Abstract Introduction and problem statement The Van Hiele theory of geometrical thinking Commognition Conclusion Acknowledgements References Footnote About the Author(s) Sfiso C. Mahlaba Department of Mathematics Education, Faculty of Education, North West University, Mafikeng, South Africa Vimolan Mudaly Department of Mathematics Education, Faculty of Education, University of KwaZulu-Natal, Pinetown, South Africa Citation Mahlaba, S.C., & Mudaly, V. (2022). Exploring the relationship between commognition and the Van Hiele theory for studying problem-solving discourse in Euclidean geometry education. Pythagoras, 43(1), a659. https://doi.org/10.4102/pythagoras.v43i1.659 Review Article Exploring the relationship between commognition and the Van Hiele theory for studying problem-solving discourse in Euclidean geometry education Sfiso C. Mahlaba, Vimolan Mudaly Received: 15 Oct. 2021; Accepted: 11 May 2022; Published: 29 July 2022 Copyright: © 2022. The Author(s). Licensee: AOSIS. This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This article is an advanced theoretical study as a result of a chapter from the first author’s PhD study. The aim of the article is to discuss the relationship between commognition and the Van Hiele theory for studying discourse during Euclidean geometry problem-solving. Commognition is a theoretical framework that can be used in mathematics education to explain mathematical thinking through one’s discourse during problem-solving. Commognition uses four elements that characterise mathematical discourse and the difference between ritualistic and explorative discourses to explain how one displays mastery of mathematical problem-solving. On the other hand, the Van Hiele theory characterises five levels of geometrical thinking during one’s geometry learning and development. These five levels are fixed and mastery of one level leads to the next, and there is no success in the next level without mastering the previous level. However, for the purpose of the Curriculum and Assessment Policy Statement (CAPS) we only focused on the first four Van Hiele levels. Findings from this theoretical review revealed that progress in the Van Hiele levels of geometrical thinking depends mainly on the discourse participation of the preservice teachers when solving geometry problems. In particular, an explorative discourse is required for the development in these four levels of geometrical thinking as compared to a ritualistic discourse participation. Keywords: commognition; Van Hiele theory; Euclidean geometry; geometrical thinking; visual mediators. Introduction and problem statement A review of the Grade 12 National Senior Certificate examination diagnostic analysis from 2016 to 2020 reveals that the average pass percentage of Grade 12 mathematics learners in South Africa is below 60% (Department of Basic Education, 2016, 2017, 2018, 2019). This comes after Van Putten, Howie and Stols (2010) felt that South African teachers are not well prepared to teach Euclidean geometry. In alleviating the situation, Machisi (2021) suggests the use of unconventional teaching approaches such as the Van Hiele theory-based teaching and learning approach which as Machisi (2021, p. 1) concluded ‘meets learners’ needs better than conventional approaches in learning Euclidean geometry’. Furthermore, in alleviating the difficulties faced by mathematics learners, Sfard (2008) proposed that teachers should aim to transform learners’ discourse participation from ritualistic to explorative discourse participation. In particular, Sfard proposes this transformation of discourse participation because she believed that learners’ mathematical thinking can be encoded from the way they communicate about mathematics. We have seen Sfard’s theory being applied in other mathematical domains in South African research such as functions (Mpofu & Mudaly, 2020), numeracy (Heyd-Metzuyanim & Graven, 2016) and equations (Roberts & Le Roux, 2019). While commognition has a potential for alleviating difficulties in all domains of mathematics, the Van Hiele theory is specifically dedicated to guide teachers on how to alleviate learning difficulties related to Euclidean geometry. The current theoretical article locates the problem in the fact that these theories are currently operating in isolation yet they have a similar purpose of improving learning in mathematics. Wang (2016) combined these two theories (commognition and Van Hiele) with a focus on the elements of mathematical discourses, but this article takes a different approach by focusing on the type of discourse participation. The aim here is to harness the power of commognition in improving geometrical knowledge through the Van Hiele levels when solving geometry problems. Thus, we discuss the tenets of each theory, then discuss how the study viewed the amalgamation of commognition and the Van Hiele theory as means of enhancing geometry understanding during problem-solving. The findings reported in this theoretical article are from a larger PhD study but this article focuses only on the relationship between commognition and the Van Hiele theory when studying problem-solving discourse in Euclidean geometry. Thus, no empirical data will be cited to in this article. The Van Hiele theory of geometrical thinking The Van Hiele theory of geometrical thinking was developed by Van Hiele-Geldof (1957) and Van Hiele (1957) towards the completion of their Doctor of Philosophy degrees at the University of Utrecht. The Van Hieles posited that children go through five levels of thought development in their geometrical learning. These levels include recognition (level 1), analysis (level 2), order (level 3), deduction (level 4) and rigour (level 5). This theory was developed within the contexts of learners who are still in their secondary school education but in this review article it is viewed from the lens of preservice teachers (PSTs). Thus, any reference to PSTs is equivalent to secondary school learners in the context of this study. These four levels of geometrical thinking also have their own descriptions that determines the type of learning that occurs in that level. These characteristics per level are summarised below: Level 1: PSTs recognise names and recognise figures as a whole (i.e. a square and a rectangle are different). Level 2: PSTs begin not only to recognise objects by their global appearance but also to identify their properties with appropriate technical language (e.g. a triangle is a closed figure with three sides). Level 3: PSTs begin to logically order these properties through short chains of deduction and understand the interrelationship between figures through their properties. Level 4: PSTs begin to develop longer chains of deduction and understand the significance and roles of postulates, theorems and proofs. Level 5: PSTs understand the role of rigour and can make abstract deductions that allow them to understand even non-Euclidean geometries. The Van Hiele theory is characterised by the existence of four characteristics summarised by Usiskin (1982, p. 4) and De Villiers (2012) as follows: Fixed sequence: PSTs progress through the levels invariantly which means that a PST cannot be at Van Hiele level n without having passed level n–1. Adjacency: at each level of thought, the intrinsic knowledge from the previous knowledge is extrinsic in the current level. Distinction: the linguistic symbols and network of relationships connecting these symbols are distinct in each level. Separation: PSTs who are reasoning at different levels cannot understand each other. These characteristics describe the manner in which PSTs are to proceed through the levels and what is important to consider in each level. Within the five levels of geometrical thinking, the most pertinent ones in the Curriculum and Assessment Policy Statement (CAPS) are the first four levels (levels 1–4) which we focused on in this study because they also apply to PSTs’ education. At level 1, geometrical figures are recognised by their visual appearance (form) only, without any reference to their properties and any relationship that might exist between them. At this level, PSTs are able to relate geometrical figures with objects they see in their everyday lives, for example a rectangle looks like a door. These activities are critical at this level as the foundation for the next level (Yi, Flores, & Wang, 2020). In level 2, geometrical figures are identified based on their properties, without considering the relationship that exists between these properties. Thus, secondary school PSTs see a geometrical figure in isolation, not related to other figures. A square can be recognised as having four equal sides and four right angles without relating the property of right angles to a triangle. As PSTs develop to level 3, they begin to see the relationships between the properties of geometry figures. They now can relate a square and a rectangle by ordering their properties and deducing one from the other. At level 4, PSTs’ thinking and reasoning are concerned with understanding the meaning of deduction and proof. They can understand the role of theorems, postulates and properties of geometrical figures when doing proofs. There are critical issues about these levels that apply to the development of thought in geometry, especially for PSTs to use in their instruction. The language and signs used at each level are distinct, such that a relationship that is true at one level might not be true at another (Van Hiele, 1959). The second issue to be aware of is that people who reason at different levels cannot understand each other. Hence, teachers need to attempt to reason at the level of learners, understand their routines and narratives to scaffold them to the next level. Teachers must continuously support learners to construct their deductive relational system in geometry (Van Hiele, 1959), without imposing the relational system of the teacher onto the learners. These levels are critical in the analysis of thinking and reasoning in geometry, because they reveal the characteristics of thinking for both learners and teachers. Since this study had its main focus on PSTs’ thinking when solving geometry problems, it seemed useful to incorporate these levels, as they are indicators of geometrical thinking. Even though these levels were not assessed directly in this study, they are pertinent in geometry problem-solving. The PSTs’ behaviour when participating in geometry problem-solving can be related to each Van Hiele level and teachers’ discourse participation when solving Euclidean geometry problems (see Table 3). Commognition This section provides the details of Sfard’s (2008) theory of commognition, with a particular focus on the aspects of the theory that relate to this study. We begin with a brief explanation of the theory in general and a few key tenets of the theory. Thereafter, we explain how the theory relates to learning and thinking, and how mathematics learning is a discourse, as we locate the current study within the theory of commognition. This section aims to describe the keywords and the language of Sfard’s commognition and how they are key in describing PSTs’ mathematical discourse when solving geometry problems. Furthermore, this section aims to explain how commognition has enabled the study to explain PSTs’ discourses and what improvements can be made to the theory in the future. Commognition in a nutshell In 2008, Sfard published a book titled Thinking as communicating: Human development, the growth of discourses, and mathematizing, which explains a theory that can guide and be used to understand mathematical learning. Sfard (2008) uses the amalgamation of ‘communication’ and ‘cognition’ to coin the term ‘commognition’, which she describes as communicating about thinking. She defines commognition by putting into perspective this amalgamation, stating that commognition ‘stresses that interpersonal communication and individual thinking are two facets of the same phenomenon’ (Sfard, 2008, p. xvii). Here, Sfard asserts that thinking is correlated to communicating, stating that ‘thinking is defined as the individualized version of interpersonal communication’ (Sfard, 2008, p. xvii), closely relating thinking to Bakhtin’s (1986, p. 126) idea of the ‘superaddressee’. Sfard emphasises that communication and thinking are inseparable. She describes the underpinnings of the theory from the significance of communication, objectification, and elements of mathematical discourse that are significant in mathematics classrooms. In her elucidation of commognition, Sfard differentiates between colloquial and mathematical discourse, where the former is considered to be everyday, spontaneous discourses and the latter is specifically related to mathematics. She posits four characteristics of the latter kind of discourse. A discourse becomes mathematical because of its word usage, visual mediators, narratives and routines. Commognition is driven by the processes of objectification which is characterised by a double elimination of using metaphors to generate new discourse. This double elimination is characterised by the processes of alienation and reification. According to Sfard (2008, p. 44), reification ‘consist[s] of substituting talk about actions with talk about objects, [while alienation] consists in presenting phenomena in an interpersonal way’. A reified talk includes utterances like “Thabo has developed the concept of geometrical proof and problem-solving” while an alienated talk includes utterances like “the sum of all interior angles of a triangle is 180°”. The results of objectification are mainly abstraction, which helps in differentiating discourses of mathematicians and that of, for example, street vendors about similar issues. It is this power of objectification that differentiates between colloquial and mathematical discourse. Sfard asserts that mathematical discourse is characterised by word use, visual mediators, narratives and routines. These characteristics are used to differentiate between three routines: rituals, deeds and exploration; here we only focus on routines and explorations. Thus, in this study we use the elements of mathematical discourse to differentiate between a ritualistic and an objectified discourse using each of their characteristics (see Table 1). All these will be discussed in the following sections in more detail. TABLE 1: The comparison between rituals and explorations in commognition. Why commognition? Commognition is a discursive theory that is utilised here for its theoretical potential to explain PSTs’ thinking during geometrical problem-solving. It is a theory that acknowledges that everyone thinks on a daily basis but that others do not have direct access to this thinking. Hence, commognition regards thinking as ‘an individualised version of interpersonal communication’ (Sfard, 2008, p. 81). According to this view, thinking cannot just be an isolated activity, but becomes ‘the act of communication in itself’ (Sfard, 2008, p. 82). This means that whatever utterances are made through discourse by an individual are a consequence of that individual’s thinking, and the best way to study that individual’s thinking is to analyse their communication in discourse. Furthermore, school learning, as it is for teacher training, should present an opportunity to extend the discourses of learners and PSTs (Ben-Zvi & Sfard, 2007). Commognition further recognises that, just like learning, thinking develops from a patterned collective activity. Commognition recognises that thinking can be objectified or disobjectified, but rests mainly on the significance of explaining mathematical thinking through disobjectified discourses. Therefore, thinking can be explained by analysing the discourses of PSTs. Thinking, as a patterned collective activity, happens through communicating with others and ourselves. Thinking is therefore dialogical (Sfard, 2008), and thinking is modified and changed as we communicate with others. Mathematics is considered a difficult subject in South African schooling. Furthermore, geometry is seen as the topic where learners perform the poorest and where even teachers struggle to teach geometry effectively (Naidoo & Kapofu, 2020; Tachie, 2020; Van Putten et al., 2010). Most teachers who do manage to get learners to pass geometry use the drilling of theorems and how to prove them. Some teachers rely mainly on the possibility of questions being repeated in the standardised tests (Machisi, 2021). To improve the dire situation of teachers with insufficient geometry knowledge, which leads to learners performing poorly in geometry, we need to approach this problem as a collective, ensuring that teachers are properly trained to teach geometry in secondary schools. We need to tap into PSTs’ thinking when they solve geometry problems to see how they think and then design proper tasks and teaching strategies to enhance their level of geometry thinking to a suitable one, where they would be able to teach geometry effectively to ensure meaning making within learners. In this way, commognition offered a window to tap into PSTs’ thinking when they solve geometry problems, to understand their thinking. In commognition, thinking is voluntary, individuals engage in thinking through their continued participation in the mathematical discourse. Hence, in this view, geometry learning for PSTs is a consequence of their continued participation in the community of mathematics, which mainly originates from the participationist theories of learning (Lave & Wenger, 1991). As a human activity, participation in the activity of communication has emotional implications, which need to be understood properly, especially if the communication is among competing peers (Heyd-Metzuyanim, 2013). Preservice teachers need to move from being ritualistic participants in the discourse of geometry to being individual geometry problem-solvers who can teach geometry effectively. Since Vygotsky (1978), the activity of being a peripheral participant in geometry discourse for PSTs begins with the help of a more knowledgeable other. Hence, the role of the lecturer (trainer) is important as a knowledgeable other. Hence, if PSTs are learning, they become more and more independent of the lecturer and their thinking as they learn does not require the aid of the lecturer, as they become independent thinkers. Objectification in discourse The use of metaphors is common in all discourses, where words are partitioned into an unfamiliar discourse because of their familiarity and their readiness to be used in that discourse (Sfard, 2008). The building of mathematical knowledge from concrete objects has long been recognised (Dienes, 1960). In the current article, it seemed significant to distinguish between PSTs’ discourse about objects and how they communicated about mathematical objects. The ‘process in which a noun begins to be used as if it signified an extra-discursive, self-sustained entity (object), independent of human agency’ (Sfard, 2008, p. 300) is known as objectification. An example, in Euclidean geometry there can be a statement such as “angle ABC is equal to angle BCA because of angles opposite equal sides” instead of “this angle is equal to that angle because this is an isosceles triangle”. In commognition, objectification is considered to encapsulate two inseparable discursive moves: reification and alienation. According to Sfard (2008, p. 44), ‘reification is the act of replacing sentences about processes and actions with propositions about states and objects’. Hence, in this article reification describes PSTs transforming their talk about the process of problem-solving into talk about objects. Reification allows PSTs to be concise about what they are communicating, which makes it more flexible and applicable in mathematical discourse. A reified version of the statement “in the majority of the tests and tasks dealing with Euclidean geometry proof in school, he regularly did well and achieved very good marks” is “he has acquired the concept of Euclidean geometry proof”. Alienation on the other hand, involves the removal of the reified discourse from the actor. Alienation refers to ‘using discursive forms that present phenomena in an impersonal way, as if they were occurring of themselves, without the participation of human beings’ (Sfard, 2008, p. 295). Alienation includes the use of passive voice in a particular mathematical sentence, for example “the angle between the tangent and the chord is equal to the angle subtended by the chord in the alternate segment”, which removes any personal attachments to the statement (Sfard, 2021). Alienation allows PSTs to engage in the discourse of geometry problem-solving in an impersonal way. These alienated geometry discourses can be thought of as theorems, axioms or postulates, etc., since they are monological1 (Bakhtin, 1986). In this view, a geometrical proof is the final stage of the process of objectification, where the human experiences and constructions are removed from the discourse; it is the stage of alienation itself. This is a hint that geometry teaching and learning should not begin with the process of proving, but that of investigation and exploration (De Villiers & Heideman, 2014). Alienation is seen as contributing to the genesis of mathematical knowledge and understanding (Morgan & Tang, 2016). Hence, once PSTs can alienate a certain mathematical discourse, they begin to understand and construct mathematical knowledge, specifically in mathematical discourse and not just colloquial discourse. Objectification has been shown to have several advantages in the process of mathematical learning. Ben-Yehuda, Lavy, Linchevski and Sfard (2005) show that objectification may lead to mathematical discourses that contribute to increased levels of mathematical performance. Objectification further makes the way we communicate about mathematics more effective and provides a method of attaching objects into our mathematical discourses. Once we objectify, we create an ‘object’ or a ‘thing’ that has permanence in our mathematical discourse, which can also be an abstract entity. Through this objectified discourse, PSTs accumulate knowledge through participating in successive mathematical discourses that increase in complexity and applicability. The reification process relates directly to the mathematical objects objectified in discourse and it allows PSTs to endorse the discourse as a mathematical one. Hence, objectification, in this case, underlies the patterned ways in which we think. However, objectification removes personal experiences of learning and thinking in the discourse. As Sfard (2008, p. 56) puts it, objectified ‘descriptions deprive a person of the sense of agency, restrict her sense of responsibility, and, in effect, exclude and disable just as much as they enable and create’. This is possibly a consequence of objectification removing the PSTs’ thinking and learning experiences and the way in which they might communicate in their everyday lives in the discourse. The objectification of mathematical discourse means that colloquial discourse is reduced into a more specific mathematical discourse, consisting of specific word usage, routines, narratives, and some visual mediators. In geometry, this can be articulated by the colloquial utterance “this angle is equal to that angle” compared to the more objectified utterance “