120 As was noted above, conceptual systems need representational tools. These tools help to support reasoning and act as a medium of communicating and sharing information. A fundamental presupposition of cognitive science is that humans think about real and imaginary worlds though internal representations. One role of representation is
helping learners express externally what they are "seeing" internally. Representational tools are thus necessary to describe external systems and to express internal ones. Lesh and Doerr (2000) explain that "the purpose of representations in this
development is not only for learners to communicate with one another; it is also for learners to communicate with themselves and to externalise their own ways of thinking so they can be examined and improved" (p. 368).
To facilitate communication, many kinds of representations are used in modelling. These may include, but are not limited to, linguistic modes in the form of verbal or written communications, visual communications including gestures, pictures, diagrams, concrete manipulatives, or computer simulations, as well as conventional notations expressed, for example, in mathematical equations. Different
representational systems will emphasise (and de-emphasise) different aspects of the concept. To clarify, Dai (2010, p. 660 Kindle edition) states that in an instructional content with curricular activity there can be three levels of representation:
● representation of subject matter as part of the curricular content in its purposes, structure, and functionality;
● representation of the informational content as part of a larger body of domain knowledge and its epistemic value and practical utilities;
● and, representation of content being learnt as a cultural way of knowing and part of social practice that produced this body of knowledge (i.e. recognising it as a particular kind of socially sanctioned meaning making or problem solving).
121 Traditionally, symbolising was seen as a unidirectional process. It generally took the form of attaching a semiotic placeholder to an already extant object. Yet, within the modelling framework assumptions regarding the co-emergence of meaning and symbolisation are introduced (Sfard, 2000; van Oerts, 2000). The relationship between learning and symbolising now has a reflexive nature in so far as symbols and their meanings are continually revisited and revised as learners re-organise their own thinking and engage in communication with others in the classroom.
Proponents of mathematical modelling generally agree that learners should be engaged in activities, reflections, and discussions that show how a symbol is used in action, rather than handing learners ready-made symbols and assuming that they can decode them in a similar manner to an expert. But, there are differences of opinion as to whether learners should be initially encouraged to invent their own symbolism as they develop a model or whether the modelling activities should be more geared towards exploring already existing
mathematical notation. Authors such as Bransford et al. (2000) argue that learners need to be initiating into already existing symbols and their meanings, whereas others such as Nemirovsky and Monk (2000) state that it is important that learners are given opportunity to invent their own symbol systems. Those who side with the latter support the general claim that it is unrealistic to expect that learners will create representations in line with the standardized
conventions that have evolved in the course of mathematical history.
3.3.7 Learners acquire knowledge through social participation
Engagement in modelling also affects the level and type of social participation. Although there are elements of Sfard's (1998) acquisition metaphor and her participation metaphor in modelling, modelling tends to fit better with a third metaphor, which is the knowledge creation metaphor of learning suggested by Paavola and Hakkarainen (2005, p. 539). These authors' argument is that knowledge
122 creation must be seen as more than the individual building his own knowledge
structures with the aim of creating a logical system of organised content with rules that allow transfer to new situations. It is also more than just being part of a culture and learning how to act in a socially sanctioned manner. Knowledge creation entails a unique quality of collaborative activity that leads to shared objects and artefacts, both intellectual and physical.
In line with the knowledge-creation metaphor, the modelling approach provides a rich and balanced blend in its consideration of the individual, the group, the subject
domain, and the cultural context. It covers the concern for the individual in that the individual has to mathematise, explore, justify, and own the knowledge. There is a concern for the group, the individual has to work within a group and negotiate arguments between groups. At the same time, there is an acknowledgement of the dynamics between individuals and groups — the group affects the individual and the individual in turn changes the dynamics of the group. And lastly, there is concern for the subject matter — the learning of mathematical principles and content.
A key point is that modelling involves collaborative learning. Collaborative learning is about a group of learners working together on a task. As an illustration, Damon and Phelps (1989, p. 9) distinguish three types of collaborative learning experiences, namely, peer tutoring, co-operative learning, and collaborative learning. These
authors make the distinctions by contrasting one another along dimensions of equality and mutuality of engagement. In their framework, peer tutoring tends to foster
dialogues that are relatively low on equality and varied in mutuality; cooperative learning foster ones that are relatively high in equality and low to moderate in mutuality; and peer collaboration fosters ones that are high in both. On the positive side, Gillies and Khans (2008) describe that some of the core intentions of
collaborative learning are to provide learners with opportunities to communicate with one another, share information, and to develop new understandings and perspectives through this kind of reciprocity. In reality, the nature and dynamics of collaborative learning can result in unintended consequences. For example, we know from research that learning in collaborative setups is affected by perceptions of power amongst
123 group members. Webb (2013, p. 22) provides a list of incidences that will undermine group performance. These include learners failing to share elaborate explanations, not asking for help when needed, disengaging from the group, suppressing others'
participation, engaging in too much conflict or avoiding it all together, not co- ordinating their communication, or engaging in negative social-emotional behaviour that impedes group functioning.
All things considered, Black-Hawkins (2014, p. 392) reminds teachers who use collaborative learning techniques to hold on to the mindset that collaboration is a resource for learning, dependent on the range, experiences, and expertise among class members, and not simply a problem to be overcome. She also adds that collaborative learning necessitates a
consideration of the emotions of learners evoked through participatory processes. She
explains that evaluating the emotions of learners with SEN is not done sentimentally, but in a systematic way during the modelling process by taking heed of expressions that are negative like fear, humiliation, anger, intolerance, and failure and of more positive ones like feelings of confidence, joy, kindness, resilience, and respect. Likewise, Grosser (2014) argues that cooperative learning argues that the focus of cooperative learning is on social interaction and not necessarily on explicit cognitive processes. It creates opportunities for actively mediating cognitive skills and metacognitive awareness