Cognitive processes during mathematical modelling

The modelling approach enhances sense making (Mousoulides, 2009). Modelling problems promote understanding thus growth in understanding, as there is no memorised answer and no ‘correct’ solution. The ability to solve algorithmic procedures does not indicate depth of understanding. Students capable of solving these types of problems often cannot connect these manipulations to the real world (Schoenfeld, 1988). Problem solving based on a mathematical modelling perspective enables the learner to develop a deeper understanding of mathematics as it allows the learner to construct a stable understanding about situations (Izsák, 2004). The

knowledge and conceptual tools developed via the modelling process is situated cognition (Lamberts & Goldstone, 2005; Lesh and English, 2005). Situated cognition refers to the knowledge emerging from authentic, context-bound activities as well as from social

constructions (Olivier, 1999). Anderson, Reder and Simon (1997) argue that learning is not bound to the specific situation of its application; knowledge can transfer between different tasks. The idea is that instruction does not have to take place in a complex social setting only; there is also value to individual learning. A negative aspect of the transmission approach is the lack of preparation for unseen problems. The transmission approach results in a learner reconstructing existing objective knowledge (Murray, Olivier & Human, 1998). Learning needs to be a personal experience where a learner creates his own subjective knowledge rather than reconstructing objective knowledge. The active reasoning and organising of information promote meaningful learning (Chamberlin, n.d.).

Encoding and decoding define the cognitive elements of the modelling process (Occelli, 2001). Encoding involves the formulation of abstract pictures from the observable reality, thus building models representing these pictures, while decoding involves referring back to the observed reality (Occelli, 2001). Occelli (2001) differentiates between two types of “fundamental loops”

(para. 10) in the modelling activity: the internal loop describes the abstraction during the modelling process and the external loop represents the general modelling context of the modelling problem. Cognitive processes integrate the abstraction and context to initiate the modelling activity. Lesh and Sriraman (2005) argue that concept development transforms into understanding and forms mathematical modelling perspectives. The different levels of

constructing mathematical understanding, which lead to mathematical knowledge, can be compared directly to mathematical modelling and the outcomes thereof.

Borromeo Ferri (2006) explains that if a teacher looks at a learner’s modelling process from a cognitive viewpoint, he can only refer to the verbal descriptions and representations, and external illustrations and representations to identify if the learner is successful at beginning and

understanding the first steps of the modelling cycle and working through the process effectively. The mathematical-modelling concept has the advantage, not only of developing and constructing mathematical knowledge but also of developing mathematical skills and mathematical thinking (Sjuts, 2005).

Mathematical understanding and continuous focus and activation of the learner’s metacognitive processes are requirements for mathematical modelling. Metacognition refers to: “The active monitoring and consequent regulation and orchestration of processes in relation to the cognitive aspects on which they bear, usually in the sense of some concrete goal or objective” (Stillman & Galbraith, 1998, p. 162). Metacognitive knowledge consists of knowledge and beliefs about factors that influence the outcomes of cognitive knowledge and understanding. Metacognition is divided into the following processes: monitoring, control, orienting and reflecting.

Metacognition entails the knowledge of monitoring a process and involves verifying and acknowledgement of one’s own diagnosis. Metacognitive teaching in a cooperative setting has shown the potential to enhance problem-solving skills and mathematical-modelling construction (Mevarech, Zion & Michalsky, 2005). If teachers are knowledgeable concerning the mental processes and the learners’ thinking habits, and misconceptions they will adjust and adapt their type of instruction to enhance thinking, understanding, knowledge construction, the development of models, and meaningful learning. Knowledge is a development and procession of thinking, learning and making sense of complex situations (Burkhardt, 2006; Mickelson, 2006). Thus the knowledge of metacognitive strategies enables the management of thinking and learning.

Jagals’ (2013) study on the role of reflection and confidence when doing mathematics and the various components of metacognition were explored. Two of Jagals’ concluding remarks

emphasises the importance of the activity of reflection: “The act of reflection stimulates the level of confidence relating to planning and monitoring of tasks” (p. 179), and “the act of reflecting possibly manipulate and vary the knowledge and feelings associated with person, strategy and task characteristics during problem solving” (p. 180). Sjuts (2005) identifies metacognitive thinkers as self-regulated learners, who plan, organise, self-instruct, self-monitor, and self-evaluate during the process of learning. Research has shown that the higher the metacognitive ability of a learner, the higher the learner’s thinking abilities and academic levels (Garofolo & Lester, 1985;

Mevarech et al., 2005). Self-regulated learners are in control of their affective processes (Sjuts, 2005). Affect determines the rate of the development of learning and especially learners’ mathematical development (Hannula, 2006; Janvier, 1996). Affect controls cognitive processes (Owens, 2008). The importance of conscious self-regulation and extreme control over a learner’s own learning process is essential in all areas of the problem-solving process. Self-regulated learning describes the ability of a learner to set his own goals, to use appropriate methods and techniques regarding the content and the goal, and to review, as well as judge, his own processes (Maaß, 2006). It forces a learner to be in charge of his metacognitive processes. When learners take responsibility for their own learning experiences, metacognitive abilities become an important factor of learning. It is important for a learner to be aware of his own thinking and understanding as they form the foundation for the construction of knowledge.