mathematics Article Mathematical Connections and the Mathematics Teacher’s Specialised Knowledge Genaro De Gamboa * , Sofía Caviedes and Edelmira Badillo Department of Teaching of Mathematics and Experimental Science, Autonomous University of Barcelona, 08193 Bellaterra, Spain * Correspondence: genaro.degamboa@uab.cat Abstract: This study seeks to explore the relationship between recent findings on mathematical connections and the Mathematics Teacher’s Specialised knowledge model. From a qualitative approach, we seek to explore the specialised knowledge mobilised during the establishment of mathematical connections in different contexts. The study takes into consideration the results from previous studies that were conducted with both prospective and in service teachers. The results suggest the need to include the notion of complexity in the mathematical connections defined by the Mathematics Teacher’s Specialised knowledge model. Similarly, they suggest that it is possible to find some ambiguities in the use of the model, so it would be enriching to consider the current classification of mathematical connections in order to improve the model as an analytical tool. Finally, the results suggest the need to include mathematical connections as part of the pedagogical content knowledge domain, since the knowledge of this domain could be mobilised during the establishment of mathematical connections. Keywords: mathematics teacher’s specialised knowledge; mathematical connections; complexity of mathematical connections Citation: De Gamboa, G.; Caviedes, MSC: 97B50; 97F40; 97G30 S.; Badillo, E. Mathematical Connections and the Mathematics Teacher’s Specialised Knowledge. Mathematics 2022, 10, 4010. https:// doi.org/10.3390/math10214010 1. Introduction Several models of teacher knowledge point to the importance of mathematical con- Academic Editor: Luis nections in describing different types of knowledge (Ball et al. [1]; Carrillo et al. [2]; Row- Carlos Contreras-González land [3]). For example, the MKT (Mathematics Knowledge for Teaching) model of Ball Received: 22 September 2022 et al. [1] considers connections as a part of teachers’ awareness, related to how mathemat- Accepted: 25 October 2022 ical ideas or concepts can be associated throughout the school years. In this sense, the Published: 28 October 2022 model uses the notion of connection to define Horizon Content Knowledge, a subdomain Publisher’s Note: MDPI stays neutral of knowledge linked to knowing the relationships between content. In the Knowledge with regard to jurisdictional claims in Quartet (Rowland [3]), connections define one of the dimensions of teacher knowledge and published maps and institutional affil- are linked to the establishment of connections between concepts, procedures, and to the iations. use of these connections for sequencing content, or recognising the conceptual appropri- ateness of an activity. In the MTSK model (Mathematics teacher’s specialised knowledge) Carrillo et al. ([2]) recognise two types of mathematical connections, intraconceptual and interconceptual. While intraconceptual connections are part of KoT (Knowledge of topics), Copyright: © 2022 by the authors. interconceptual connections are recognised in the subdomain of KSM (knowledge of the Licensee MDPI, Basel, Switzerland. structure of mathematics). This article is an open access article Although the previous models recognise and take into consideration the mathematical distributed under the terms and connections in the teacher’s knowledge, they do not contemplate the characterisation of conditions of the Creative Commons mathematical connections addressed in more recent research (Businskas [4]; de Gamboa Attribution (CC BY) license (https:// et al. [5]; de Gamboa et al., [6]; Evitts [7]; Eli et al. [8]; Hatisaru [9]; Rodríguez-Nieto [10]; creativecommons.org/licenses/by/ García-García and Dolores-Flores [11,12]). In this regard, this study aims to incorporate 4.0/). Mathematics 2022, 10, 4010. https://doi.org/10.3390/math10214010 https://www.mdpi.com/journal/mathematics Mathematics 2022, 10, 4010 2 of 24 the results of this recent research into the analysis of knowledge mobilised by mathematics teachers, in line with other recent studies such as Hatisaru ([9]) who uses an extended model for mathematical connections as an analytical tool and in relation to mathematical knowledge for teaching. In previous studies (de Gamboa et al. [5]; de Gamboa et al. [6]; Caviedes et al. [13]), we have used MTSK model to analyse the mathematical knowledge of both preservice and in service teachers. The results of using MTSK model in the analysis of teachers’ knowledge show that the model is useful for identifying mathematical connections. However, the results of the aforementioned studies also show some difficulties in using MTSK model to classify the knowledge mobilised by teachers when establishing mathematical connections. The difficulties observed are linked to possible ambiguities in identifying the intra or inter- conceptual character of connections established by preservice and in service teachers. This, in turn, is associated to ambiguities in characterising the knowledge mobilised by them, e.g., KoT or KSM. Thus, in order to investigate such ambiguities in detail, the present study aims to answer the following questions: How does the classification of mathematical connections reported in recent research relate to the types of knowledge defined in MTSK model? What can research on mathematical connections contribute to MTSK model enrichment? 1.1. The Mathematics Teacher’s Specialised Knowledge Model—MTSK Taking as a reference the contributions of Shulman ([14]), it could be said that teacher knowledge has been a subject of research since the 1980s. The author points out that a deep knowledge of mathematical content must be complemented by a knowledge of learning and teaching mathematics, since a purely mathematical or purely didactic knowledge is not sufficient for the development of mathematical teaching skills (Shulman, 1986). In order to make mathematics comprehensible to students, Shulman ([14,15]) warns that it is necessary for teachers to have a problem-solving tendency and to have the necessary reasons to understand why they solve mathematical tasks in a certain way and not in another way. Thus, teachers also need to know different ways of solving and different ways of teaching the content to their students. Since Shulman’s work, other models have emerged seeking to address the demands of the task of teaching mathematics. One of these models is the MTSK model, which considers specialisation as the core of the mathematics teacher’s knowledge in all its domains, subdomains, and categories (Aguilar et al. [16]). From this specialisation, the model understands mathematics as a “systemic knowledge network structured according to its own rules” (Carrillo et al. [2] (p. 6)). A good knowledge of this network, of the rules and of the characteristics of the process of creating mathematical knowledge, would allow a teacher to teach the contents in a connected way and to validate his own and the students’ conjectures (Carrillo et al. [2]; Carrillo et al. [17]). The MTSK includes two main dimensions: Mathematical Knowledge (MK) and Pedagogical Content Knowledge (PCK). Within MK are included KoT (knowledge of topics), KSM (knowledge of the structure of mathematics), and KPM (knowledge of mathematical practice). KoT describes what and how mathematics teachers know the content they teach, being a combination of the knowledge students are expected to learn and a deeper, formal and more rigorous understanding (Carrillo et al. [2]). KoT includes different categories of knowledge: definitions (e.g., what is an integer?); properties and their foundations (e.g., properties of powers); the phenomenology or contexts of use; procedures and their justifications (e.g., when, how and why to use formulas or decomposition procedures); and systems of representation. Intra-conceptual connections are also part of KoT “on the grounds that they are located in the proximity of a single concept, and can be regarded as enriched knowledge of a single content item” (Carrillo et al. [2] (p. 8)). The KSM describes teachers’ knowledge of connections between different mathemati- cal topics (Montes et al. [18]). Four categories of connections are considered: connections of increasing complexity, simplification, auxiliary and transversal. The connections of increasing complexity (complexity) and simplification relate to elementary and advanced Mathematics 2022, 10, 4010 3 of 24 knowledge. Advanced knowledge allows teachers to deal with elementary mathematics from an advanced perspective. Elementary knowledge is treatment of advanced mathe- matics from an elementary perspective (Montes et al. [18]). For example, a simplification connection can be established when a teacher uses the Riemann sum and the surface grid in order to calculate areas by means of an elementary-advanced knowledge relationship (Flores-Medrano [19]). A complexity connection can be established when a teacher uses the calculation of the area of similar rectangles to introduce the comparison between linear and quadratic growth. Auxiliary connections are related to the involvement of a topic in a longer process (Carrillo et al. [2]). According to (Flores-Medrano [19]), a useful aspect for their identification, or even to generate examples of them, is the identification of prior knowledge of a concept and process, although “ . . . not all prior knowledge would form an auxiliary connection with the object in question. Sometimes they may be examples of complexity or simplification connections (e.g., knowledge of whole number fractions as prior knowledge of algebraic fractions)” (Flores-Medrano [19] (p. 52)). Finally, transversal connections refer to knowledge that underlies the establishment of relationships between several topics with common features (Montes et al. [18]). For example, the connection between the measurement of a magnitude using different units of measurement, and the inversely proportional relationship between the size of the unit of measurement and the nu- merical value obtained from the measurement. The four categories of connections “do not intersect with each other and allow to have a global view on the possible interconceptual connections” (Flores-Medrano [19] (p. 53)). Knowledge about defining, making conjectures, proving or solving problems are a part of the KPM. Thus, this subdomain acquires an organising role of the mathematical knowledge that makes up the KoT and the ways of operating them (Carrillo et al. [2]). For example, the different registers of representation that area calculation admits (geometric, symbolic) can be used to validate or demonstrate that a result is correct. In turn, the knowledge and use of registers of representation is related to knowledge of different procedures, properties, and justifications, directly impacting the relationships between KPM-KoT. On the other hand, in PCK we find KFLM (knowledge of the features of student learn- ing), KMT (knowledge of mathematics teaching) and KMLS (knowledge of mathematics learning standards). The former encompasses the knowledge associated with the inherent characteristics of learning mathematics, focusing on the mathematical content (as the object of learning) and not on the learner (Carrillo et al. [2] (p. 11)). It includes four categories: knowledge about theories of learning mathematics (e.g., Van Hiele’s levels); strengths and weaknesses in learning mathematics; the way learners interact with mathematical content (e.g., often solving area tasks using formulas); knowledge of the motivations and expectations that learners have when confronted with particular content (e.g., how problem solving often generates rejection). For its part, KMT refers to knowledge about teaching that is intrinsically linked to content and considers three categories: knowledge about teaching theories (e.g., accumulated knowledge from experience or anthropological theory of didac- tics); knowledge about teaching resources (e.g., tangram or geoboard for teaching area), their potential and limitations (e.g., it is not possible to construct an equilateral triangle on an orthometric geoboard); knowledge about strategies, techniques, tasks and examples (e.g., the usefulness of using different representation registers for teaching content). Finally, KMLS includes knowledge about the expected learning outcomes for a given school level (through specific measurement instruments); knowledge about the content to be taught at a given school level; and the sequence that mathematical content follows throughout formal education. 1.2. Recent Results in the Characterisation of Mathematical Connections In recent years, a growing line of research has been developed on the study and char- acterisation of mathematical connections (Businskas [4]; Evitts [7]; Eli et al. [8]; Rodríguez- Nieto [10]; Duval [20]; de Gamboa and Figueiras [21]; Rodríguez-Nieto et al. [22]). In Mathematics 2022, 10, x FOR PEER REVIEW 4 of 29 Mathematics 2022, 10, 4010 4 of 24 Nieto [10]; Duval [20]; de Gamboa and Figueiras [21]; Rodríguez-Nieto et al. [22]). In the results of such research, two levels of complexity can be observed in terms of the types of the results of such research, two levels of complexity can be observed in terms of the connections described. In a broad sense, a differentiation is made between extra-mathe- types of connections described. In a broad sense, a differentiation is made between extra- matical connections and intra-mathematical connections. Extra-mathematical connections mathematical connections and intra-mathematical connections. Extra-mathematical connec- occur when mathematics is related to contexts outside of mathematics, as it is the case of tions occur when mathematics is related to contexts outside of mathematics, as it is the case modelling activities. Intra-mathematical connections occur when there is a relationship of modelling activities. Intra-mathematical connections occur when there is a relationship between representations, definitions, concepts, procedures, and propositions within the between representations, definitions, concepts, procedures, and propositions within the context of mathematics. Within the broad set of intra-mathematical connections, two types context of mathematics. Within the broad set of intra-mathematical connections, two types are differentiated are differentiated (de (de Gamboa Gamboa and and Figueiras Figueiras [19]). [19]). Intra-mathematical Intra-mathematical conceptual conceptual connec- connec- tions and tions and intra-mathematical intra-mathematicalconnections connectionsrelated relatedtotoprocesses. processes.The Theformer formerconnections connections re- refer fer to relationships between different conceptual elements. The to relationships between different conceptual elements. The latter are connections that ex- latter are connections that explicitly plicitly emphasise emphasise transversal transversal processes processes to to mathematical mathematical activity, activity, suchsuchas as problem-solv- problem-solving heuristics, communication of mathematical information, argumentation andand ing heuristics, communication of mathematical information, argumentation justifica- justification. tion. In this context, this study considers three categories of In this context, this study considers three categories of mathematical connections: extra- mathematical connections: extra-mathematical mathematical connections connections (EMC), intra-mathematical (EMC), intra-mathematical conceptual (IMCC), conceptual connections connections and (IMCC), and intra-mathematical connections intra-mathematical connections related to processes (IMCRP). related to processes (IMCRP). De Gamboa De Gamboa and andFigueiras Figueiras([21]), ([21]),when whenanalysing analysing inin a classroom a classroom context context thethe mathe- mathemat- matical connections that emerged in practice, observed that ical connections that emerged in practice, observed that these were produced as networks these were produced as net- works of other more elementary connections. They also of other more elementary connections. They also observed that these connections were observed that these connections were explicitly explicitly triggered triggered by thebyteacher the teacher and student and student interventions. interventions. In context, In this this context, connec- connections tionsbecan can be triggered triggered by students’ by students’ mathematically mathematically inconsistent inconsistent ideas,ideas, which which act asactanasindica- an in- dicator tor of the ofexistence the existence of a consistent of a consistent mathematical mathematical connection connection that canthatlead canto lead to clarify clarify these these mathematically inconsistent ideas. Thus, mathematical mathematically inconsistent ideas. Thus, mathematical connections are relations between connections are relations be- tween two two mathematical mathematical objects objects O1 and 𝑂𝑂1Oy2 , 𝑂𝑂and 2 , and cancan bebe formedbybyone, formed one,two twoor or more more connec- connec- tions, which tions, which we will call we will call elementary elementary connections. connections. In In this this study, study, aa mathematical mathematical object object is is understood as understood as any any material materialor orimmaterial immaterialentity entityinvolved involvedinin the mathematical the mathematical activity activityin thethe in sense of OSA sense of OSA (Godino (Godinoet al.et([23]). We focus al. ([23]). on those We focus types types on those of connections that match of connections that our understanding match of elementary our understanding connections of elementary and which connections and derive which from the recent derive from theliterature recent related torelated literature mathematical connections. to mathematical Figure 1Figure connections. shows1how shows elementary how elementaryconnections intro- connections duce relations introduce with other relations mathematical with other mathematical objectsobjects (e. g. , 𝑂𝑂(e.g., 1 and 𝑂𝑂 O31).and TheOcoordination between 3 ). The coordination elementary between connections elementary defines the connections mathematical defines connection the mathematical betweenbetween connection 𝑂𝑂1 andO𝑂𝑂12and . There O2 . are different There typestypes are different of elementary of elementary connections. connections. In the In following, the following, we wedescribe describethethetypes typesof elementary of elementary connections connections that thatareare used usedboth bothininthisthisstudy studyand andin in others others that characterise types of connections. Figure 1. Figure 1. Example Exampleofofthethe structure of aofconnection structure as a as a connection network of elementary a network connections of elementary (O: math- connections (O: ematical object). mathematical object). Equivalent representation: Equivalent representation:The Theelementary elementary connection connection is established is established between between two two rep- representations resentations of same of the the same mathematical mathematical object object ( R1 →(𝑅𝑅1R→ ). 2 ).𝑅𝑅The 2 The connection connection relates relates equiv- equivalent alent representations representations (ER), both (ER), when whenrepresentations both representations belongbelong to theregister to the same same register (e.g., (e.g., when a when a teacher connects 5 −2 5 and 5 −(+2) 5 ) to emphasise the base of teacher connects −2 and −(+2) ) to emphasise the base of the power) (Adu-Gyamfi [24];the power) (Adu- Gyamfi [24]; Businskas [4];Businskas [4]; Dolores-Flores Dolores-Flores and Garcia-Garcia and Garcia-Garcia [25]). [25]). Mathematics 2022, 10, 4010 5 of 24 Alternate representation: The elementary connection is established between two rep- resentations of the same mathematical object ( R1 → R2 ). The connection relates alternate representations (AR), when there is a change in the register (e.g., representing numbers geometrically). It is proposed by Businskas ([4]) and is related to the notion of conversion proposed by Duval ([20]). Common feature of representation: The elementary connection is established between two objects O1 → O2 that share a common feature, without being equivalent. These elementary connections are triggered by erroneous and ambiguous interpretations of mathematical objects by the students. The common features that sustain the elementary connections can be related to commonalities in the representation (CFR), for instance, when 8 32 32 and 8 are connected. It is related to the category Categorical proposed by Eli et al. ([26]). Common feature of definition: As in the previous case, the elementary connection is established between two objects O1 → O2 that share a common feature, without being equivalent. The common features that sustain the connection can be related to their definition (CFD), for instance when a student makes a mistake such as (−1)36 = −1 × 36 and the teacher connects both operations by stressing their commonalities and differences. It is related to the Categorical category proposed by Eli et al. ([26]). Metaphorical projection: The elementary connection is established between two objects O1 → O2 and they can be related to some metaphorical projection (MP) in which the metaphor is used literally. For instance, when −4 − (−8) is interpreted as a metaphor of being in basement 8 and going up 4 floors. It corresponds to the Metaphor category proposed by Rodríguez-Nieto ([10,22]) Procedure: The elementary connection is established between a concept and a proce- dure (C→P) that can be used when dealing with the concept. For instance, when a teacher suggests that using the rules for operating with integers to obtain a notation without double symbols is a good resource to make the calculation (Dolores-Flores and Garcia-Garcia [25]; Businskas [4]). Equivalent procedure: The elementary connection is established between two proce- dures ( P1 → P2 ) that are useful for solving the same task. For instance, when a teacher (−5)7 conducts a discussion on the different procedures that can be used to solve (de (−5)7 Gamboa et al. [6]). Justification: The elementary connection is established between two propositions ( PR1 → PR2 ), where PR1 stands for a premise and PR2 stands for a conclusion. It refers to justifications (JU) of practices in mathematics. For instance, when a teacher discusses the difference between inductive reasoning and deductive reasoning emphasizing the meaning of a mathematical proof (de Gamboa et al. [6]). Implication: The elementary connection is also established between two propositions ( PR1 → PR2 ); it refers to implications (IM) or if-then arguments (Businskas [4]). Particular case: The elementary connection is also established between two proposi- tions ( PR1 → PR2 ), it can refer to the application of a property to a particular case (PC). It corresponds with the category Part-Whole proposed by Businskas ([4]) and Dolores-Flores and Garcia-Garcia ([25]). Generalisation: The elementary connection is also established between two proposi- tions ( PR1 → PR2 ), it can refer to generalisations (GE) of a property to a broader set. For p instance, when a teacher generalises aaq when p ≥ q to the case when p < q. (Businskas, [4]). The above-mentioned classification of elementary connections corresponds to a rein- terpretation of the extended model for mathematical connections proposed by (Rodriguez- Nieto. [22]), which includes some of the results of our previous research. In the case of procedural connections, we also propose the equivalent procedure. The reason for this is that we have found some examples of connections that do not correspond to the definition of procedural connection, but refer to explicit relations between two different procedures that can be used in the same activity. In the case of the Part-Whole connections proposed by Businskas ([4]) and Dolores-Flores and García-García ([25]) we have decided to separate Mathematics 2022, 10, 4010 6 of 24 them into two different types, particular case and generalisation. This is because we have observed in previous studies (de Gamboa, [6]) that generalisation connections can be more complex in nature than particular case connections, and therefore we believe that they should be considered separately. 2. Methods The present study has an exploratory and comprehensive purpose on the connections established and the type of knowledge mobilised by in service and preservice teachers. Therefore, the study is situated in an interpretative paradigm and follow a qualitative approach (Cohen et al. [27]; Bryman [28]). Qualitative research is concerned with inter- pretation and exploration that guides researchers toward understanding and explaining different phenomena and events (Creswell and Garrett [29]). In this case, the aim is to explore how the classification of mathematical connections reported in the literature relates to the types of knowledge described in MTSK model. A content analysis (Krippendorff [30]) is carried out which contemplates data that were collected in two broader studies, and which relate to mathematics teachers’ knowledge. In one study (Caviedes et al. [13]), the knowledge that preservice teachers (PSTs) bring into play when solving area tasks that require making mathematical connections is characterised. In another study (de Gamboa [5], de Gamboa et al. [6]), the knowledge in use of an in service teacher is characterised when she favours the establishment of mathematical connections in the teaching of a unit on integers. The decision to consider two different studies is motivated by two reasons. First in both studies difficulties were observed when characterising the teacher’s knowledge during the establishment of connections using the MTSK tools. Mainly, difficulties were observed in the classification of intra and inter conceptual connections; specifically, in KoT-KSM subdomains. Second, the above-mentioned studies allow for a complementary analysis of the mathematical connections established by in service and preservice teachers at different educational stages (Primary and Secondary). Moreover, the mathematical topics taught by in service and preservice teachers are of a different nature. In study 1, a topic corresponding to a geometrical context is taught, while in study 2, a topic corresponding to a numerical context is taught. Finally, the types of data analysed are also different. While in the first case they correspond to written resolutions of questionnaires, in the second case they correspond to recordings of classroom sessions. All of the above diversity, in terms of data, contexts, topics, levels and teachers’ experience allows for a complementary analysis that leads to more robust results in terms of the relationship between the establishment of connections and the specialised knowledge of teachers who teach mathematics. The first study analysed the responses to a written questionnaire, on the area of flat figures, from a group of 147 preservice teachers (PST) in the third year of the Primary Education Degree. PSTs had one week to answer the questionnaire and send it in pdf or word format. The questionnaire consisted of a total of eight tasks: three tasks that respond to contexts of equal distribution or comparison and reproduction of shapes, and where the use of formulas was prohibited (Tasks 1, 2 and 3); two measurement tasks involving the use of different procedures (Tasks 4 and 5); a task of classification of statements and another of definition of the concept of area (Tasks 6 and 7); finally, a task of analysis of students’ answers (Task 8). In this manuscript, tasks 4 and 8 of the questionnaire are analysed. Task 4 (Figure 2) was designed by the researchers and PSTs had to solve it using two or three different procedures. Task 8 (Figure 3) is based on a previous study (Caviedes et al. [31]) and PSTs had to analyse a high school student’s resolution of an area task. In task 4 PSTs established a greater number of mathematical connections, as the task required to be answered using different procedures. This allowed PSTs to use procedures of a geometric and numerical nature in a joint manner. In the case of task 8, when PSTs were asked to analyse students’ answers, mathematical connections linked to their mathematical and pedagogical knowledge were established. This did not happen in the rest of the tasks. Mathematics 2022, 10, x FOR PEER REVIEW 7 of 29 Mathematics 2022, 10, 4010 7 of 24 mathematical and pedagogical knowledge were established. This did not happen in the rest of the tasks. Figure 2. Figure 2. Task Task44of ofthe thequestionnaire. questionnaire. Task 88 proposed Task proposed thethe following followingsituation: situation:“Tània “Tània and and Laia areare Laia in the 2nd2nd in the yearyear of Sec- of ondary Education Secondary (ESO). Education In maths (ESO). class, In maths the the class, teacher asksasks teacher them to calculate them the area to calculate of two the area of Mathematics 2022, 10, x FOR PEER REVIEW squares two using squares more using than more oneone than procedure. procedure. Tània Tàniahashasdifficulty difficultyfor forcalculating calculating the the area 8ofof 29 area of one of one of the the squares. squares. Laia Laia has has been been able able to to solve solve the the whole whole task task using using different different procedures. procedures. What knowledge What knowledge do do you you think think Tània Tània needs needs toto be be able able to to solve solve the thetask? task? How How could could the the teacher help teacher help her? her? Justify Justify your your answer”. answer”. Figure Figure 33 illustrates illustrates the the students’ students’ answers. answers. Figure3.3.Figure Figure Figureaccompanying accompanying Task Task 8. 8. The analysis was carried out with the use of MAXQDAplus software in order to iden- tify the elementary connections established by preservice teachers (Figure 4). These ele- mentary connections are established in written justification and in the resolution process followed by PSTs. Connections are stablished between representations, procedures and properties (KoT), and can be auxiliary (KSM) and linked to knowledge about teaching Mathematics 2022, 10, 4010 8 of 24 The analysis was carried out with the use of MAXQDAplus software in order to identify the elementary connections established by preservice teachers (Figure 4). These elementary connections are established in written justification and in the resolution process followed by PSTs. Connections are stablished between representations, procedures and properties (KoT), and can be auxiliary (KSM) and linked to knowledge about teaching strategies (KMT) and students’ difficulties/strengths (KFLM). Once the connections have Mathematics 2022, 10, x FOR PEER REVIEW 9 of 29 been identified, a frequency count is carried out. This is done using the aforementioned software. Figure Figure 4. 4. Example Example of of the the analysis analysis performed performed in in study study 1. 1. The second study The second studyanalysed analysedthe thediscursive discursive interactions interactions between between the teacher the teacher and aandclassa class of 12–13 year-old students during the teaching of powers and of 12–13 year-old students during the teaching of powers and integers. The data used integers. The data used to to perform perform the the analysis analysis areare video video andand audio audio recordings recordings of of eight eight regular regular class class sessions sessions of of 60 60 min min and and their their corresponding corresponding transcriptions. transcriptions. To To collect collect the the information, information, non-participant non-participant observation observation was was used used (Caldwell (Caldwell andand Atwal Atwal [32]; [32]; Cohen Cohen et et al. [27]). Therefore, al. [27]). Therefore, thethe class class group group waswasobserved, observed,listened listenedtotoand andvideo-recorded, video-recorded, andandnotes noteswere taken, were without taken, withoutinter- in- vening tervening in the in thedesign design of of the thesessions, sessions,thetheactivities activitiesproposed, proposed,or orthe thedevelopment development of of the the sessions. sessions. TheThe data-processing data-processing consists consists of of viewing viewing all all eight eight class class sessions sessions from from the the teaching teaching unit unit on on integers. integers. The The aim aim ofof this this first first viewing viewing is is to to identify identify andand define define episodes episodes in in which which connections connections appear. appear. Each Each session session was was viewed viewed at at least least twice twice by by two two researchers researchers thatthat took took notes notes of relevant relevant moments in the videos in which potential connections may emerge. The average average duration duration of of the the episodes episodes waswas 55 min min and and 5555 s. s. In both studies, the analysis is divided into two phases In both studies, the analysis is divided into two phases (Figure (Figure 5).5). The The first first phase phase is is divided into three stages. The first stage consists of the identification and characterisation divided into three stages. The first stage consists of the identification and characterisation of mathematicalconnections, of mathematical connections,following following thethe categories categories of elementary of elementary connections connections pre- presented sented in Section 1.2. These elementary connections can form more complex connections in Section 1.2. These elementary connections can form more complex connections if the relation relationship between them is of an explicit nature: O1 −−−−→𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 O2 . if the relationship between them is of an explicit nature: 𝑂𝑂1 �⎯⎯⎯⎯� 𝑂𝑂2 . In the case of the first study, given the nature of the data (only written justifications) it is not possible to look for evidence of complex connections (formed by more than one elementary connection). The elementary connections identified in PST justifications do not present explicit relations between them, but are established separately. In the second study, the three elements (𝑂𝑂1 , 𝑂𝑂2 and relation) are identified from explicit interventions of the teacher and students, so it was possible to identify more complex connections. The connections in study 1 are named QC (questionnaire connections) and the connections in study 2 are named CC (classroom connections). The second stage consists of characterising the knowledge mobilised by PSTs and the Mathematics 2022, 10, x FOR PEER REVIEW 10 of 29 analysing teachers’ written and oral discourse. In the second phase, these ambiguities are Mathematics 2022, 10, 4010 9 of 24 analysed, emphasising the characterisation of connections related to more than one sub- domain of knowledge, e.g., KoT or KSM. Figure 5. Figure 5. Phases Phases of of the the analysis analysis process. process. 3. Results In the case of the first study, given the nature of the data (only written justifications) it is not possible 3.1. Global Resultstooflook for evidence the First and Second ofStudy complex connections (formed by more than one elementary connection). The elementary connections identified in PST justifications do In the first not present study explicit (characterisation relations between them, of PSTs’ knowledge but are established mobilised in the separately. resolution In the second and analysis of area tasks), a total of 169 elementary intra-mathematical study, the three elements (O1 , O2 and relation) are identified from explicit interventions conceptual con- nections are identified, in the first stage of Phase 1 of the analysis. These of the teacher and students, so it was possible to identify more complex connections. The connections are identified in the resolution process followed by PSTs in Task 4, and in connections in study 1 are named QC (questionnaire connections) and the connections in the analysis process of student study 2 are responses named CCfollowed (classroom byconnections). PSTs in Task 8 of the questionnaire. Only 5 out of 11 typesTheof elementary intra-mathematical second stage consists of characterising connections are identified. the knowledge No complex mobilised by PSTs connec- and the teacher when these connections are established, using the tools of the MTSKcoordinate tions are identified, as the PSTs’ written justifications did not explicitly ele- model. Finally, mentary connections to form more complex connections. Table 1 shows the third stage identifies the relationships between the types of knowledge mobilised that the smallest number when of connections mathematical established connections are by PSTs corresponds established. The resultsto of particular the first case phaseconnections. of analysis This means showed thatthat connections associated the classification given by MTSKwith the use presents model of different geometrical ambiguities properties when of analysing the figures teachers’ were and written moreoraldifficult to establish discourse. by PSTs. In the second In addition, phase, it can be observed these ambiguities that are analysed, the rest of thethe emphasising intra-mathematical characterisation ofconnections connectionsofrelated conceptual to moretype present than a similar fre- one subdomain of quency. knowledge, e.g., KoT or KSM. Table 3. 1. Intra-mathematical questionnaire connections (QC) identified in study 1. Results 3.1. Global Results of the First and Second Study Type of IMCC Number In the first study of Connections (characterisation Total and of PSTs’ knowledge mobilised in the resolution Connections analysis of area tasks), a total of 169 elementary intra-mathematical conceptual connections 2, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, are identified, in the first stage of Phase 1 of the analysis. These connections are identified ER 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 39 in the resolution process followed by PSTs in Task 4, and in the analysis process of student 167, 168, 169 responses followed by PSTs in Task 8 of the questionnaire. Only 5 out of 11 types of 1, 6, 91, 92, 93, 94, 95, 106, 107, 108, 109, 110,connections 111, 112, 113,are 114, 115, 116, 117, 118, AR elementary intra-mathematical identified. No complex 33 connections are 119, 120,121,identified, 122, 1238, as 124, 125, the 126, written PSTs’ 127, 128,justifications 129, 130, 131 did not explicitly coordinate elementary 7, 8, 12, 50, 51, 52, 53, 54,to55,56, connections form 57, 58,complex more 59, 60, 61, 62,63, 64, 65, connections. 66, 67,68, Table 1 shows69,that 70, 71, the smallest number of P 45 72, 73, 74, 75, 76, 77, 78, 79, connections 80, 81, 82,by established 83,PSTs 84, 85, 86, 87, 88, to corresponds 89,particular 90, 103 case connections. This means 3, 4, 5, 9, 13,that 14, 15,16, 17, 18, associated connections 19, 20, 21, 22, with23,the 24,use 25, of 26,different 27, 28, 29, 30, 31, 32, properties geometrical 33, 34, of the figures EP 43 of the 35, 36, 37, 38, were39, more 40, 41,difficult 42, 43, 44, 45, 46, 47,by48, to establish 49, 104, PSTs. 105 In addition, it can be observed that the rest PC 10, 11, 96, 97,intra-mathematical 98, 99, 100, 101, 102connections of conceptual type present a similar frequency. 9 In the second study (characterisation of the knowledge a teacher uses in a classroom context when making mathematical connections), 34 complex connections triggered and constructed by discursive interactions between the teacher and students were identified. These complex connections consisted of elementary connections that could vary from 1 to Mathematics 2022, 10, 4010 10 of 24 Table 1. Intra-mathematical questionnaire connections (QC) identified in study 1. Type of IMCC Connections Number of Connections Total 2, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, ER 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 39 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169 1, 6, 91, 92, 93, 94, 95, 106, 107, 108, 109, 110, 111, 112, 113, AR 114, 115, 116, 117, 118, 119, 120,121, 122, 1238, 124, 125, 126, 33 127, 128, 129, 130, 131 7, 8, 12, 50, 51, 52, 53, 54, 55,56, 57, 58, 59, 60, 61, 62,63, 64, 65, P 66, 67,68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 45 84, 85, 86, 87, 88, 89, 90, 103 3, 4, 5, 9, 13, 14, 15,16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, EP 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 43 46, 47, 48, 49, 104, 105 PC 10, 11, 96, 97, 98, 99, 100, 101, 102 9 In the second study (characterisation of the knowledge a teacher uses in a classroom context when making mathematical connections), 34 complex connections triggered and constructed by discursive interactions between the teacher and students were identified. These complex connections consisted of elementary connections that could vary from 1 to 6. For example, CC13 and CC31 were made up of one elementary connection, while CC12 was made up of six elementary connections. Table 2 shows 34 complex connections involving a total of 84 elementary connections in their formation. It can be seen that most of the connections identified were intra-mathematical conceptual connections (23), followed by process-related connections (8) and extra-mathematical connections. The frequency of elementary connections is detailed in Table 3. Table 2. Complex classroom connections (CC) identified in study 2 and number of elementary connections that form them. Number of Elementary Connections 1 2 3 4 5 6 Total EMC 13, 31 4 3 Categories of 7 connections IMCRP 27, 33, 29 5, 20 1 8 17 2, 3, 6, 8, 11, 16, 18, 19, 9, 12, IMCC 10, 28, 34 15 14 23 21, 23, 24, 25, 26, 30, 32 22 Total 7 18 2 2 2 3 34 Table 3. Types of elementary connections identified in study 2. Type of Elementary Connection ER CFR CFD MP P EP JU IM PC GE Frequency of connections 8 9 8 4 17 3 15 2 3 15 3.2. Mathematical Knowledge Mobilised by PSTs and Teachers In the second stage of Phase 1 of the analysis, PSTs knowledge mobilised during the establishment of mathematical connections, in the resolution and analysis of area tasks (Study 1), is characterised. Five types of elementary intra-mathematical conceptual connections are distinguished. For each connection, the knowledge mobilised by PSTs is identified. For this, the categories defined by each of the subdomains of MTSK model and described in Section 1.1 are considered. Mathematics 2022, 10, 4010 11 of 24 Table 4 shows that the subdomain that is most mobilised is KoT. Likewise, it is possible to observe that when connections related to procedure-concepts (P), equivalent procedures (EP) and properties (PC) are stablished, the knowledge mobilised by PSTs is mainly KoT and KFLM. Furthermore, it is observed that KMT is mobilised when con- nections with procedures-concepts (P) and with equivalent procedures (EP) occur. Since all KMT connections are also included in KoT, it is possible to consider a relationship between conceptual-procedural knowledge (P) and KMT mobilisation. Something similar occurs with KSM, since two of the three connections present in that subdomain are found in KoT, specifically in equivalent procedural knowledge (EP) and procedural-conceptual knowledge (P). The evidence of KFLM mobilisation, during the establishment of connec- tions related to procedures and properties, allowed us to identify that PSTs mobilise their KFLM associated with knowledge about students’ strengths/difficulties (Figure 5). Thus, connections for KFLM (Table 4) are related to procedural and conceptual knowledge that can be useful for the resolution of the task. For example, in connections QC10 and QC3, PST 3 points out that the student who solves the task by making mistakes needs to acquire knowledge about the properties of squares and right triangles, as well as the decomposition of figures into other figures (Figure 6). Although it is true that PST 3 does not make explicit the properties of the square, we can infer that it refers to the fact that this figure can be divided into two isosceles right triangles, by tracing one of its diagonals, as this is what was shown in the resolution to be analysed. Table 4. Specialised knowledge mobilised during the establishment of mathematical connections (QC) in study 1. Categories of Specialised Knowledge and Number of Mathematical Connections Related to Them IMCC Connections KoT KSM KMT KFLM 2, 132, 133, 134, 135, 136, 137, 138, 39, 40, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, ER 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169 1, 6, 91, 92, 93, 94, 95, 106, 107, 108, 109, 110, 111, 112, 113, 114, 125, 116, AR 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131 8, 7, 50, 51, 52, 53, 53, 55,56, 57, 58, 59, 60, 61, 62,63, 64, 65, 66, 67,68, 69, P 8, 12, 103 8 7, 8, 60, 61, 62,63, 64 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89. 90 3, 4, 9, 13, 14, 15,16, 17, 18, 3, 5, 13, 14, 15,16, 17, 18, 3, 9, 13, 14, 15,16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 19, 20, 21, 22, 23, 24, 25, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 26, 27, 28, 29, 30, 31, 32, 26, 27, 28, 29, 30, 31, 32, EP 3 33, 34, 35, 36, 37, 38, 39, 33, 34, 35, 36, 37, 38, 39, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 40, 41, 42, 43, 44, 45, 46, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49 47, 48, 49, 104, 105 47, 48, 49 10, 11, 96, 97, 98, 99, 100, PC 10, 11, 96, 97, 98, 99 101, 102 3, 5, 13, 14, 15,16, 17, 18, 3, 9, 13, 14, 15,16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 19, 20, 21, 22, 23, 24, 25, 3, 4, 9, 13, 14, 15,16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 32, 26, 27, 28, 29, 30, 31, 32, EP 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 3 33, 34, 35, 36, 37, 38, 39, 33, 34, 35, 36, 37, 38, 39, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49 40, 41, 42, 43, 44, 45, 46, 40, 41, 42, 43, 44, 45, 46, Mathematics 2022, 10, 4010 12 of 24 47, 48, 49, 104, 105 47, 48, 49 PC 10, 11, 96, 97, 98, 99, 100, 101, 102 10, 11, 96, 97, 98, 99 “The knowledge that Tània needs to internalise is diverse…we need to understand the properties of the square…In addition, one concept that will be very useful for us are the spins, to observe that the black geometric figure is also a square…another would be the Pythagorean theorem, as the formula is necessary to discover the diagonal and to find out the area of the black square. … fractions, as they are a key element to be able to solve this exercise… the properties of triangles, more specifically right triangles… these can help us when splitting the figure, in order to know the area of the black square”. Figure6.6.Analysis Figure Analysis of of PST PST 33 to to aa student’s student’sresolution resolution(first (firstpart). part). Inthe In thecase case of of connection connection QC8, QC8,the thePST PSTmentions mentionsthe need the needto to know know and operate and with operate with thePythagorean the Pythagorean theorem, theorem, itit is is inferred inferredthatthatthis thisisisrelated relatedtoto thetheproperties propertiesof triangles of triangles (QC10),since (QC10), sincewhen when these these are are rectangles, rectangles,the thelength lengthofofthe hypotenuse the hypotenuse cancan be be obtained obtainedby by means of this theorem. In connection QC7, the PST points out that means of this theorem. In connection QC7, the PST points out that the student needs the student needs to to acquire knowledge about fractions due to their usefulness in dividing a acquire knowledge about fractions due to their usefulness in dividing a whole into parts ofwhole into parts of equal equal area area andand in in calculatingthe calculating thefraction fractionofof aa number number (1/3(1/3ofof12). Finally, 12). in in Finally, connection connection QC14, the PST refers to “spins”, it is inferred that she refers to rotational QC14, the PST refers to “spins”, it is inferred that she refers to rotational movements movements thatthat allow to recompose a figure into a different one. allow to recompose a figure into a different one. Figure 7 shows the KMT mobilised by PST 3. Thus, the connections for KMT (Table Figure 7 shows the KMT mobilised by PST 3. Thus, the connections for KMT (Table 4) 4) are mainly associated with teaching strategies to guide a student. For example, connec- are mainly associated with teaching strategies to guide a student. For example, connection tion QC8 relates to conceptual-procedural knowledge about Pythagorean theorem, which QC8 relates to conceptual-procedural knowledge about Pythagorean theorem,13which Mathematics 2022, 10, x FOR PEER REVIEW of 29 can can be used as a teaching strategy. Connection QC16 relates to the identification of errors be used as a teaching strategy. Connection QC16 relates to the identification of errors about about the diagonal and side of a square. Connection QC17 relates to the use of various the diagonal and side of a square. Connection QC17 relates to the use of various procedures procedures to help students to correct errors. to help students to correct errors. “To help her, we have to concretise all those concepts that we are not clear about…to do this, we will observe how the diagonal of the square does not measure the same as one of its sides (as justified in her answer) …we have several options, one…is to have her position the side of the square parallel to observe that the diagonal is longer. An- other would be to give her a ruler to measure the side… when dealing with the Py- thagorean theorem, the first option is better… then the measure of the diagonal can be found through the formula… she makes an incorrect fraction. In order for her to observe the error, we will make her divide the side into three segments and put in each segment how many centimetres there are… She will then be able to observe vis- ually, or by means of a sum, that 1/3 of 12 cm is not 8 cm. Finally, we will intervene in the division of figures into small triangles… we will ask a good question: What is the geometric figure with which we can divide the square (only using that figure)? From here on, we can see that Tània will try to give an answer by trial and error”. Figure Figure 7. 7.Analysis AnalysisofofPST PST3 3totoa student’s resolution a student’s (part resolution two). (part two). ItIt is is inferred inferred that that PST PST 33 proposes proposes aa manipulative manipulative type type of of strategy, strategy, as as she she points points out out that that the student the student can can compare compare the the sideside andandthethe diagonal diagonal of of thethe square square byby placingboth placing bothofofthem them in parallel, in parallel, in order in order to observe to observe their their lengths. lengths. PSTPST 3 alsopoints 3 also pointsout outthat that such such aa proce- procedure dure would help to work on the Pythagorean theorem. Connections would help to work on the Pythagorean theorem. Connections QC3 and QC5 are associated QC3 and QC5 are associated with procedural knowledge about decomposition of figures, with procedural knowledge about decomposition of figures, as PST 3 indicates that it isas PST 3 indicates that it is possible possible to guidetotheguide the student student throughthrough a procedure a procedure that that involves involves measuring measuring each each of the of the three segments into which the side of the large square is divided. In turn, three segments into which the side of the large square is divided. In turn, connection QC18 connection QC18 relatesrelates to the to the use ofuse of questions questions that aimthattoaim to evoke evoke alternative alternative procedures. procedures. Inrespect, In this this re- PST spect, PST 3 asks a question that aims to evoke procedures for decomposing figures. 3 asks a question that aims to evoke procedures for decomposing figures. Regarding KoT, it is possible to observe that PSTs mobilise this subdomain in relation to representations, properties and procedures. Thus, the knowledge mobilised in the con- nections corresponding to KoT (Table 4) is associated with the use of geometric and sym- bolic representations, either jointly or separately. For example, connection QC1 is related to decomposing and recompositing procedures performed by PST 7 (Figure 8) on triangles 1 and 2, in order to subsequently apply the area formula. Connection QC6 would be re- Mathematics 2022, 10, 4010 13 of 24 Regarding KoT, it is possible to observe that PSTs mobilise this subdomain in relation to representations, properties and procedures. Thus, the knowledge mobilised in the connections corresponding to KoT (Table 4) is associated with the use of geometric and symbolic representations, either jointly or separately. For example, connection QC1 is related to decomposing and recompositing procedures performed by PST 7 (Figure 8) on triangles 1 and 2, in order to subsequently apply the area formula. Connection QC6 would be related to the use of the decomposition of figures into congruent units, in order to obtain the area in an additive way. Connection QC10, with the use of KoT associated with the properties of polygons, for example, when PST 7 uses the area formula, it is inferred that she recognises that a triangle is half of a square with the same base and height that contains it. Another example is evident in connections QC3, QC4 and QC9, which are related to PST 7’s strategic use of procedures, e.g., cutting two-dimensional space to compare areas, fractioning the unit of measurement to facilitate the process of measuring areas, and rearranging a figure into a new figure. From this, it is inferred that QC11 connection is associated with the properties involved in area measurement processes Mathematics 2022, 10, x FOR PEER REVIEW (area conservation). Likewise, it is possible to observe that connection QC3 is related14 of to 29 more elementary procedures for the calculation of area, in this sense it is linked to KSM. Figure 8. Figure 8. Resolution of PST Resolution of a PST to Task 7 to Task 4. 4. Connection QC8 Connection QC8isislinked linkedtotoconceptual-procedural conceptual-procedural knowledge. knowledge. Thus, Thus, QC8QC8 connec- connection tion is auxiliary in KSM and procedural in KoT. It is observed that PST is auxiliary in KSM and procedural in KoT. It is observed that PST 7 identifies that the legs7 identifies that thetriangle of legs of 3triangle 3 correspond correspond to hypotenuses to hypotenuses of other of other right right triangles triangles (not given(not given by the by the exercise). exercise). This allowsThisher allows her knowledge to use her to use her knowledge of Pythagorean of Pythagorean theoremthe theorem to calculate to calculate length of the the length legs of the (seen legs (seen asofhypotenuses as hypotenuses of other other triangles). Thus,triangles). Thus,ofthe the calculation calculation areas of areas of right triangles of righttotriangles allows allows that evoke content to evoke servescontent that serves as a support as aresolution for the support for of the the resolution task. In theofcase the task. of In the case connection of connection QC12, the right QC12, the located triangles right triangles located onallow on the geoboard the geoboard PST 107allow her to evoke to evoke knowledge knowledge of Pick’s(Figure of Pick’s theorem theorem9), (Figure through9), through a conceptual-procedural a conceptual-procedural relationship,rela- and tionship, which and which is linked is linked to KoT-KSM. to KoT-KSM. Something similarSomething occurs withsimilar occurs connection with referring QC103, connectionto QC103, the use of referring to the use offormula the semiperimeter the semiperimeter formulaoffor for the calculation the calculation of areas. areas. Mathematics 2022, 10, x FOR PEER REVIEW 15 of 29 Mathematics 2022, 10, 4010 14 of 24 Figure 9. Figure 9. Resolution Resolutionofofa PST PST to 107Task 4. 4. to Task In the In the second secondstudy, study,MTSK MTSKmodel modelisisused usedtotocharacterise thethe characterise knowledge knowledgemobilised mobilised by by a teacher when making mathematical connections in a classroom practice context. a teacher when making mathematical connections in a classroom practice context. From the From the complex connections triggered by the teacher, the knowledge she mobilises in complex connections triggered by the teacher, the knowledge she mobilises in making such making such connections is characterised. Table 5 shows the results of the types of connections is characterised. Table 5 shows the results of the types of knowledge mobilised knowledge mobilised by the teacher in the establishment of each complex connection. by the teacher in the establishment of each complex connection. Table 5. Relationship between the establishment of mathematical connections (CC) and types of knowledge mobilised by the teacher in study 2. Type of Categories of Specialised Knowledge and Number of Mathematical Connections Related to Them Connection KOT KSM KPM KMT KFLM KMLS EMC 4 4, 31 13, 31 13, 31 3, 6, 8, 9, 10, 11, 12, 2, 6, 10, 11, 12, 14, IMCC 14, 16, 19, 21, 22, 23, 9, 10, 16, 20, 22 15, 16, 18, 21, 22, 6, 12, 16, 21, 23, 24, 25, 30, 34 23, 24, 26, 30, 32 25, 26, 28, 30 IMCRP 1, 5, 7, 17, 20, 29 1 1, 5, 17, 20, 27, 7 5, 33 29, 33 About 57 out of 84 elementary connections identified were triggered by teacher’s interventions and 27 were triggered by students’ interventions. Table 6 shows the types of knowledge mobilised in the establishment of the elementary connections that were triggered by the teacher interventions. Table 6. Relationship between the establishment of elementary connection and the specialised knowledge mobilised by the teacher in study 2. Type of Elementary Categories of Specialised Knowledge and Number of Mathematical Connections Related to Them Connection KOT KSM KPM KMT KFLM KMLS 27, 33, 36, 39, 40, 35, 36, 37, 39, 40, ER 33 43 43 CFR 84 34, 44, 45, 46, 47 CFD 78 7, 54, 74, 78 74, 76, 78 MP 11, 80 38, 80 38, 80 P 9, 10, 70 10, 70 EP 26, 56, 57, 65, 71 66 66 71 5, 15, 20, 21, 49, 5, 15, 49, 52, 58, JU 20, 22, 49, 60, 68 15, 49, 68 60 52, 60, 68, 77 75, 77 IM 41 41 PC 1, 18 4, 8, 12, 13, 17, 12, 13, 17, 29, 31, GE 4 4, 28, 29, 83 17, 83 19, 28, 29, 31 82 Mathematics 2022, 10, 4010 15 of 24 The analysis of the knowledge mobilised by the teacher in each connection allows us to observe some patterns in the mobilisation of this knowledge. First, it can be observed that the two most mobilised subdomains are KoT and KMT. Second, it can be observed that when connections related to processes (CRP) are established, the knowledge mobilised by the teacher is mostly KoT and KPM. Likewise, it can be observed that KPM is mainly mobilised when intra mathematical conceptual connections (IMCC) and connections related to processes (CRP) occur. If we take into account that the number of CRP connections is one-third of the IMCC, we can infer a relationship between the establishment of CRP connections and the mobilisation of KPM type of knowledge by the teacher. The evidence of the teacher’s mobilisation of KPM during the establishment of CRP connections allowed us to identify two patterns of knowledge. First, the knowledge mobilised by the teacher during the establishment of connections CC1, CC5, CC27 and CC33 emphasised general characteristics of mathematical practice (KPM). During connection CC1 the teacher emphasised the difference between demonstration and inductive Mathematics 2022, 10, x FOR PEER REVIEW 17 intuition of 29 (Figure 10). During connection CC5 the teacher emphasises the importance of knowing how to explain where mathematical results come from (Figure 11). During connection CC27, the teacher CC27, thepoints outpoints teacher the importance of knowing out the importance how to explain of knowing how tothings, explainwhile things,inwhile connection in CC33 the teacher emphasises the importance of thoroughness in communication connection CC33 the teacher emphasises the importance of thoroughness in communica- by stating “we tionshould answer by stating “we in the same should terms answer thatsame in the the problem asks terms that theus to. When problem asksthe usproblem to. Whenasks us to do divisions that are whole quotient then we speak that language. You the problem asks us to do divisions that are whole quotient then we speak that language.haven’t told You haven’t told us anything about decimals”. us anything about decimals”. Because suspicion is one thing, and being sure is another […] we are going to make our first mathematical demonstration. We are going to prove that our suspicions are true. [Suspicion proven. Now, a conclusion that I like very much. If something happens18once, Mathematics 2022, 10, x FOR PEER REVIEW of 29 twice or four times, it doesn’t necessarily always happen. Figure 10. Figure 10. Fragment Fragment of ofthe theteacher’s teacher’sintervention during intervention thethe during connection CC1. connection CC1. Igor: What I don’t understand is why two minuses make a plus. […] Laia: Because, yes, there’s no explanation. Teacher: That’s where I come in, it does have a bit of an explanation, it’s just that it’s not easy, but it does have an explanation. Figure 11. Figure 11. Fragment Fragmentofofthe thesequence sequenceofof interventions that interventions defined that connection defined CC5. connection CC5. Second, Second,the theknowledge knowledge mobilised mobilisedin connections in connectionsCC17, CC20CC20 CC17, and CC29 and CC29is related to is related thethe to strategic useuse strategic of of procedures. procedures.In the casecase In the of connections of connections CC17 CC17andand CC29 CC29the teacher the teacher emphasised the emphasised the importance importanceof ofbeing beingable abletotodecide decidewhy whyoneoneprocedure procedure is is more more appropri- appropriate ate than another in a given context. For example, during connection than another in a given context. For example, during connection CC17, the teacher CC17, the teacher assessed assessed the appropriateness of a procedure based on matching negative the appropriateness of a procedure based on matching negative factors when performing factors when performing natural natural powers powers ofbase: of negative negative “It’s base: true, “It’s so withtrue,Ivan’s so with Ivan’s trick, trick,out it turns it turns that out hey . . . that hey… What Pedro says is true, but it’s tougher”. In the same way, during What Pedro says is true, but it’s tougher”. In the same way, during connection CC29, the connection CC29, the teacher teacher valued thevalued the appropriateness appropriateness of a use of a specific specific of theuseassociative of the associative property property by stating “you and Clara should associate in a different way [ . . . ] I find it a bit easier andand by stating “you and Clara should associate in a different way […] I find it a bit easier that is that is why I have shown it to you, but it is not necessary”. Finally, during the CC20 con- why I have shown it to you, but it is not necessary”. Finally, during the CC20 connection, nection, the teacher emphasises the importance of looking for alternative procedures in the teacher emphasises the importance of looking for alternative procedures in case of case of blocking by commenting “You have to be resolute, and if you don’t remember the blocking by commenting “You have to be resolute, and if you don’t remember the powers, powers, fine, nothing happens, but imagine”. fine, nothing happens, but imagine”. The analysis the teacher knowledge mobilised when her interventions trigger ele- The analysis the teacher knowledge mobilised when her interventions trigger elemen- mentary connections allowed us to observe some patterns in the mobilisation of mathe- tary connections allowed us to observe some patterns in the mobilisation of mathematical matical knowledge, related to the establishment of elementary connections in the class- knowledge, related to the establishment of elementary connections in the classroom. First, room. First, it is identified that the types of knowledge mobilised when establishing com- it monis identified that the types feature connections (CFR of knowledge and CFD) were mobilised mostlywhenlinkedestablishing to pedagogical common feature content connections knowledge (KMT and KFLM). Second, it can be observed that the mobilisation of KPM (KMT (CFR and CFD) were mostly linked to pedagogical content knowledge is linked to elementary connections of justification and generalisation. Finally, it can be ob- served that the elementary connections of representation are related to the mobilisation of two types of knowledge that, in general, were mobilised at more points in time, namely KoT and KMT. Finally, it can be observed that in the establishment of the elementary con- Mathematics 2022, 10, 4010 16 of 24 and KFLM). Second, it can be observed that the mobilisation of KPM is linked to elementary connections of justification and generalisation. Finally, it can be observed that the elemen- tary connections of representation are related to the mobilisation of two types of knowledge that, in general, were mobilised at more points in time, namely KoT and KMT. Finally, it can be observed that in the establishment of the elementary connections of justification all types of knowledge were mobilised except KMLS. A Critical View on Mathematical Connections within MTSK Model We seek to observe whether the nature of the connections identified allows us to pinpoint the need for a refinement of MTSK model. To do so, we identified connections that involved knowledge that could admit different classifications, according to the current definition of MTSK model. Regarding the first study, potential ambiguities were identified in three intra-mathematical conceptual connections. All the ambiguities identified are associated with difficulties in using the MTSK to classify the knowledge mobilised by PSTs. Specifically, in the three cases that present ambiguities, difficulties are observed when using the KoT- KSM subdomains, which indicates connections associated with a conceptual-procedural knowledge could be part of more than one subdomain of knowledge in the MTSK. This is the case of connections QC3, QC8 and QC12. In this sense, knowledge about the properties of right triangles, as well as the procedures that allow their area to be calculated, would be linked to what and how mathematics teachers know the content they teach (Carrillo et al. [2]), and would therefore be part of the KoT. Similarly, if we consider, for example, that the properties of right triangles are prior knowledge that allows us to evoke another type of procedure (Flores-Medrano [17]) we could classify the use of the Pythagorean theorem Mathematics 2022, 10, x FOR PEER REVIEW 19 of 29 (QC8) as an auxiliary connection within KSM (Figure 12). Similarly, if we consider that the characteristics of the triangles constructed in geoboard can be a previous knowledge that allows us to evoke, later on, Pick’s theorem (QC12), this theorem could be positioned as an as an auxiliary auxiliary connection, connection, or or as as aa complexity complexity oneone if if we we take take into into account account the the temporality temporality and complexity and complexity of of it. it. However, However, given given that that in in both both cases cases the the action action of of “evoking” “evoking” is is taken taken into consideration, into consideration, both both theorems theorems are are positioned positioned in in the the category category of of auxiliary auxiliary connections. connections. Likewise, ifif itit is Likewise, is taken taken into into consideration consideration thatthat both both theorems theorems areare mobilised mobilised at at aa procedural procedural level, the level, the associated associated knowledge knowledge cancan be be linked linked toto KoT, KoT, since since itit can can be be interpreted interpreted as as being being related to related to properties properties and and their their underlying underlying principles, principles, definitions definitions and and procedures procedures (Carrillo (Carrillo et al. et al. ([2]). ([2]). Figure 12. Figure 12. Example Example of of aa resolution resolution process process followed followed by by PST PST 66. 66. Another ambiguous example is the procedure that involves decomposing surfaces (Figure 13) to simplify the area calculation process (QC3). In this case, if such a procedure is considered to be of a prior and elementary nature in the calculation of areas, it could be positioned as a simplification connection within KSM. If it is considered that such a pro- cedure is a prior knowledge that allows, subsequently, to evoke the area formula, it could Mathematics 2022, 10, 4010 17 of 24 Another ambiguous example is the procedure that involves decomposing surfaces (Figure 13) to simplify the area calculation process (QC3). In this case, if such a procedure is considered to be of a prior and elementary nature in the calculation of areas, it could be positioned as a simplification connection within KSM. If it is considered that such a procedure is a prior knowledge that allows, subsequently, to evoke the area formula, it Mathematics 2022, 10, x FOR PEER REVIEW 20 of 29 could well be an auxiliary connection. In this case, we have taken it as prior and elementary knowledge. Moreover, it is also possible to interpret that knowing different ways of decomposing figures, in order to calculate their area, is part of a knowledge of characteristic procedures for the calculation of areas and therefore could be linked to the KoT. Figure 13. Figure 13. Example Example of of aa resolution resolution process processfollowed followedby byPST PST110. 110. With regard With regard toto the the second second study, study,potential potential ambiguities ambiguities are are identified identified inin 13 13 of of the the 34 34 connectionsidentified. connections identified.These Theseambiguities ambiguitiesare arerelated relatedtoto difficulties difficulties in in using using MTSKMTSK to clas- to classify sifyknowledge the the knowledge mobilised mobilised by theby teacher. the teacher. In particular, In particular, in the in the 13 cases 13 cases withwith ambigui- ambiguities, ties, difficulties difficulties are observed are observed when when using using KSM KSM subdomain,especially subdomain, especiallywhenwhenpossible possible inter- inter- conceptual connections conceptual connections were were observed. observed. In In three three complex complex connections connections (CC9, (CC9, CC17 CC17 and and CC29), elementary CC29), elementary connections connections were were identified identified between between operations operations withwith integers integers (powers (powers and multiplication) and and andwith withthe theuse useofofthe associative the associative property property (Figure 14).14). (Figure At first, it was At first, it considered that the teacher was deepening connections of inter-conceptual type, and was considered that the teacher was deepening connections of inter-conceptual type, and therefore knowledge therefore knowledge belonging belonging to to KSM KSM was was mobilised. mobilised. ThisThis type type ofof connection connection can can bebe interpreted as interpreted as an an auxiliary auxiliary typetype of of connection, connection, since since aa mathematical mathematical property property (associative (associative property)is property) is used used with with thethe calculation calculation of of negative negative base base powers. powers. However, However, itit was was observed observed that, although the teacher made detailed explanations related to the connection, the that, although the teacher made detailed explanations related to the connection, the use use made of the associative property was of a procedural type, and was therefore more related made of the associative property was of a procedural type, and was therefore more related to to properties properties and and their their underlying underlying principles, principles, than than to to the the interlinking interlinking systems systems which which bind bind the subject (Carrillo et al. [2]), so it was decided to classify the knowledge the subject (Carrillo et al. [2]), so it was decided to classify the knowledge mobilised Mathematics 2022, 10, x FOR PEER REVIEW mobilised as29KoT. 21 of as KoT. For For example, example, duringduring the interactions the interactions definingdefining the CC9 the CC9 connection, connection, (−2)5 , while calculating while calculating the (−2) , the teacher teacher 5 tries totries to organise organise the students’ the students’ proposals proposals to introduce to introduce a property a property of neg- of negative base powers, which aims to justify a property whose use will be mostly ative base powers, which aims to justify a property whose use will be mostly procedural. procedural. Let’s go first to the associative property and then see if there are other, easier ways. […] I can group them 2 by 2 as I want. So then I can put it like this or I can put it like this. That is, we can associate the signs as we want. […] I have a base with a minus sign, […] and an exponent that is odd, in the end, I don’t need to go through all this trouble, in the end it’s going to give me a minus sign, all of them? Figure14. Figure 14.Excerpt Excerpt from from the the teacher’s teacher’sintervention interventionduring duringthe connection the CC9. connection CC9. Thesame The same ambiguity ambiguity between betweenKoT KoTand andKSMKSMoccurs occursininconnection connection CC17, CC17,where wherethethe teacher introduces an assessment of the use of a procedure proposed by a teacher introduces an assessment of the use of a procedure proposed by a student for student for the the calculationof calculation of(−1) compared (−1) 36 36 comparedto tothe theuse useof of the the associative associative property property to to group groupthetheprod- product uct of the 36 factors into 18 pairs of products that would give a positive result (Figure of the 36 factors into 18 pairs of products that would give a positive result (Figure 15). 15). As As in the previous case, this could be interpreted as an auxiliary inter-conceptual connec- in the previous case, this could be interpreted as an auxiliary inter-conceptual connection tion indicating the mobilisation of KSM. However, the procedural role played by the con- indicating the mobilisation of KSM. However, the procedural role played by the connection nection relates more to knowledge of the properties that apply when operating with inte- relates more to knowledge of the properties that apply when operating with integers, and gers, and can therefore be classified as KoT. can therefore be classified as KoT. Mathematics 2022, 10, x FOR PEER REVIEW 22 of 29 Mathematics 2022, 10, 4010 18 of 24 S: For example, we put the thirty six -1. And for example it would be − times − equals +, + times − equals −, − times − equals +, and so on. T: What Peter says is true, but it’s like much heavier. […] So with these two (pointing to the first two brackets) we make two + signs […] And with these other two—(pointing to the 3rd and 4th) I’ll make another + sign […] Well, we have 18, and as they are all +, what we can do is the associative rule of Pedro signs and then we only have + left. Figure15. Figure 15.Fragment Fragment of of the the sequence sequence of ofinteractions interactionsthat thatdefined definedthe connection the CC17. connection CC17. InInthe the case case of ofconnection connection CC29, CC29,when performing when performingthe operation (−(+9):(+3))((+3) the operation −(−7)) (−(+9):(+3))((+3) −the teacher (−7)) again suggests the teacher the use ofthe again suggests theuse associative property byproperty of the associative stating thatby “At this that stating point“At I think this weI have point thinkone weinhave whichoneyou in and Clara which you(student) and Clarawould like to associate (student) would like differently” to associate and therefore differently” thetherefore and same ambiguity the same occurs as before. ambiguity In the occurs as case of connections before. In the case CC21, CC23 of connections and CC26, CC21, CC23there and are connections CC26, there arebetween different connections operations between that could different be interpreted operations that could as be inter-conceptual connections related to the generative role of mathematical interpreted as inter-conceptual connections related to the generative role of mathematical items in the construction items of other items in the construction (Carrillo of other itemset al. [2]).etHowever, (Carrillo it was identified al. [2]). However, that thethat it was identified knowledge mobilised by the teacher in these connections was not the knowledge mobilised by the teacher in these connections was not part of KSM, part of KSM, as the as the teacher’s interventions had a descriptive and procedural character. In the teacher’s interventions had a descriptive and procedural character. In the case of connec- case of connec- tionsCC21 tions CC21andandCC26, CC26, there there were were connections connectionsbetween betweenmultiplication multiplication and empowerment and empowerment of integers. In connection CC21, the teacher answered a student’s question (Figure 16) by of integers. In connection CC21, the teacher answered a student’s question (Figure 16) by showing an example of how a power of negative exponent can be obtained. In this case, it showing an example of how a power of negative exponent can be obtained. In this case, can be interpreted that the teacher’s intervention establishes a connection of increasing it can be interpreted that the teacher’s intervention establishes a connection of increasing complexity, as positive exponent integer powers are related to a more complex idea, such Mathematics 2022, 10, x FOR PEER REVIEW complexity, as positive exponent integer powers are related to a more complex idea, 23 of 29 as such as negative exponent integer powers. However, the knowledge mobilised by the teacher negative exponent integer powers. However, the knowledge mobilised by the teacher was was classified as KoT, as she limits herself to showing an example without delving into classified as KoT, as she limits herself to showing an example without delving into how how powers of integer base and positive exponent are related to those of negative expo- powers of integer base and positive exponent are related to those of negative exponent. nent. S: If p is smaller than q, then it would be a negative number? T: If I had for example 23 /27 , hey Victoria, is that really 2−4 ? right? Is that what you are saying? And my answer to you is yes. Do you want me to prove it to you? Figure16. Figure 16.Fragment Fragment of of the the sequence sequence of of interactions interactionsthat thatdefined definedthe connection the CC21. connection CC21. 𝑎𝑎 InInthe the case case ofofthetheCC23CC23connection, connection, thethe difference difference between operations between of the type operations of the𝑎𝑎 = type a a 1=y 1𝑎𝑎 y− a𝑎𝑎−= a0 = was discussed 0 was in class. discussed This discussion in class. This discussionwas triggered by a mistake was triggered by a made mistake 𝑎𝑎 0 a 0 by a student made by a studentwhen when simplifying a fraction simplifying a fractionas . In this as . case, In the this teacher case, the responded teacher to the responded 𝑎𝑎 0 a 0 tostudent by saying “Anna, it’s ones, because actually what you’re getting… If I would leave the student by saying “Anna, it’s ones, because actually what you’re getting . . . If I a 0 here, it means it’s kind of hidden, right? Yes, and you’ll have zero for this whole wagonthis would leave a 0 here, it means it’s kind of hidden, right? Yes, and you’ll have zero for (−5 · −5 whole wagon · −5 (−· −55·…−) 5and· − zero 5· − for 5 . .whatever .) and zero it is?”. It could beitinterpreted for whatever is?”. It could thatbe in interpreted this case there is a transversal inter-conceptual connection between division and subtraction that in this case there is a transversal inter-conceptual connection between (moreand division specifically(more subtraction between their neutral specifically elements), between their since neutraltheelements), idea (in this casethe since intuitively idea (indevel- this case oped) of a developed) intuitively neutral element of a is common neutral to many element mathematical is common operations. to many However, mathematical given operations. the procedural However, givennature of the context the procedural nature and ofof the the teacher’s context andintervention, of the teacher’s it was decided to it intervention, classify was decidedthis knowledge to classify this as KoT. knowledge as KoT. InInconnections connections CC11, CC11, CC25 CC25 and andCC32CC32there therewerewereconnections connections between between different nu-nu- different mericalsets: merical sets:natural, natural, integers integers andand decimals; decimals;and andininconnection connection CC2CC2 there there waswasa connection a connection betweenthe between the concepts conceptsofofgreatest greatest common common divisor and and divisor least least common commonmultiple. In the same multiple. In the way as in the previous cases, it can be interpreted that these same way as in the previous cases, it can be interpreted that these were inter-conceptual were inter-conceptual con- nections andand connections thatthat the knowledge the knowledge mobilised mobilisedby theby teacher in theseincases the teacher thesewas of KSM cases was type. of KSM However, in these last four cases it was also observed that the type. However, in these last four cases it was also observed that the teacher’s interventions teacher’s interventions were focused were focused on aon procedimental a procedimental treatment of theofcontents, treatment the contents, whichwhich is why is itwhy wasitdecided was decided to classify the knowledge mobilised by the teacher as KoT. to classify the knowledge mobilised by the teacher as KoT. Moreover, the classification of Moreover, the classification of ambiguitybetween ambiguity betweenKSM KSM and and KoT KoT was wasin infavour favourofofthe theuse useofofKSM. KSM. In In thethe casecaseof the CC1 of the CC1 connection, there is a connection between the least common connection, there is a connection between the least common multiple and the greatest com- multiple and the greatest common mon divisor. divisor. In thisIncase, this case, whatwhat determined determined the classification the classification of theofknowledge the knowledge mobi- by mobilised lised the by the teacher wasteacher was the the depth depth with which withthewhich the connection connection was made was made17), (Figure (Figure 17), byof a by means means of a mathematical demonstration that in the case of a number b being a multiple of a number a, it will always be the case that 𝑚𝑚. 𝑐𝑐. 𝑚𝑚(𝑎𝑎, 𝑏𝑏) = 𝑏𝑏 y que 𝑚𝑚. 𝑐𝑐. 𝑑𝑑(𝑎𝑎, 𝑏𝑏) = 𝑎𝑎. The demonstration of the two results shows how the two concepts are related, which goes beyond the knowledge of properties and procedures and is related to a deep understand- ing of the multiplicative structure of integers. Mathematics 2022, 10, 4010 19 of 24 mathematical demonstration that in the case of a number b being a multiple of a number a, it will always be the case that m.c.m( a, b) = b y que m.c.d( a, b) = a. The demonstration of the two results shows how the two concepts are related, which goes beyond the knowledge Mathematics 2022, 10, x FOR PEER REVIEW 24 of 29 of properties and procedures and is related to a deep understanding of the multiplicative structure of integers. Why? Because suspicion is one thing, being sure is another. Now everybody put down your notebooks, because we are going to do our first mathematical demonstration. We are going to prove that our suspicions are true. Let’s start with the m.c.d. […] There can be no more common divisors, that is, a is the maximum of the common divisors. Suspect proven. Figure17. Figure 17.Fragment Fragment of of the the teacher’s teacher’s interventions interventionsininthe theconnection connectionCC1. CC1. InInthe thecase caseofof connection CC16, thethe difference between thethe operations andand 5 connection CC16, difference between operations −25−2 (−2)5 (−2) 5 was discussed. The discussion was triggered by the intervention was discussed. The discussion was triggered by the intervention of a student who proposed of a student who proposed that that both expressions both expressions were equivalent wererepresentations equivalent representations of the samesince of the same operation, operation, the result ofsince boththe result is the of both same. In her is the same. Inthe response, herteacher response, the teacher emphasised emphasised that that of the mere fact theobtaining mere fact of obtaining equal results does not imply that the operations equal results does not imply that the operations are equal. Although the connection are equal. Although thecan connection can be classified as inter-conceptual, the teacher’s be classified as inter-conceptual, the teacher’s intervention shows knowledge referred to intervention shows knowledge criteria establishing referredoftoequivalence establishingbetween criteria of equivalence operations, mathematical between mathematical which relates opera- to how tions, which relates to how different operations are linked, and therefore relates more to different operations are linked, and therefore relates more to KSM than to KoT. KSM than to KoT. In the case of connection CC22 (Figure 18), there was a discussion on how to solve In the case of connection n CC22 (Figure 18), there was a discussion on how to solve operations of type aa𝑎𝑎n𝑛𝑛. During the discussion, the students proposed different procedures operations of type related to two properties . During the discussion, of the integers, a the students = 1 and a0 =proposed different 1. The teacher procedures explains that all 𝑎𝑎𝑛𝑛 𝑎𝑎a related the to twoprocedures proposed properties of arethe integers, correct = 1 and and𝑎𝑎 makes some 0 1. The teacher 𝑎𝑎 =comments. explainsitthat Although could all be the proposed Mathematics 2022, 10, x FOR PEER REVIEW considered to beprocedures a connection are that correct and makes emphasises thesome comments. properties Although of powers it could 25 ofitbe of integers, 29was considered decided to be a connection to classify it as a KSM, that emphasises since the properties it not only connects the of powers conceptsof integers, it was of multiplication decided and divisionto classify it as abut of integers, KSM,alsosince it not makes only connects explicit how they theare concepts of multiplication connected, emphasising a and division property of integers, that may not be but veryalso makes explicit transparent for thehow they are students, connected, such as a0 = emphasising 1. a property that may not be very transparent for the students, such as 𝑎𝑎0 = 1. Because there are many operations in maths that give the same result and come from different places, shall I tell you one? 2 and 2? 4? 10 − 6? 4. Hey, you have the same result, but they come from signs, from different things. Well, this is an example. Dimitri, and the last one. But at the end of the day it’s the same as doing the above. No, no, no…. I totally disagree, because, here (b) the base is? −2, and on the other hand, here the base is…? 2. They’re two totally different powers, which, well, for some reason give the same result. Figure18. Figure 18.Fragment Fragment of of the the teacher’s teacher’sinterventions interventionsininthe theconnection CC22. connection CC22. 4.4.Discussion Discussion Theabove The above results results show show thatthat the the establishment establishmentofofmathematical mathematical connections connections by by the the teacherand teacher and PSTs PSTs isis linked linked to to the the mobilisation mobilisationofofspecific specific types types of of specialised specialised knowledge. knowledge. First,ititcan First, canbe be observed observed that thatknowledge knowledgeassociated associated with with mathematical mathematical topics (KoT), topics math- (KoT), math- ematicsteaching ematics teaching (KMT), (KMT), and and features featuresofofstudents’ students’learning learning (KFLM) (KFLM) is mobilised is mobilised in most in most ofofthe themathematical mathematical connections connections established. established.This Thisisisrelevant, relevant,since thethe since characterisation characterisationof of KMT and KFLM subdomains does not include the establishment KMT and KFLM subdomains does not include the establishment of mathematical connec- of mathematical connec- tions,but tions, butwithin within the the MTSK, MTSK, these these connections connectionsare areonly onlypart partofof mathematical mathematical knowledge knowledge (MK). For mathematical connections to be included in the practice (MK). For mathematical connections to be included in the practice of PSTs, and of PSTs, and used andand used promoted by teachers, it is necessary that such connections are interpreted promoted by teachers, it is necessary that such connections are interpreted from pedagogi- from pedagog- icalcontent cal contentknowledge knowledge (PCK). (PCK). In In particular, particular,common commonfeaturefeatureconnections connections (CFR andand (CFR CFD)CFD) and connections involving equivalent procedures and connections involving equivalent procedures (EP). (EP). Theanalysis The analysisofofthe thedata datafrom fromthe thesecond secondstudy study allowed allowed us us to to observe observe specific specific relation- rela- tionships ships between between connections connections related related to process to process (CRP)(CRP) andandthe the teacher’s teacher’s mobilisation mobilisation of of KPM. KPM. In this relationship, it is observed the way in which KPM can be linked to the de- velopment of transversal competences in students, such as those related to problem solv- ing, justification and communication of mathematical results. Likewise, the mobilisation of KPM, when the teacher establishes CRP connections, represents a clear example of rel- evant situations in teaching practice where KPM can play a leading role. The identification Mathematics 2022, 10, 4010 20 of 24 In this relationship, it is observed the way in which KPM can be linked to the development of transversal competences in students, such as those related to problem solving, justifica- tion and communication of mathematical results. Likewise, the mobilisation of KPM, when the teacher establishes CRP connections, represents a clear example of relevant situations in teaching practice where KPM can play a leading role. The identification of these situations and connections can be useful for the development of training activities for teachers, aimed at the development of KPM with emphasis on its specific dimension for teaching practice. The analysis of ambiguities in the characterisation of mathematical knowledge (mo- bilised by the teacher and PSTs during the establishment of mathematical connections), allowed us to observe difficulties in the use of MTSK model, when identifying the knowl- edge mobilised during the establishment of connections. First, the proposed differentiation between intra- and inter-conceptual connections, contained in KoT and KSM respectively, is linked to certain difficulties that emerge when classifying the knowledge mobilised by the teacher and PSTs. In this sense, when comparing the different ambiguities identified, it can be observed that the evidence of the knowledge mobilised by the teacher and PSTs when making inter-conceptual connections is more linked to the knowledge of procedures associated with both, the calculation of areas and whole numbers (KoT) and not to the structure of mathematics itself (KSM). On the contrary, the evidence of the knowledge mobilised in connections CC1 and CC22 does reach a level of complexity and elaboration that allows us to affirm that the teacher in study 2 has a knowledge of the structure of mathematics (KSM). This is due to the fact that the connections are detailed from their internal structure. Thus, it is possible that mentioning or describing the emergence of a connection is not sufficient evidence of the mobilisation of knowledge related to the structure of mathematics. It is possible that the interpretation of inter-conceptual connections does not necessarily imply the mobilisation of KSM, as the inter-conceptual character could be determined by a procedural competence on a particular concept. For knowledge about the structure of mathematics to be mobilised, it would be necessary to detail and characterise the connections at a structural level (de Gamboa and Figueiras ([21]). For example, the explicit mobilisation of knowledge about how concepts are related. Thus, the analysis shows that KSM subdomain would have an associated understanding of the structure of concepts and how they are related, which coincides with the definition proposed in Carrillo et al. ([2]), although not with the characterisation of the KSM subdomain. The differentiation between intra- and inter-conceptual connections, and the separa- tion of concepts as disjoint units, contradicts the idea of connection and the perspective of mathematical knowledge in which the MTSK is positioned: “network of systemic knowl- edge structured according to its own rules” (Carrillo et al. [2] (p. 6)). Thus, including intra-conceptual and extra-mathematical connections in the KoT subdomain, leaving inter- conceptual connections in the KSM domain, could be problematic. This is because in many cases the difference between intra- and inter-conceptual connections is very subtle, which can lead to an inappropriate use of the KSM tools. For example, in teaching area, when introducing the idea of a square unit of measurement to measure the area of a rectangle, multiplication is used to efficiently count the number of square units of measurement that cover the rectangle. Therefore, it can be argued that there is an inter-conceptual connection between area concept and the concept of multiplication, which could be classified as an auxiliary connection, within KSM sub-domain. However, the use of multiplication in the calculation of areas is more procedural than structural in nature, so it could be argued that it is an intra-conceptual connection in which the characteristic procedure for the calculation of areas of flat figures is introduced, and this would be an example of KoT. In the case of simplification and complexity connections, it is possible that the knowl- edge linked to their establishment is of two types, KoT or KSM. Flores-Medrano ([19]) mentions, as an example, that knowledge about complexity connections would correspond to a teacher’s interpretation of the meanings associated with fractions of whole numbers in basic education, and when working with fractions of differentials in secondary and Mathematics 2022, 10, 4010 21 of 24 higher education. However, this knowledge associated with meanings could be somewhat confusing, since when talking about meanings, reference is also being made to phenomenol- ogy or contexts of use of a given mathematical object (KoT). In the case of simplification connections, the example proposed in Carrillo et al. ([2]) of relating algebraic expressions to numerical expressions can be part of KoT when it allows a teacher to assess the equiva- lence between two algebraic expressions, or it can be part of KSM if there is an elaborate justification of the relationship between algebraic work, with indeterminate quantities, and arithmetic work with real numbers. In addition, it could be argued that KFLM is also mobilised when the teacher uses this numerical substitution strategy to show students a first approach to assessing equivalence between algebraic expressions, which can be useful for students to avoid common errors when manipulating algebraic expressions, such as ( a + b )2 = a2 + b2 . On the other hand, auxiliary connections present two problems. From the point of view of classification of connections, the use of one mathematical concept (e.g., equations) to develop another mathematical concept (e.g., roots of polynomials), can be interpreted as procedural in nature, and therefore more like knowledge related to KoT than to KSM. From a teacher’s knowledge point of view, knowing different mathematical ideas that can be used when working with a concept is more related to KoT -understood as a “thoroughgoing knowledge of mathematical content (e.g., concepts, procedures, facts, rules and theorems) and their meanings” (Carrillo et al. [2] (p. 7)) than to KSM -understood as the teacher’s knowledge of connections between mathematical items (Carrillo et al. [2] (p. 8)). In this sense, and in accordance with the above, it is possible that what determines the type of knowledge that is mobilised is not the inter- or intra-conceptual character of the connections that are produced, but the level of elaboration of the connection by the teacher. This elaboration provides evidence of a knowledge that goes beyond knowing that relationships exist, as it includes a knowledge of how such relationships are produced. Therefore, the analysis of ambiguities in characterising the knowledge mobilised by the teacher and PSTs when establishing mathematical connections shows examples of the way in which recent research on mathematical connections (e.g., Businskas, [4]; Ro-dríguez-Nieto et al. [10]; de Gamboa and Figueiras [21]; Rodríguez-Nieto et al. [22]; Dolores-Flores and Garcia-Garcia [25]) can contribute to a future refinement of MTSK categories. First, the research does not distinguish between intra- and inter-conceptual connections, which is consistent with the difficulties in differentiating between them, as in the case of the connections discussed above. This suggests the need to refine the distinction that MTSK model makes between KoT and KSM. In the definitions of KSM that can be read in recent publications (Carrillo et al. [2]; Carrillo et al. [17]; Flores-Medrano [19]), KSM is defined by knowledge of inter-conceptual connections. As noted in the previous paragraphs, this definition can be problematic. Mainly because the delimitation of concepts can be fuzzy, making it difficult to differentiate between intra- and inter-conceptual. Second, the characterisation of complex connections as networks of elementary connections, as well as the different types of these elementary connections (representational, common feature, procedural, justification or generalisation) point to different levels of complexity for connections (de Gamboa et al. [5]). In turn, this implies different levels of complexity in the mobilised knowledge. 5. Conclusions The performed analysis allowed us to identify relationships between the subdomains of MTSK model and various types of mathematical connections. The identification of these relationships shows the great usefulness of MTSK model to analyse the knowledge mobilised by teachers when they participate in situations rich in mathematical connections, which confirms the results of de Gamboa et al. ([33,34]). Moreover, these relationships show that recent results in the characterisation of mathematical connections can be useful for better understanding some of the subdomains of MTSK model. Mathematics 2022, 10, 4010 22 of 24 The relationship between the mobilisation of KPM and the emergence of IMCRP-type connections is particularly relevant, as these connections give explicit examples of how KPM can be mobilised in classroom practice. This can be useful to identify classroom situations that can contribute to the characterisation of KPM, which is still under construction (Carrillo et al. [2]). Furthermore, the relationship identified between the establishment of elementary connections (of the EP, CFR and CFD types), and the mobilisation of knowledge linked to KMT and KFLM subdomains is also relevant. This is because this relationship points to the importance, for teachers, of knowing the types of connections that are linked to student’s recurring errors and possible teaching strategies. The problem identified in the intra- and inter-conceptual distinction, in the analysis of the knowledge mobilised by teachers, suggests that it is necessary that the characterisation of KSM can explicitly include knowledge about the way in which concepts are related. The predominance of a procedural use of connections is identified as an indicator of KoT subdomain, although they may be interpreted as inter-conceptual connections. For its part, KSM is associated with the use of mathematical connections aimed at understanding how mathematical ideas are related in a teaching-learning context, which is consistent with Hatisaru’s findings ([9,34]). The analysis carried out in the two studies shows that the notion of complexity associated to connections should also be considered in the characterisation of KSM, in order to enrich and facilitate its use as an analytical tool. The enrichment of KSM subdomain can help to understand the relationship between different subdomains of MTSK model, in line with those proposed by Badillo et al. ([35]). In this way, the connections between concepts, characterised in KSM, would benefit from taking into account the notion of complexity in their characterisation. Primarily, because it would be possible to further clarify the knowledge mobilised by teachers during the establishment of mathematical connections. The levels of complexity of the connections would be related, for example, to the way in which different concepts and properties can form a mathematical connection. In this sense, the connections that mention or describe procedures would be insufficient to characterise the complexity that underlies the structure of a complex mathematical connection. In any case, we consider that further studies are needed to problematise the use of MTSK in the establishment of mathematical connections. Finally, we consider that in order to refine the characterisation of the subdomains of MTSK model, in terms of mathematical connections, more studies are required with a variety of data collection instruments that allow us to account for the complexity of the knowledge mobilised by in service and preservice teachers when establishing mathematical connections. For example, it is necessary to incorporate the use of clinical interviews that, triangulated with the data available to us, allow us to deepen the subdomains of knowledge that teachers use when establishing mathematical connections, both in the resolution of professional tasks and in classroom practice. Author Contributions: Conceptualisation, G.D.G., S.C. and E.B., methodology, G.D.G., S.C. and E.B.; validation, G.D.G., S.C. and E.B.; formal analysis, G.D.G., S.C. and E.B.; investigation, G.D.G., S.C. and E.B.; writing—original draft preparation, G.D.G., S.C. and E.B.; writing—review and editing, G.D.G., S.C. and E.B.; supervision, G.D.G., S.C. and E.B. All authors have read and agreed to the published version of the manuscript. Funding: PID2019-104964GB-I00 (MICINN-Spain and GIPEAM, ANID-Chile/2018-72190032. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to their containing information that could compromise the privacy of research participants. Conflicts of Interest: The authors declare no conflict of interest. Mathematics 2022, 10, 4010 23 of 24 References 1. 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