation is what and how a mathematics teacher knows or should know about mathematics”. Geometry in compulsory education is commonly identified with the handling of quantities, formulas and geometric shapes, focusing on measurement in primary education, and becoming analytical in secondary education. However, it is not usually related to asking questions, obtaining models and identifying relationships and structures [ 2 ]. On the other hand, various studies have revealed the difficulties experienced by prospective primary teacher (henceforth PPT) in this area [ 3 – 5 ], which could come from their own training in primary education and continue later in their professional development. These antecedents show us the importance of addressing the problem of teaching and learning geometry in primary education, and the need to focus on investigating what and how a primary education teacher knows or should know about mathematics when teaching geometry. Thus, this study involves the observation of the practice of a teacher of the last year of primary education while teaching geometry, in particular, Euclid’s fifth Mathematics 2021,9, 2805. https://doi.org/10.3390/math9212805 https://www.mdpi.com/journal/mathematics Mathematics 2021,9, 2805 2 of 18 postulate, focusing on the specialised knowledge that this teacher mobilizes when teaching the observed classes. The interpretation of this knowledge is crucial to understanding how this teacher knows mathematics, which, in turn, provides us with information to partially answer the question: how should one know mathematics in order to teach a class of the last cycle of Primary Education? Based on this analysis, we provide a complementary perspective by focusing on “details” associated with the teacher’s class management, such as situations or moments that emerge in the classroom as a result of explanations or interactions with students, and the use of the materials available in the classroom. Thus, we broaden our focus of attention to include these “details”, considering the specialised knowledge that all this evokes in the researcher, and allowing us to reflect on the specialised knowledge that could support an alternative management of the classroom situation, without assessing why it is not employed. This new analysis of the specialised knowledge evoked in the researcher is linked to the teacher’s class management, as it arises from the teacher’s own practice, and gives greater emphasis to our role as teachers trainer and researchers, in the sense that the purpose of our research is to improve teacher training. To understand this new approach, let us look at the following example from one of the teacher’s classes observed under this study. When asking students about straight lines and their relative positions, in order to elicit their previous knowledge, a girl simulated with her fingers two segments that intersected at a point. The student then spread her fingers apart vertically to obtain a representation of two crossed segments that did not, however, intersect, therefore making the third dimension ostensibly explicit. As researchers, we consider this student’s gesture, which was left unaddressed by the teacher, an opportunity to ask ourselves what specialised knowledge would support an alternative management of this situation. Answering this question lends a greater role to our theoretical sensitivity [ 6 ] and to the triangulation of experts [ 7 ]. The specialised knowledge evoked in the researcher by the opportunities [ 8 – 10 ] is a construction of the researcher interwoven with the situation managed by the teacher, which faithfully reflects its nature. In this study, we present part of a broader investigation with access to all of the geometry classes taught by a teacher in the fifth year of primary education. 2. Mathematics Teachers’ Specialised Knowledge: MTSK as a Theoretical Study Lens We adopt the definition of knowledge provided by [ 11 ], considering the knowledge of a teacher as information available to solve problems, in the broad sense of the expression. [ 12 ] highlight two fundamental elements: information available to use that discards all that does not make sense for the task being carried out; and information that is not necessarily correct that allows us to position ourselves in the search for the understanding of knowledge (it may not be showed in the observed episode, but not being showed does not necessarily imply that it is not available). In this study, this question becomes of special interest, as, in observing the class, we try to identify: (a) the specialised knowledge mobilized by the teacher, without judging whether or not it is correct; and (b) the specialised knowledge evoked in the researcher by the opportunities. On the other hand, as we are observing practice, the fact that certain knowledge is not mobilized does not mean it is not possessed, as it may constitute knowledge that the teacher decides not to use. Based on the work of [ 13 , 14 ], ref. [ 15 ] refined their proposals and formulated the analytical model Mathematics Teachers’ Specialised Knowledge (MTSK), which considers specialization as the axis of knowledge of the teacher of mathematics in all its domains, subdomains and categories, and does not refer to any other science or profession, considering such professional knowledge to be what the teacher needs and uses [ 11 ] by the nature of mathematics teaching. On the other hand, it is rooted in mathematics itself, leaving out aspects of Pedagogy and general Psychology, it includes beliefs and the affective domain, and it covers all educational levels, from early childhood education [ 16 , 17 ] to university education [18]. Mathematics 2021,9, 2805 3 of 18 MTSK divides knowledge into three main domains: Mathematical Knowledge (MK), Pedagogical Content Knowledge (PCK), and, finally, beliefs on mathematics and mathematics teaching and learning, which are not considered in this article (Figure 1). Figure 1. Mathematics teachers’ specialised knowledge (elaborated from [15]). Mathematical Knowledge (MK) integrates the knowledge of the mathematical discipline that is taught, in our case, geometry. It considers three sub-domains: (deep) knowledge of the mathematical content itself and its intra-conceptual relationships (Knowledge of Topics, KoT); knowledge of the inter-conceptual relationships between these knowledges (Knowledge of the Structure of Mathematics, KSM); and knowledge of how to produce and proceed in mathematics (Knowledge of Practices in Mathematics, KPM). The Knowledge of Topics (KoT) considers the knowledge of mathematical contents, processes and procedures in themselves, approached from what [ 19 ] conceives as deep knowledge of fundamental mathematics. Taking as references the areas (numbers and operations, algebra, geometry, measurement, data analysis, and probability) proposed by [ 20 ], it achieves an international characterization that might vary slightly according to the curriculum of each country. A teacher must know contents, processes and school procedures, and the different situations in which they are presented, not only in relation to the mathematical discipline, but also as interrelated elements, understanding their properties and the processes that justify them, and the possible representations and procedures, from how to when and why. We distinguish four specific categories: Definitions,properties and foundations;phenomenology;registers of representation and procedures. The category of definitions, properties and foundations considers the set of properties that make a topic definable and give it sense and meaning, in addition to alternative ways of doing so. For example, in this category, we can place knowledge of the properties of regular polygons, which, in turn, define them, or the definition of a convex angle, as opposed to a concave angle, derived through its properties. Within this sub-domain, the model considers intra-conceptual connections, that is, relationships between concepts or processes belonging to the same topic. For example, the definition of perpendicularity between straight lines, which implies the division of the plane into four regions, or the relationship that exists between the definition of angle Mathematics 2021,9, 2805 4 of 18 and their classification according to position and measure (e.g., opposite angles, congruent angles), which allows us to demonstrate the property of the sum of the magnitudes of the interior angles of any triangle. Phenomenology, the second category we consider in the KoT sub-domain, encompasses the teacher’s knowledge of the phenomena relating to a topic that can generate mathematical knowledge and its uses and applications. We can interpret phenomenology from two different perspectives, one that identifies the knowledge that a teacher has about the applications that a topic can have, and one that the teacher has about models attributable to a topic. For example, we can consider the Pythagorean theorem as knowledge that allows us to construct algebraic irrational numbers, as we can also consider in this category the differentiation between the place value of figures in a number expressed in the Decimal Numbering System (DNS) and the grouping value; that is to say, in the number 2748, we can say that the number 7 occupies the hundreds place, which means that when regrouping from thousand to thousand, 7 groups of a hundred have remained without generating a group of a thousand. Registers of representation bring together the teacher’s knowledge of the different ways in which a topic can be represented [ 21 ], and the transformations between them, as well as associated notation and vocabulary. For example, if we treat a straight segment, we can draw it based on its properties, define it verbally with those properties, find elements of reality that could represent them, and use algebraic language to describe it analytically in relation to a coordinate system, whether Cartesian, rectangular, polar, etc. Included here is the knowledge that allows us to move from one representation to another without loss of its properties or characteristics, even within the same register, identified by [ 22 ] as mathematical flexibility. In addition, there is also the knowledge that certain mathematical constructs are not representable if not through cognitive metaphors [23]. In the procedures category, we have know-how, the practical knowledge of mathematics, which includes the indicators of how, when and why and the characteristics of the result. For example, knowledge of the decimal metric system and its structure would be behind the knowledge of all the previous indicators in the procedures for changing units of measurement, and in the consequent differences between the changing of units corresponding to the measurement of the magnitudes of length, surface area and volume (we consider the magnitudes of length, surface area and volume as measurable characteristics of geometric constructs approached from the composition and decomposition of said constructs, which is why we include them in the knowledge of geometry [3]). Knowledge of the Structure of Mathematics (KSM) contemplates connections, referring to the teacher’s knowledge of the relationships between different contents [ 12 ]. This subdomain has its origin in the work of [ 24 ], also considered for the development of Horizon Content Knowledge (HCK) subdomain in MKT model [ 14 ]. HCK considers all types of connections, including intra-conceptual ones and connections to content from other disciplines (HCK with regard to topics [ 25 ]). MTSK includes intra-conceptual connections in KoT subdomain (see KoT description above), and inter-conceptual ones in KSM, but connections to content from other disciplines are excluded. We distinguish four categories, referred, as indicates before, all of them to inter-conceptual connections, within this subdomain. Connections of increased complexity relate current contents with others that will be part of later moments in education. We can see clear examples in kindergarten teachers or the first grade of primary education when the teacher works on the designation of objects and sets and on classification with an eye on, for example, natural, rational or real numbers. Connections of simplification relate current contents with others that have already been covered, though not necessarily from the previous school year. An example of this would be the relationship that be drawn between the definition of the geometric figure of a circular sector and the interpretation of a fraction of the surface of a circle delimited by two radii (part of a whole, measurement). Transverse connections are those existing between different contents that have qualities in common. For example, the relationship that can be established between the distance function on a plane and in space, and the definition Mathematics 2021,9, 2805 5 of 18 of the heights of a triangle and the equivalence existing between the area calculated with one or another height. Auxiliary connections are established between knowledges that are related to each other as support for a certain purpose. For example, using the division of a circumference into arcs of equal length in order to construct a regular polygon, or using two parallel segments to generate all the quadrilaterals that have at least two parallel sides. Knowledge of Practices in Mathematics (KPM) is directly related to Shulman’s syntactic knowledge [ 13 ]. These practices refer to the actions put into play when a mathematical action is carried out. Thus, we will consider argumentation [ 26 ] in the knowledge of the processes associated with problem-solving as a means of producing mathematics and knowledge of ways of validating and demonstrating, that is, mainly how results are demonstrated, the fundamental difference in illustrating one or more cases in which a statement is fulfilled, and the role of examples and counterexamples in the validation of results and in the generation of definitions (forms of validation). The role of symbols and use of formal language, hierarchy and planning as a way of proceeding with the resolution of mathematical problems,the particular procedures for mathematical work, such as modelling, and the necessary and sufficient conditions for generating definitions are the remaining indicators of this sub-domain. Table 1 shows the categories and indicators of the domain MK. Table 1. Categories and indicators of the MK domain [15]. Subdomains Categories/Indicators 1 Knowledge of Topics KoT Categories Procedures: How, why and when to do, and characteristics of the result Definitions, properties and foundations Phenomenology Registers of representation Knowledge of the Structure of Mathematics KSM Connections of increased complexity Connections of simplification Transverse connections Auxiliary connections Knowledge of Practices in Mathematics KPM Indicators Knowledge of the processes associated with problem-solving as a means of producing mathematics Knowledge of ways of validating and demonstrating Role of symbols and use of formal language Hierarchy and planning as a way of proceeding with the resolution of mathematical problems Particular procedures for mathematical work Necessary and sufficient conditions for generating definitions 1 : Categories and indicators are tools proposed by the analytical model MTSK to operationalize the identification and characterization of each subdomain. Categories are identified in all subdomains, except for KPM, which remains at the indicator (are expressed in italics in order to distinguish from categories) level, as it is still in the characterization process. In Pedagogical Content Knowledge (PCK), we distinguish three sub-domains, referring to the knowledge of mathematics teaching, the characteristics of the learning process of students and learning standards. Knowledge of Mathematics Teaching (KMT) considers knowledge of aspects relating to the teacher’s main activity: Teaching, professional practice directly linked to the classroom, although this model also considers other contexts and situations specific to the work of the teacher and that have the same importance as part of the teacher’s development as an education professional [ 27 ] (p. 46). This subdomain arose as a result of the reflection on Knowledge of Content and Teaching (KCT) in the MKT model [ 14 ]. For example, this subdomain considers the knowledge of metaphors [ 23 , 28 ] as a powerful resource for the teaching learning process [ 19 ] (to see more details of its evolution and foundation see [ 15 ]). Mathematics 2021,9, 2805 6 of 18 We distinguish three categories within this sub-domain. The first is theories of mathematics teaching associated with mathematical content, which can be personal or formal, and involve knowledge based on observation and experience, or published research relating to mathematics teaching. The category of teaching resources (physical and digital), on the other hand, represents knowledge of the mathematical characteristics (mathematical potential and limitations) of resources and materials. For example, the limitation of the non-isometric rectangular geo-plane in representing equilateral triangles. Finally, strategies, techniques, tasks and examples lead us to knowledge that the teacher has of the adequacy of these elements given the teaching intention at a given moment, including both the potential for the teaching of mathematics that certain sequences of tasks and examples may have and their limitations, and the obstacles generated in a particular group of students. Knowledge of Features of Learning Mathematics (KFLM) considers the knowledge of aspects relating to the learning of the students, and the chosen categorization is supported by the initial proposal of [ 13 ]. Here, we distinguish four categories. The first, theories of mathematical learning, whether they are formal or personal theories of the observed teacher, considers the teacher’s knowledge about the students’ ways of learning mathematical content, including theories on cognitive development for mathematics. The second, strengths and weaknesses in learning mathematics, represents the teacher’s knowledge about the strengths of students concerning mathematical content, but also about errors, obstacles and difficulties in the students’ mathematical thinking. Also included in this category would be knowledge about strengths deriving from potentialities that may be exploited according to the class group, for example, considering whether they have a facility for a certain type of representation, such as symbolization for the calculation of the area of a surface, and its use to obtain this measure. The ways students interact with mathematical content relate to the teacher’s knowledge about the procedures and strategies of the students, including knowledge about the language and representations that they usually use. The fourth and last category refers to emotional aspects of learning mathematics, including the interests and expectations of students with respect to mathematics, as well as whether they consider a specific content, process or procedure easy or not. Knowledge of Mathematics Learning Standards (KMLS) considers knowledge of both the official curriculum in force in each country at a given time and the standards defined by research groups or professional mathematics teaching and learning associations (such as the National Council of Teachers of Mathematics—NCTM). The temporal aspect of this sub-domain is motivated by the curriculum itself, that is, what content is situated in a given course. We distinguish three categories within this sub-domain. Expected learning outcomes include the teacher’s knowledge of what is expected, in the context of the official curriculum, for a student to learn in a given course. Expected level of conceptual or procedural development refers to knowledge about the level to be reached for that content at a specific academic time. An example of this are the different contexts in which the subtraction of numbers is treated, and the different properties that are gradually added, from natural numbers to real numbers. Sequencing of topics a certain school moment differs from the KSM by referring to a temporal rather than conceptual question. Table 2shows the categories of the domain PCK. Mathematics 2021,9, 2805 7 of 18 Table 2. Categories of the PCK domain [15]. Subdomains Categories 1 Knowledge of Mathematics Teaching KMT Theories of mathematics teaching Teaching resources (physical and digital) Strategies, techniques, tasks and examples Knowledge of Features of Learning Mathematics KFLM Theories of mathematical learning Strengths and weaknesses in learning mathematics Ways students interact with mathematical content Emotional aspects of learning mathematics Knowledge of Mathematics Learning Standards KMLS Expected learning outcomes Expected level of conceptual or procedural development Sequencing of topics 1 : Categories are tools proposed by the analytical model MTSK to operationalize the identification and characterization of each subdomain. 3. Methodology The trigger for this study is our interest as researchers and teacher educators in understanding the specialised knowledge desirable for teaching geometry in primary education. To understand this knowledge, we consider that the classroom of a teacher with many years of experience is a privileged environment, and, therefore, we access Jimena’s classroom with her 25 students in the 5th year of primary education. Jimena is a teacher specialised in social sciences, with 35 years of experience divided between early childhood education and primary education. She has a special interest in the teaching of geometry at various stages, believing that it is largely forgotten in various courses, as it is usually taught towards the end of the year, and that it is one of the topics that can be most contextualized in the students’ environment. Our aim was to understand the specialised knowledge needed to teach geometry. However, during the study, we observed that focusing only on the interpretation of the knowledge mobilized by the teacher was not enough. During the observation, various situations occurred, sometimes attended to by Jimena, sometimes not, but always a consequence of the teacher’s management of the class. We consider that, if she had decided to attend to them, geometric concepts typical of the educational stage could have been addressed, which, together with the knowledge actually mobilized, would allow access to further specialised knowledge needed to teach geometry at that stage. For this reason, we introduced the methodological element of opportunities, understood as those moments or situations that arise in the class, as a consequence of the actions of the teacher, and that allow the researcher to reflect on the specialised knowledge that would have supported an alternative management of those same situations. Thus, emerges the knowledge evoked in the researcher by the opportunities, as specialised knowledge that the researcher interprets as being able to support an alternative management of the opportunities arising in the class. This interpretation is the consequence of a dialogic relationship [ 10 ] between the theoretical proposals, which make up the theoretical sensitivity of the researcher [ 6 ], and the reality of the fifth-year primary education class that we observed. The triangulation of experts [ 7 ] is a key validation tool in this interpretation. Consistent with this methodological positioning, we formulate the following research objectives: (a) identify the specialised knowledge for the teaching of geometry that Jimena mobilizes in her class, in particular, when she deals with Euclid’s fifth postulate; and (b) interpret the specialised knowledge that could have supported an alternative management of Jimena’s class when dealing with such content. We position ourselves in a paradigm close to the interpretive one [ 29 ], aware of the special role we play as researchers. We adopt an instrumental type case study research design [ 30 ], as we intended to understand a singular reality (Jimena’s classroom with her 5th-year primary school students as a particular case for studying the specialised knowledge for teaching geometry), in depth, and in its natural environment (the classroom Mathematics 2021,9, 2805 8 of 18 within its school, without modifying its natural course). This deep understanding is achieved via an initial observation of the specialised knowledge mobilized, subsequent interpretation of the knowledge evoked in the researcher by the opportunities, and validation by the triangulation of experts [7]. The information gathering technique used for this study is non-participant classroom observation, which is carried out in the class described, using a video recording of the class and taking field notes [ 31 ]. For this study, we have selected the beginning of the first lesson on geometry in 5th year of primary education. This lesson stops at a contextualization that the textbook proposes at the beginning of the topic, centered on a reading that tells the story of a hypothetical disciple of Euclid dealing with the following question, “Did you know that, through a point outside a path, only one parallel path can be drawn?”. The analysis of the information begins with the transcription of the session. In it, the vocabulary and expressions used by the actors are faithfully reproduced, including the gestures of the students or the teacher when they are considered significant for the analysis. We have carried out a detailed interpretive analysis [ 32 ], taking as a reference the corresponding field notes that, by themselves, are already the researcher’s interpretation [ 31 ]. We consider evidence of knowledge when the researcher interprets that there is sufficient certainty to indicate that the informant possesses such knowledge [ 33 ]. We also see an indication of knowledge when there is a suspicion of it, which could become evidence upon further investigation. In this study, in addition, we rely on opportunities, as already defined. The analysis of specialised knowledge identified in Jimena’s classroom has been carried out following the natural course of the class, in two phases. In the first phase, we have identified the knowledge mobilized by Jimena, from indications and evidences. Such degrees of certainty allow us to respond to the first objective of the investigation: To understand the specialised knowledge in geometry that the teacher mobilizes during our observation. In the second phase, we have identified the opportunities, and we have interpreted the specialised knowledge that would allow alternative management (knowledge evoked in the researcher by the opportunities), responding to the second objective of the investigation. The research team participates collaboratively, accomplishing the triangulation of experts [ 7 ] in the whole process. The analysis instrument has been the MTSK model itself, considering its domains, sub-domains, categories and indicators [15]. Finally, we integrated both analyses to obtain the mapping of the MTSK in geometry in relation to the content selected in the observed 5th-year class. In this way, we increasingly moved away from the specific data to finally obtain a global vision of the whole. 3.1. Description of the Session: Only One Parallel Path Can Be Traced through a Point Outside a Path We present a part of the transcript of the first lesson on geometry in the 5th year of primary education that Jimena is teaching (Table 3). They begin by reading the introductory text of the corresponding chapter of the textbook, where a fictitious conversation between Euclid and a supposed disciple is dramatized, in which they argue about the affirmation that, through a point outside a path, only one parallel path can be traced. Mathematics 2021,9, 2805 9 of 18 Table 3. Part of the transcript of the first lesson on geometry. (They start by reading the introductory text). Teacher (T): Where does it say it? There, at the end, at the end, “By the way, did you know that, through a point outside a path, only one parallel path can be drawn?” Let’s see, if someone thinks for a moment to draw, because now we are dealing with geometry and need to think about what we are seeing. A path (indicates a path with two fingers) and then a point, and he says that I can only draw one parallel straight line. Is that true? Student (S): Yes. T: Doesn’t anyone see a path in their head? S: Yes, I do. T: It’s so simple . . . A path, a point (draws two parallel segments on the board, simulating a path, and an exterior point). S: Ah, I thought it was a path (pointing to the drawing on the board). T: Sure, outside... How? Have I made a mistake? Have I not read it well . . . Yes. Outside, out of the path. It says that we can only draw one parallel straight line. S: Yes, yes. There it is a parallel path (points to the drawing of the two parallel segments without considering the point). T: Let’s see, S., a path, if at an outside point, outside the path, we can only draw one parallel line. Yes or no? I’m going to start drawing a lot of lines (draws lots of segments that go through that point). S: Straight lines? T: They all go through the point, don’t they? (points out the one that seems parallel) So on to infinity, I can draw an infinite number of straight lines that pass through the point, but he says that there is only one that is parallel (points to it), and I am going to draw it like that with a wavy line so you can see it. T: Well, that’s one of the things that Euclid thought. S: But, Miss, there’s another way, from the path, draw a line down. T: From the path, I draw a line down (does it), but it is not parallel. S: Ah, no, that’s perpendicular. T: So, Euclid was right. Look, it seems like a tongue twister, but we only have to draw what we are told, either in our brains or on paper. In geometry, there’s a lot to draw, isn’t there? So, is it true what Euclid says, that, on a path, if we draw an exterior point, there is only one line parallel to that path? Yes? Are you sure? Can there be two parallel lines at the same point?” ( . . . ) 3.2. The Specialised Knowledge in Jimena’s Class for Teaching Geometry We organize the analysis of the session described in the previous section in two parts. In the first, we highlight the specialised knowledge that the teacher, Jimena, mobilizes when she teaches Euclid’s fifth postulate. In the second part, we describe what the opportunities are, and what specialised knowledge would support an alternative approach to them, which we call specialised knowledge evoked in the researcher by the opportunities. The set of knowledge elements that we express in an interpretive way in the following sections are shown in a synthesized way in Table 4. This table is organized to indicate the sub-domains and MTSK category in the rows and the specialised knowledge mobilized by the teacher and the knowledge evoked in the researcher in the columns, in relation to the corresponding sub-domain and category. Mathematics 2021,9, 2805 16 of 18 first approach to characterizing the specialised knowledge desirable for teaching Euclid’s fifth postulate. We consider that this work raises new research opportunities for the future, as, in fact, observing knowledge from the potential perspective of opportunities would consequently result in a new approach to research supported by the framework of MTSK. Based on the case study of this article, we have understood that, by including in the observation the specialised knowledge evoked in the researcher by the opportunities, this has become part of the specialised knowledge that supports and creates a foundation, in the sense of [ 19 ], for the knowledge treated in the classroom. There are various studies that have used the specialised knowledge evoked in the researcher by the opportunities, including at other educational levels, such as early childhood education [ 10 ] or in the training of primary education teachers [ 9 , 36 ]. These studies use it as a necessary construct for the design of training tasks for PPT, based on real classroom practice. Starting from the consideration of the teacher’s professional tasks as the backbone of their initial training, [ 37 ] define a training task as one that takes professional tasks as a reference, and implies the systematic analysis of real classroom situations that make practice the focus of training for PPT (professional tasks constitute the teacher’s system of activities characterizing their practice [ 38 ]). To design training tasks for initial teacher training, [ 36 ] propose the selection of powerful fragments of specialised knowledge that can be reflected on in a student teacher training classroom. Considering the specialised knowledge evoked in the researcher by the opportunities, they include, in the literal transcription, elements that can enrich reflection on additional aspects of specialised knowledge for teaching geometry in an initial teacher training classroom. The training task is thus made up by adding activities oriented to stimulate the PPT to mobilize elements of specialised knowledge from the different MTSK sub-domains. The use of the specialised knowledge evoked in the researcher by the opportunities in the design of training tasks allows the possible alternative management of specific situations, to promote the mobilization of specialised knowledge, to be worked on with PPT. Author Contributions: M.d.M.L.-G., Conceptualization, Formal Analysis, Funding Acquisition, Investigation, Methodology, Resources, Validation Supervision, Writing—Original Draft, Writing— Review and Editing; M.C.M.-C.; Conceptualization, Formal Analysis, Investigation, Methodology, Resources, Validation, Supervision, Writing—Review and Editing; L.C.C., Conceptualization, Supervision, Writing—Review and Editing; V.J.B.-C., Formal Analysis, Investigation, Methodology, Resources, Validation Supervision, Writing—Review and Editing. All authors have read and agreed to the published version of the manuscript. Funding: Work supported by the CEU Cardenal Spínola Centre for University Studies, Seville, Spain, and the University of Seville, Spain. Institutional Review Board Statement: The study was conducted according to the guidelines of the Declaration of Ethical Committee for Experimentation of the University of Seville. Informed Consent Statement: Written informed consent has been obtained from the volunteers to publish this paper. Acknowledgments: We specially thank CEU Cardenal Spínola Centre for University Studies, Seville, Spain; University of Seville, Seville, Spain for their support. We thank Jimena and her students, without whose generous participation, this study would not have been possible. Also, we thank to the project ‘Specialised knowledge of mathematics’ teachers and teacher training’ (RTI2018-096547- B-I00, from the Ministry of Science, Innovation and Universities, Spain). It is also linked to the Ibero-american MTSK Network of the Asociación Universitaria Iberoamericana de Posgrado (AUIP), and to the Research Centre COIDESO (University of Huelva, Spain). Conflicts of Interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 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