The theory of commognition is cognate from the work of Wittgenstein, (1953) and Vygotsky (1978). All they were not contemporaries, these two luminaries espoused similar ideas on the nature and processes of learning. To them, learning and thinking are means by which one communicates mathematical concepts with others and oneself in an objectified manner (Daniels, 2001).
Commognition derives from one of the participation theories which posit learning as the process of individualising the mathematical discourse (Sfard, 2007; Sfard, 2014). It is in that context that commognition was developed from socio-cultural learning theory, because in its development. there is acknowledgement that learning of Mathematics is both societal and individual (Caduri & Heyd-Metzuyanim, 2015). The change in societal communication results in a change in communication with other individuals. In commognition, it is expected that participation progresses from ritualised routines to exploratory routines (Sfard, 2014).
As a concept, commognition is a combination of “communication” and “cognition”. In the commognition perspective, talking and thinking are viewed as part of communication (Sfard, 2008); and thinking is viewed as an individualised form of communication activity with oneself (Sfard, 2007; Sfard, 2008). Thinking is largely expressed in the form of talking and through writing. Individuals change by being involved in various activities. For instance, participation of learners in the mathematical discourse is measured by their talking, gestures and writing in the particular sub-discourse they are engaged in. Therefore, learners participate by talking and bringing forth their reasoning in class or group discussion; or in solving problems in their writing.
According to the commognitive perspective, the aim of learning Mathematics is to enable learners’ membership of the community of Mathematics (Nachlieli & Tabach, 2012). In such a context, learning is viewed as the process of becoming a competent participant in mathematical discourse or practice. This participation is evident in communication by a learner. 3.3.1 Foundational principles of commognition
Commognition has its foundation on three human activities and their constitutive principles (Sfard, 2014). Where communication changes in mathematical activities, there is a concomitant
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transformation in the way we do things. Secondly, discourses function as propagators of innovation and repositories of complexity. Thirdly, learning happens at either object-level or meta-level. Object-level learning denotes accumulation of endorsed narratives, while meta- level learning is about a whole transformation of the endorsed narrative. The three foundational principles of commognition are discussed below.
3.3.1.1 The co-constitutive nature of discourses and other human activities
As an auto-poietic, Mathematics grows from within (Nachlieli & Tabach, 2012; Sfard, 2015). Mathematical activities and developments contribute to the growth of Mathematics as a field of study. As human beings engage in mathematical activities, a need for change arises. When humans were at ease with natural numbers, counting their wealth and so on, they arose a need of representing debts and the negative numbers were born out of that need. These, and a host of other human activities, are responsible for innovation within the field of Mathematics, while also generating new narratives and transforming other narratives as well. For example, the introduction of the hyperbola graph has changed the perception that a graph should always be continuous.
3.3.1.2 Discourses function as propagators of innovation and repositories of complexity The ability to communicate enables the change and complexity to be preserved (Sfard, 2014). Verbal communication allocates a semblance of reflective communication. This communication is both intra-personal and interpersonal. The exchange of views among different mathematicians and reflective thought of individuals leads to the preservation and application of the changes and innovation to new situations. Change and innovations also produce compression, when symbols are used to minimise or reduce communication by saying less (Caspi & Sfard, 2012). Instead of writing a set of squares and listing all of them, we can write 𝑥2; 𝑥 𝜖ℝ. This then changes mathematical discourse from ‘talk about process’ to ‘talk about objects’. In this regard, communication helps to say much with less, while preserving what would have been gained.
3.3.1.3 Two levels of discursive learning
In object-level development, there are new narratives. For example, when learners are introduced to exponential functions subsequent to exposure to the quadratic function, there are new objects for them to contend with. These objects include the formula/equation, intercepts, asymptote, ordered pairs and types of graphs to be produced. The graph is completed by drawing a line that joins all the points. The rules used in the exponential functions sub- discourse remain the same as those of plotting points on a quadratic, and the development is cumulative. At the introduction of exponential functions, learners are already familiar with the plotting of points. One major characteristic of object-level development is that it is possible for
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learners to construct narratives without the interlocutor. This is facilitated by allowing learners to explore the mathematical object, as well as making conjectures and generalisations.
For grade 10 learners who have plotted points in earlier grades, it is possible for them to make generalisations on the linear function without the assistance of the interlocutor because the growth is cumulative. Points on linear functions always form a straight line. All linear functions have this characteristic, and it is not difficult for learner to make or prove a conjecture and make generalisations with less help from the interlocutor or teacher. Aother example of learning without the interlocutor relates to observing that the constant in a general function represents the y-intercept. The constant in functions is the value of the graph’s intersection with the y-axis. Learners can observe the behaviour of the constant and expand their discourse. There are no contradictions between the rules of sketching a linear graph and those of plotting points. Learners can also realise that algebraically, when x is zero, the y value is always the constant. In object-level learning, it is possible for learners to generate endorsed narratives without the help of the interlocutor on condition that the previously endorsed narratives (realisation tree) are in place (Nachieli & Tabach, 2012).
For meta-level development rules, the interlocutor becomes the important factor. Meta-level development is characterised by a change in rules, and new objects change rules of endorsement. In meta-level development, there are apparent contradictions between the newly- introduced narratives and previously accepted narratives. The introduction of Calculus brings about meta-level development in that new objects, differential rules, the gradient function, and stationery points are introduced; just to name a few. There is no way that learners can use their previous knowledge to unlock the rules of differentiation. The interlocutor should explain the gradient function, the meaning of stationery points, and differential rules. In object-level development, learning is cumulative and there are no new rules, as is the case for meta-level development. The interlocutor comes in mainly to narrow the gap between meta-rules and the learners’ current development. The purpose of narrowing the gap is to enable learners’ advancement to the next possible level. This may be true for most of meta-level learning of mathematical objects. For example, the introduction of differential rules, stationery points, as well as the behaviour of the graph as it approaches horizontal and vertical infinite points (Sfard, 2015).
3.3.2 Applicability/ Relevance of commognition in this study
The special type of communication in Mathematics is guided by the characteristics of commognition. These are the words used, the interpretation of the visual mediators, the endorsed narratives and the routines. People engage in mathematical activities so that they may become a part of a community of mathematicians (Sfard, 2012). The use of words, interpretation of visual mediators and the narratives students endorse determine the routines
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that learners operate on. In this study, has introduced an analytical tool known as the Discourse Profile of the Hyperbola and Exponential Function (DPHEF). The tool was useful in the analysis of learners’ mathematical discourse, some of which was found to be ritualised since these learners based their mathematical actions on what they saw others (e.g. teachers) doing. On the other hand, other learners were able to lay claim to being part of the community of mathematicians by exhibiting exploratory routines by using words and interpreting visual mediators accordingly.
Sfard (2015) emphasises the centrality of the interlocutor in commognition, who facilitates reflective imitation. Accordingly, the learner does (imitates) what he or she sees the interlocutor doing. The more the learner becomes successful, the more s/he reflects carefully on the process s/he was involved in, and s/he comes up with conjectures and proves them. Given this state of affairs, what Sfard (2007) refers to as mathematical objects is what Vygotsky refers to as concepts. Mathematical objects are discursive constructs that replace ‘talk about processes’ with ‘talk about things’ (Sfard, 2015). Therefore, mathematical objects are used to say much with less. For example, instead of writing “the combination of two sets of marbles one with 6 (six) marbles and the other with 5 (five) is eleven marbles” we write “6 + 5 = 11”. This form of mathematical configuration of learning in commognition theory is presented as the attainment of objectification, when the talk about objects replaces talk about processes (Sfard, 2008).