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.

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,

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 (

The Van Hiele theory of geometrical thinking was developed by Van Hiele-Geldof (

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 (

Fixed sequence: PSTs progress through the levels invariantly which means that a PST cannot be at Van Hiele level

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,

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,

This section provides the details of Sfard’s (

In 2008, Sfard published a book titled

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 (

The comparison between rituals and explorations in commognition.

Elements of discourse participation | Ritual | Exploration |
---|---|---|

Closing condition or goal | Relationship with others (improving one’s position concerning others) | Description of the world (production of endorsed narrative about the world) |

By whom the routine is performed |
With (scaffolded by) others | No need for scaffolding; can be performed individually |

Others (authoritative discourse) | Others and oneself (internally persuasive discourse) | |

Applicability (changing the when, keeping the how constant) | Restricted: the procedure is highly situated | Broad: the procedure is applicable in a wide range of situations |

Flexibility (changing the how, keeping the when constant) | Almost no degree of freedom in the course of action | The procedure is a whole class equivalence of different courses of action |

Correctability | Cannot be locally corrected; has to be reiterated in its entirety | Parts can be locally replaced with equivalent subroutines |

Acceptability condition | The activity has to be shown to adhere strictly to the rules defining the routine procedure; the acceptance depends on other people | The narrative produced through the performance must be sustainable in such a way that the acceptance is independent of other people |

Words and mediators use | Phrase-driven use of keywords as descriptions of extra-discursive mediators | Objectified use of keywords as signifying objects in their own right |

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

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,

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,

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,

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, ^{1}

Objectification has been shown to have several advantages in the process of mathematical learning. Ben-Yehuda, Lavy, Linchevski and Sfard (

Sfard (

The notion of signifiers is important, also, in considering mathematics as a discourse. In commognition, ‘signifier’ refers to any primary object that encapsulates its realisation procedures (Sfard,

Furthermore, Sfard (

The key to identifying the realm in which a discourse belongs is the keywords used by interlocutors in the dialogue. If you hear people dialogising and the keywords ‘tangent’, ‘chord’ or even ‘straight line’ occur, we know this dialogue belongs within the constraints of geometrical discourse in the high school curriculum. Mathematical discourses in schools are similar. There is no way one can talk of ‘quadratic equations’ or ‘the tan-chord theorem’ in any subject other than mathematics. Even though some word usages in mathematics appear in colloquial discourse, they form part of the formal mathematical discourse that helps in understanding mathematical concepts. The significance is highlighted by Sfard (

These are the visible objects that can be used by interlocutors in communicating (Sfard,

In Sfard’s (

any sequence of utterances framed as a description of objects, of relations between objects, or of processes with or by objects, that is subject to

Narratives in discourse can be thought of as ideas that need to be discussed and endorsed mathematically, and once a certain narrative is endorsed, it is considered a theory (Sfard,

Routines are significant and special in mathematical practice. Routines, according to Sfard (

Mathematics education has been characterised by different beliefs about teaching and learning. A recent development was commognition, a belief that equates learning to communicating about thinking, which becomes a ‘legitimate peripheral participant’ in mathematical discourse (Lave & Wenger,

Rituals are socially oriented actions performed to conform to society. One engages in rituals to avoid punishment, please someone or for gain certain rewards (Lavie, Steiner, & Sfard,

The main driver of rituals is societal expectations. When one feels obliged to perform in a certain way to please someone, then one engages in rituals. If learners in Mrs X’s classroom were directing all answers to her without giving reasons and the teacher was the one endorsing the answers as correct or incorrect through giving applause for positive evaluation, these learners’ participation in this discourse was solely motivated by getting applause or positive evaluation from the teacher (Heyd-Metzuyanim & Graven,

As opposed to rituals, explorations are not aimed at pleasing or conforming to societal expectations, but at advancing theory. Particularly, routines can be characterised as explorations if they produce narratives contributing to a mathematical theory instead of tangible objects (Sfard, ^{2}

Too often learners are considered ‘good’ at mathematics because they conform very well to society’s expectations, and those who do not conform are usually labelled as ‘outcasts’ and ‘weak at mathematics’ (Heyd-Metzuyanim & Graven,

Comparing cognitive conflict and commognitive conflict.

Elements of the conflict | Cognitive conflict | Commognitive conflict |
---|---|---|

The conflict is between | The interlocutor and the world | Incommensurable discourses |

Role in learning | Is an optional way for removing misconceptions | Practically indispensable for meta-level learning |

How is it resolved? | By student’s rational effort | By student’s acceptance and rationalisation (individualisation) of the discursive ways of the expert interlocutor |

A brief explanation of the relationship between the two theories as conceived in the study suffices. In particular, this relationship is described based on the type of discourse participation evident from PSTs and how this type of discourse participation was related to each level of the Van Hiele theory. Firstly, the relationship is explained based on teachers’ failed behaviours during geometry problem-solving instead of their successes, because this will raise awareness of the severity of the need to develop competent geometry teachers in South Africa. Teachers’ behaviours, described in

Relationship between preservice teachers’ discourse participation and the Van Hiele levels of geometrical thinking.

Level of geometrical thinking | Ritualistic discourse participation. (Characterised by colloquial discourse) (Routines scaffolded by others) | Explorative discourse participation. (Characterised by objectified discourse) (Routines performed individually) |
---|---|---|

1 | PST uses visual cues without corroborating them with properties, theorems or definitions to identify mathematical objects. Sometimes fails to link descriptions with their visual mediators. | PST corroborates visual cues, theorems, properties and definitions to identify mathematical objects. Shows a good understanding of linking descriptions with their corresponding visual mediators. |

2 | PST uses properties to identify mathematical objects as they appear but does not connect these properties to perform routines, endorse narratives or produce endorsable visual narratives. Sometimes assumes properties and definitions of mathematical objects based on the visual appearance of the diagrams. | PST uses properties to identify mathematical objects as they appear and can connect these properties with the performance of routines and endorsing narratives. Never uses visual appearance of diagrams to conclude about properties or definitions of mathematical objects. |

3 | PST relates figures using their properties but still relies on remembering procedures and algorithms to proceed with performing routines and endorsing narratives nor can they substantiate their conclusions. They also fail to see the link and relationship between properties of geometrical objects. Thus, they struggle to form even short deductive chains of arguments. | PST relates figures using their properties and uses explorations to produce discursive mathematical objects ( |

4 | PST attempts to follow strict procedures and previous experiences to describe abstract mathematical objects and construct narratives through logical deduction but fails to substantiate the subroutines leading to the production of the narrative. | PST explores different approaches interconnectedly to invent problem-solving strategies and construct narratives through logical deduction. They can substantiate all subroutines followed in producing the narrative and are flexible in their routine performance. |

A PMT who is operating at a ritualistic Van Hiele level 1 not only struggles with the visual identification of geometrical objects using their appearances but also fails to link descriptions with their visual appearances. This means that this PMT is not able to identify a theorem or property using the appearance of the diagram in a particular problem. This PMT is then not ready to proceed to the other levels of geometrical thinking and must stay in that level until they have mastered the explorative Van Hiele level 1. Wang (

A commognitive analysis of pre-service teachers’ geometrical thinking development through Van Hiele levels of geometrical thinking.

The transformation in discourse is a critical and necessary condition for mathematical learning (Heyd-Metzuyanim,

Upon conducting this literature study, it was found that development to higher Van Hiele levels was dependent on the discourse participation of the PSTs according to

The identification of geometrical figures by their appearances only is the critical determining skill in level 0 because only the recognition of the figure is required. The PSTs who could not identify geometric figures by their appearances during discourse and those who relied on scaffolding from the interviewer lack this critical geometrical skill. Furthermore, in advanced geometry, theorems can also be identified through the appearance of the diagrams and PSTs who could not identify particular theorems that were available in the diagram lack visualisation. From the literature findings, the lack of visualising theorems can hinder PSTs’ problem-solving but once a scaffold was given by the interviewer, they were able to complete subroutines that involved the theorem as exploration. Thus, PSTs who lack this visualisation skill should not be allowed to progress to level 1 because visualisation is a prerequisite in level 1. They must stay at level 0 as indicated by the red arrow in

The defining characteristic in Van Hiele level 1 is that PSTs should have moved from not only recognising objects by their appearances but also linking the objects to their properties. However, interrelating the properties of objects is still not developed at level 1. The theoretical findings indicate that PSTs who cannot name properties of geometrical objects without scaffolding and those who used colloquial discourse to name these properties must not move to Van Hiele level 3. Furthermore, PSTs who mentioned incorrect properties to gain social acceptance, relying on the usage of ritualistic discourse and past experiences should also remain in ritualistic Van Hiele level 2. These PSTs have not mastered the ability to link a geometrical figure with its property and thus they should remain in ritualistic Van Hiele level 2 until this skill is mastered. Those PSTs who are promoted to explorative Van Hiele level 3 are those who can individually link a particular property to a particular geometrical figure individually. These PSTs can use objectified discourse to mention and link geometrical properties with their geometrical figures which according to the theoretical findings relates to explorative discourse participation. This link can allow PSTs to produce endorsed narratives about the geometrical world and these PSTs can be promoted to explorative Van Hiele level 3 because they show mastery of the skill of linking geometrical figures with their properties using explorative discourse.

At level 3 PSTs do not just link properties with their geometrical figures but can logically order and interrelate properties to understand the relationship between geometrical figures. Mastery of level 3 means that PSTs are getting ready for logical deduction required in proofs. At this level, PSTs should not be reliant on scaffolding to link properties and still be promoted to level 4 because that shows that they have not mastered Van Hiele level 3. Furthermore, the formulation of meaningful definitions is critical in level 3 and if PSTs still rely on scaffolding they are not ready for level 4. The understanding of properties and theorems is critical in proofs and if one has not mastered the link between properties and the relationship between figures, then one is not ready to move to level 4. This is because PSTs will produce a whole proof without giving proper reasoning for their arguments or statements with correct reasons. Only PSTs who show logical understanding of how properties of different figures link and are able support their narrative individually using objectified discourse should be allowed to proceed to level 4 which is the last Van Hiele level required in the CAPS.

Level 4 requires that PSTs be able to use experiences from the previous levels to understand the role of properties, theorems and the links therein when doing geometry proofs. At this level, PSTs now begin to develop longer arguments to perform geometrical proofs and can successfully substantiate each argument with an endorsed mathematical narrative. Those PSTs who continually rely on the same procedure applied in exactly the same way instead of being flexible in their strategies are not ready for level 4 and they should be demoted to level 3. At this level, relying on procedures, subroutines and scaffolding from others to obtain a proof does not guarantee that one will be able to prove similar problems in the future; independence is required to master level 4. Justifying statements during proofs using phrase-driven and colloquial discourse instead of objectified discourse also shows that one has not mastered level 3; thus, they must be demoted to level 3. A PST who performs geometrical proofs independently and uses explorative discourse when talking about geometrical proofs can be thought of as ready to teach geometry at Grade 12 level in the CAPS.

The authors declare that no competing interest exists.

S.C.M. completed the article individually and V.M. was the supervisor of the PhD study and contributed by checking the accuracy of the information in the article.

This article followed all ethical standards for research without direct contact with human or animal subjects.

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Data sharing is not applicable to this article, as no new data were created or analysed in this study.

The views expressed in this article are of the authors only and do not represent the official position of the two institutions.

A monologically understood world is objectified and corresponds to a single and unified authorial consciousness.