Characterising pre-service primary school teachers’ discursive activity when defining

International Journal of Instruction July 2023 ● Vol.16, No.3 e-ISSN: 1308-1470 ● www.e-iji.net p-ISSN: 1694-609X pp. 117-130 Citation: Martín-Molina, V., Toscano, R., Fernández-León, A., Gavilán-Izquierdo, J., & González- Regaña, A. J. (2023). Characterising pre-service primary school teachers’ discursive activity when defining. International Journal of Instruction, 16(3), 117-130. https://doi.org/10.29333/iji.2023.1637a Article submission code: 20220617164924 Received: 17/06/2022 Revision: 16/12/2022 Accepted: 07/01/2023 OnlineFirst: 01/04/2023 Characterising Pre-Service Primary School Teachers’ Discursive Activity when Defining Verónica Martín-Molina Universidad de Sevilla, Spain, veronica[email protected] Rocío Toscano Universidad de Sevilla, Spain, [email protected] Aurora Fernández-León Universidad de Sevilla, Spain, auro[email protected] José María Gavilán-Izquierdo Universidad de Sevilla, Spain, gavila[email protected] Alfonso J. González-Regaña Universidad de Sevilla, Spain, ago[email protected] This paper studies how pre-service primary school teachers construct and select mathematical definitions through the analysis of their discursive activity. Specifically, the theory of commognition (Sfard, 2008) is employed to determine whether the existence of different meta-rules always leads to the existence of a commognitive conflict. Moreover, we study the reasons that give rise to the commognitive conflicts found and whether they are resolved. To this end, we studied the discourse of 45 pre-service primary school teachers while they answered several questions on defining geometric solids. The data in this study consisted of audio recordings of their discussions and their written answers. In this paper, three vignettes showing different meta-rules are presented. In the first, discussions regarding the characteristics of a definition promoted the appearance of different meta-rules that existed in incommensurable discourses, which meant the existence of a commognitive conflict. This conflict highlights the fact that certain pre-service teachers confuse the processes of describing and defining. Both the second and third vignettes featured the appearance of two different meta-rules. However, in both cases, those meta-rules could coexist in the same discourse, and therefore a commognitive conflict could not be inferred. Keywords: commognitive framework, commognitive conflict, discourse, mathematical practice of defining, meta-rule, pre-service primary school teacher 118 Characterising Pre-Service Primary School Teachers’ … International Journal of Instruction, July 2023 ● Vol.16, No.3 INTRODUCTION The practice of defining is essential in mathematics. It is as important as finding the proof of a theorem (Freudenthal, 1973), and learning the relevant definitions should hence form an essential part of the process of learning mathematics (Avcu, 2022; Miller, 2018; Tabach & Nachlieli, 2015; Zaslavsky & Shir, 2005). Therefore, learning to define should constitute an important part of mathematics education (Mariotti & Fishbein, 1997; Zaslavsky & Shir, 2005), but this mathematical practice is often neglected (de Villiers, 1998) and remains “typically underemphasized in school mathematics; instead, students often experience definitions as received from an authority” (Kobiela & Lehrer, 2015, p. 425). Learning to define should also form an important part of mathematics teacher education (Leikin & Zazkis, 2010; Miller, 2018; Sánchez & García, 2008) and, during that learning, “explicit connections [should] be made between university mathematics and school mathematics” (Leikin & Zazkis, 2010, p. 465). Indeed, it is very important that pre-service teachers learn how to use or propose definitions because that knowledge may influence their future choice of pedagogical approaches to teaching mathematical definitions (Avcu, 2022). In geometry, a field in which mathematical communication skills are particularly needed (Kusumah et al., 2020), teachers should encourage students to participate in the process of defining because “mathematicians and mathematics educators alike have often criticised the direct teaching of geometry definitions with no emphasis on the underlying process of defining” (de Villiers et al., 2009, p. 190). In order to be able to evaluate the definitions that students propose, teachers should be able to grasp the differences between different types of definitions (de Villiers et al., 2009). Unfortunately, previous studies report that mathematical definitions constitute a difficult topic for pre-service teachers (e.g., Miller, 2018), so it is important to study the definitions they produce to gain more information regarding their knowledge of mathematical definitions (Avcu, 2022). To this end, Miller (2018) categorised the content and the form of pre-service primary school teachers’ (PPTs’) definitions of quadrilaterals and Avcu (2022) categorised middle school mathematics teachers’ definitions of certain quadrilaterals by using Zazkis and Leikin’s (2008) framework. The definitions of three-dimensional objects have been much less explored than the definitions of two-dimensional objects, thus one of the objectives of our study was to fill this gap by studying the definitions that PPTs propose for three-dimensional geometrical objects. According to Tall (1991), one of the aspects that characterises the transition from elementary to advanced mathematical thinking is the transition from describing to defining, and hence the objective included determining whether PPTs distinguished between describing and defining the three-dimensional geometrical solids when proposing their definitions. One way to obtain information on someone’s conception of a mathematical definition is to present them with several definitions of mathematical objects and ask them to choose which one they would prefer as a definition (Zaslavsky & Shir, 2005). Moreover, the ability to access information in our current society means that in-service and pre-service Martín-Molina, Toscano, Fernández-León, Gavilán-Izquierdo & González-Regaña 119 International Journal of Instruction, July 2023 ● Vol.16, No.3 teachers have access to a wide variety of definitions (some of them incorrect) and they have to decide which one is the most suitable for their classroom. These two reasons motivated another objective of our study, which involves determining which criteria PPTs use when they have to choose a definition. In summary, our study focuses on characterising how pre-service primary school teachers defined and selected definitions for three-dimensional mathematical objects. Context And Theoretical Background The mathematical practice of defining, as other mathematical practices, is a social practice which is essential in mathematical knowledge construction (Rasmussen et al., 2005). This social nature makes sociocultural approaches conducive to investigating this mathematical practice. Among the various sociocultural approaches, the theory of commognition (Sfard, 2008) has proven useful to study this and other mathematical practices. For instance, this theory has been used in the study of mathematical practices such as modelling (Viirman & Nardi, 2019) and proving (Schüler-Meyer, 2020). In the case of the practice of defining, this sociocultural approach has recently been employed in the studies by Biza (2021), Fernández-León et al. (2021), Schüler-Meyer (2020), and Tabach and Nachlieli (2015). It should be mentioned that this theoretical perspective has led to many other major results in mathematics education (see, for instance, Gallego- Sánchez et al., 2022; Ioannou, 2018; Nachlieli and Tabach, 2022; Nardi et al., 2014; Sfard, 2021b; Thoma & Nardi, 2017). In the following, the theory of commognition is described as an operational framework for the study of students’ discursive activity when defining. Sfard (2008) coined the term “commognition” by joining the terms “communication” and “cognition”. The theory of commognition regards thinking as a particular type of interpersonal communication; specifically, the communication that one has with oneself. Furthermore, this theory considers mathematics as a particular type of discourse that can be characterised through four characteristics: keywords, visual mediators, narratives, and routines. The first characteristic, that of keywords, refers both to mathematical words (such as angle and prism) and to colloquial words with a mathematical meaning (such as “leaning” to mean oblique). The visual mediators of physical entities are physical objects (such as a graph or an equation) that participants in the discourse (discursants) use as part of their communication to clarify their performances. Another important feature that defines the mathematical discourse are its narratives, which are sentences (spoken or written) about objects, relations between them, or activities with or by objects. Unlike other discourses, the objects in mathematics, called mathematical objects, do not exist independently of the discourse, but such objects are brought into being when the discursants talk about them (Sfard, 2021b). Furthermore, if a narrative is accepted by the discursants, that is, it is regarded as true, then the narrative is called an “endorsed narrative” (such as definitions or theorems). The fourth and last characteristic that defines the mathematical discourse are the routines, which are repetitive patterns that characterise the actions of the participants in the discourse (such as the way they 120 Characterising Pre-Service Primary School Teachers’ … International Journal of Instruction, July 2023 ● Vol.16, No.3 solve equations or the strategies they employ to prove theorems). In Lavie et al. (2019), routines are described as a task-procedure pair, in which the task captures information about how the task performer views a certain task situation and the procedure refers to the actions this performer undertakes to tackle the task situation. Note that, in this context, a task situation is “any setting in which a person considers herself bound to act – to do something” (Lavie et al., 2019, p. 159). We emphasise here that the task focuses on the performer’s interpretation, whereas the procedure focuses on what the researcher interprets from the task performer’s actions in a given task situation (Nachlieli & Tabach, 2022). In the theory of commognition (Sfard, 2008), human communication is defined as an activity governed by rules in which the participants act and react in an organised way according to a well-determined set of options. Sfard (2008) establishes that two types of rules can be identified in human communication: object-level rules and meta-level rules (henceforth, meta-rules). Object-level rules are rules regarding the regular behaviour of objects (such as “the sum of the angles of a square is 360º”) and meta-rules are rules that appear when the discursants try to produce or justify narratives at object level (such as “to prove the previous statement, it is appropriate to divide the quadrilateral into two triangles along one of its diagonals”). In this theory, learning is considered a change in the discourse and is manifested by changes in any of the four characteristics of the discourse. These changes may be caused by the resolution of commognitive conflicts, which are situations that arise when the discursants are participants of incommensurable discourses. Incommensurable discourses are those governed by different meta-rules (Sfard, 2021a) and “that differ in their use of words and mediators or in their routines” (Sfard, 2008, p. 299). In other words, commognitive conflicts emerge in those scenarios where participants of incommensurable discourses try to communicate across such discourses, which may produce a feeling of uneasiness (Sfard, 2021b). The study of commognitive conflicts is an outstanding topic that has increased its presence in recent investigations (González- Regaña et al., 2021; Ioannou, 2018; Nachlieli & Heyd-Metzuyanim, 2022; Sánchez & García, 2014) since commognitive conflicts constitute a major source of mathematical learning (Sfard, 2021b). In this paper, the great potential of certain theoretical constructs of the commognitive framework has been leveraged to characterise PPTs’ activity when defining and selecting definitions. To be precise, the research questions that inform about the particular aims of this study are: (1) did the existence of different meta-rules always lead to the existence of a commognitive conflict? (2) When there was a commognitive conflict, what were the reasons that gave rise to it and was it resolved? METHOD Participants and context The participants in this study consisted of 45 PPTs enrolled in an undergraduate degree in Primary Education. Specifically, these pre-service teachers were taking a first-year mathematics course that each week had a two-hour session on theory and exercises and a Martín-Molina, Toscano, Fernández-León, Gavilán-Izquierdo & González-Regaña 121 International Journal of Instruction, July 2023 ● Vol.16, No.3 one-hour session on problem solving. In problem-solving sessions, the students worked in groups of 3 or 4 members. In one of the problem-solving sessions, the data collection instrument was presented, and the students were asked to collaborate in the study. There were 45 students who voluntarily decided to participate. These students were organised into 12 working groups (called G1, G2, …, G12 in the study), each comprising 3-4 students. Data collection instrument The main objective of the data collection instrument, as mentioned in Fernández-León et al. (2021) and González-Regaña et al. (2021), was to promote PPTs’ discussion when defining and selecting geometric solids. The data collection instrument was of the form of a worksheet and included a two-dimensional representation of three geometric solids that was designed with the dynamic software GeoGebra (Figure 1). The geometric solids were a cube, a quadrangular, oblique, and convex prism, and a quadrangular, oblique, and concave prism. These were chosen because they have some properties in common (for example, all are quadrangular prisms) and hold other distinctive properties (for example, the cube is regular, while the others are not). The worksheet also included nine questions. The first four questions were related to the description of geometric solids and the identification of their similarities and differences: 1. In the three previous solids, you can identify basic elements such as faces, vertices, edges, etc. What properties or characteristics of these elements can you observe in each solid? 2. From among the above properties or characteristics, can you identify those that only two solids have in common? 3. Among the properties or characteristics of Question 1, can you identify any property that the three solids have in common? 4. Is there any property of any of the solids that differentiates it from the other two? The following four questions dealt with the construction of definitions of these solids and reflection thereon. 5. Define each of these solids. 6. Can you give another definition of any of the solids? 7. Is one of your definitions valid for another solid as well? For example, is the definition of solid 1 also valid for solids 2 or 3? 8. Could you give a definition that is valid for two of the solids? And for three? Finally, there was a question where the PPTs had to select a definition from among those they had previously built. 122 Characterising Pre-Service Primary School Teachers’ … International Journal of Instruction, July 2023 ● Vol.16, No.3 9. Of the two definitions that you have given for each solid in Questions 5 and 6, which one would you choose? Why? Figure 1 Representation of the three geometric solids included in the worksheet (Fernández-León et al., 2021, p. 306) Data collection Data collection took place during a one-hour problem-solving session. A paper copy of the worksheet was given to each group and the PPTs were asked to discuss the proposed questions and to reach an agreement, which they had to write down on the paper copy. Additionally, each of the 12 groups received an audio recorder and instructions to facilitate their identification in the transcript. At the end of the session, 12 paper copies with the written answers of each group were collected, together with the audio recorders with their conversations (approximately 12 hours, 1 hour per group). Analysis Once the audio recordings were transcribed, the analysis was carried out in two phases. In the first phase, the transcripts and written answers were analysed to identify categories in keywords, narratives, and visual mediators. These categories were based on similarities and differences found within each characteristic. Such categories were then analysed to identify meta-rules and routines in the discourse. In the second phase, based on the categories obtained in the first phase, the discourse was analysed in search of possible commognitive conflicts. This was carried out by locating differences between the discursive characteristics of the PPTs of each group, and then those differences were carefully analysed to decide whether commognitive conflicts could be inferred. Furthermore, the conflicts that appeared were analysed to determine the reasons that led to them and whether they had been resolved. Findings obtained in the first phase of analysis can be found in previous work (Fernández-León et al., 2021; Gavilán-Izquierdo et al., 2019; González-Regaña et al., 2021). In this paper, we focus on the findings obtained in the second phase of the analysis. FINDINGS In this section, three vignettes are presented, showing situations in which three groups of PPTs employed different meta-rules while they were answering the questions of the Martín-Molina, Toscano, Fernández-León, Gavilán-Izquierdo & González-Regaña 123 International Journal of Instruction, July 2023 ● Vol.16, No.3 worksheet. In all three vignettes, more than one meta-rule appeared, but a commognitive conflict appeared in only one of them and that conflict remained unresolved. All the vignettes are described in detail below, including excerpts from the transcripts and written answers, and the context in which the discussion occurs. In each excerpt, the PPTs are referred to as S1, …, S4, regardless of the group to which they belong, and their teacher is referred to as T. Vignette 1. Defining versus describing The task situation created by the fifth question of the worksheet required the PPTs to define the three solids for the first time, while the previous four questions asked them to describe the solids. No information regarding the structure or characteristics of a definition was provided to the PPTs to determine whether they distinguished between defining and describing. One of the discussions concerning how to answer Question 5 of the worksheet appears in the following excerpt from the transcript of group G7: 207 S1: If we talk about prisms, it is a hexagonal prism, but… 208 S4: We have already defined it, in… 209 S3: No, we have said what it has. 210 S4: Exactly, the characteristics. Sure, that it is a cube with these characteristics… 211 S2: Teacher, what do we have to write in Question 5? 212 T: My question to you is: what do you consider a definition to be? Saying which are the characteristics? Saying the name? What? 213 S2: The name and… the characteristics… is the definition. 214 T: That is what I am asking you. 215 S1: I think so… 216 T: Then, talk about it among yourselves and what you decide is what you have to write down. In this excerpt, four PPTs discussed what information should be included in the definition of the first solid. The misalignment between the different meta-rules that they used to construct such a definition has allowed us to infer the existence of a commognitive conflict between the discourses of the PPTs. On the one hand, line 208 includes a meta-level narrative that shows what S4 understood by process of defining, which he considered equivalent to describing. In the next line, a meta-level narrative from S3 shows a different meta-rule to define that contradicts S4’s meta-rule, since S3 stated in line 209 that the description given in Question 1 was not a definition. From this last narrative, a discussion arose and, since the PPTs did not come to an agreement, they finally turned to the teacher (an expert) asking for her help. However, the teacher did not explicitly tell them what they had to write down, but rather motivated them to reflect on the meta-rules that they might consider when defining (lines 212, 214, and 216). Finally, students S1 and S2 adopted a common meta-rule. This meta-rule has been inferred from the narrative at meta level that appears in line 213, in which S2 claimed that a definition of a solid should include its name and its characteristics. A resolution of the commognitive conflict cannot be deduced from the data since the PPTs stopped their discussion and simply wrote down the following answer to Question 5: “Solid 1: A cube 124 Characterising Pre-Service Primary School Teachers’ … International Journal of Instruction, July 2023 ● Vol.16, No.3 is a geometric body with six equal faces with a square base whose angles are right angles”. Vignettes 2 and 3. What is the best definition? PPTs’ meta-choices In the task situation generated by Question 9 of the worksheet, the PPTs had to choose a definition for each solid from the ones they had been asked to construct in Questions 5 and 6. They also had to justify their choices, which generated discussions in several of the groups. The following vignettes show two situations where different meta-rules do not lead to the existence of a commognitive conflict. In vignette 2, the PPTs in group G11 had the following discussion when justifying their chosen definition: 456 S1: Because it is the most complete one. 457 S3: Sure, here we have completed it more. 458 S4: We have chosen this definition because it is the most complete. You said something before. 459 S1: About what? 460 S4: About this definition. You said that… 461 S1: …that it is the most specific one. 462 S4: Should I write that it is the most complete and specific? In the previous excerpt, all the PPTs agreed on which definition to choose (the first one they had constructed). Student S1 justified this choice by stating in line 456 that they should choose that one because it was “the most complete” definition, to which S3 and S4 agreed in lines 457 and 458, respectively. Later, S4 reminded S1 that he had previously said something, so S1 added that their chosen definition was also “the most specific”. Therefore, the meta-level narratives of lines 456-458 informed us about the meta-rule that the PPTs first proposed to answer Question 9, which consisted of choosing the definition that had the greatest amount of information, even if some of it was unnecessary to a mathematician. In line 461, S1 introduced a different meta-rule: the definition had to be “the most specific”. An analysis of this group’s answers to previous questions shows that they understood that a definition for a particular solid was “specific” when that solid satisfied it, but the rest did not. Therefore, a definition can be “the most complete” and not “the most specific” and vice versa. However, the PPTs did not appear to realise that they had proposed two different meta-rules and accepted S4’s proposal of writing both together “the most complete and specific” (line 462). Since both meta-rules can coexist in the same discourse, a commognitive conflict could not be inferred. In vignette 3, we show a situation in which the existence of more than one meta-rule prompted a discussion about which criterion to employ when choosing a definition. Specifically, the three PPTs in group G2 had the following conversation: 251 S2: I would put both together, but that’s me. 252 S3: Figure of six faces, twelve edges, eight vertices, six faces with square shape and its… Martín-Molina, Toscano, Fernández-León, Gavilán-Izquierdo & González-Regaña 125 International Journal of Instruction, July 2023 ● Vol.16, No.3 253 S1: That one, right? 254 S3: Yes. 255 S2: I would choose the definitions that we have written in Question 6. […] 258 S2: Why? 259 S3: Because it is more complete. 260 S2: Because it is describing with all the characteristics that the figure has. 261 S1: Because, if it tells you it has six faces, you don’t know what those six faces look like. 262 S2: Because Question 1, what we are saying, the previous question what we are saying is that it has space and that it is a polyhedron with six faces, which can include… 263 S3: And also so that a child can identify the figure much better with the definition of Question 6 than with [the one of the] other question […] The first line of this excerpt (line 251) is a narrative at meta level, since it shows the meta-rule that S2 initially proposed to answer Question 9: merging the two definitions that they had previously constructed for solid 1. S1 and S3, without explicitly stating the meta-rule that they would use to choose the best definition, proposed to choose the definition from Question 6 (lines 252-254). S2 accepted this decision in line 255, without any debate about why. To justify their choice, S3 stated that she preferred that definition because it was the most complete one (line 259), and then added that it was also the definition that would allow a child to “better” identify solid 1 (line 263). To support S3’s justification in line 259, S2 said that the chosen definition was the one that described solid 1 with “all” its characteristics (line 260). In line 261, S1 supported this statement from S2, adding that, if they chose the definition in Question 5, information on what the faces of solid 1 were like would be missing. Therefore, the PPTs had made explicit the meta-rule that had guided their choice: choosing the definition that included the largest number of characteristics of solid 1, which they called “the most complete” definition. In this case, the existence of two different meta-rules for choosing a definition does not seem to have led to a commognitive conflict. This is because both meta-rules coexist in the same discourse since the meta-rule proposed by S2 (merging both definitions) may have been motivated by the fact that S2 considered that the maximum amount of information should appear in a definition. S2 might have abandoned her meta-rule simply because she considered that it was not a valid answer to Question 9, which requires choosing between two definitions. In this case, it seems that a social norm (you have to “obey” what the question requests) has prevailed over the meta-rule that S2 originally wanted to use. In this work, social norms are considered to be those aspects of social interactions in class that become normative, such as the obligation to explain all reasoning and to try to understand the reasoning of others (Yackel et al., 2000).