The cognitive structure in the human mind is associated with the concept image which is all those mental representations and processes regarding the concept (Tall, 1988). Tall (1988) also states when a learner meets an old concept in a new context, it is the concept image with all its assumptions and not the concept definition that reacts to the new task. The roles of emotions, consciousness and physical environments have effects on mathematical thinking, therefore influencing the forming and development of concepts (Thagard, 2008). Negative emotions, consciousness and physical environments have a negative effect and positive emotions,
learning milieu determines the cognisance of a learner. Mayer and Hagarty (1996, p. 34) summarize the cognitive processes during the problem-solving process:
i. Translating: a learner constructs a mental representation.
ii. Integrating: the learner makes assumptions and recognitions during the construction of mental representations.
iii. Planning: devising a plan for solving the problem iv. Executing: carrying out the plan and computations
Polya’s (1973) four steps to the problem-solving process involve: understanding the problem, developing a plan based on connections made, carrying out the plan, and evaluating that solution by looking back. The process above is very similar to that of Polya’s, yet Polya’s shows more reflective characteristics.
Building blocks of mental representations are perceptions, past actions and the result of reflecting on, and abstracting from, already existing representations. These will often involve identifying and retrieving a sample or prototype (Lutz & Huitt, 2003). Tall (1988) refers to the sample type or prototype as a concept image. Abstract objects emerge at intersections in development of mathematical knowledge when some new process is introduced and applied by another already existing and known process. Abstraction is the activity when a learner becomes aware of similarities amongst objects, symbols and concepts. The stages in abstraction are listed as follows: internalisation, condensation, reification, generalisation, synthesis and abstraction (Mitchelmore, 2002). Abstract objects mediate between the product of lower process and higher level manipulation (Sfard & Linchevski, 1994). Structural development is considered to be a difficult operation as it is more abstract and advanced stage of conceptual development.
Mathematical awareness and assumptions influence the extent of seeing mathematical structure in situations during model development (Zbiek & Conner, 2006). A conceptual theory is closely
associated with model building. The conceptual theory forms as a learner revises, refines and extends his ways of thinking. During this revising, refining and extending conceptual tools such as constructing, describing and explaining are developed. As models are continually revised and revisited, new ways of thinking are developed. This local conceptual development is similar to Piaget and Van Hiele’s theories of conceptual development (see Lesh & Harel, 2003). Their theories emphasise mathematical thinking being the interpretation and observation of situations.
The product of an individual’s conceptual operations and development is mathematical knowledge (Voigt, 1996).
The development of conceptual systems involves more than accumulating relations, operations and principles. They occur along a variety of dimensions. These dimensions can be specific to general, concrete to abstract, simple to complex, intuitive to formal and situated to de-
contextualised (Lesh & Lehrer, 2003). When a learner reflects on and abstracts from past experiences and perceptions he builds on existing representations to form new
conceptualisations. These cognitive structures are connected by conceptual schema and mediate interactions between conceptual understanding of situations and symbolic representations (Izsák, 2004). These processes are reversible and are organised by logic. The stages of logical modelling are composed by conceptualisation, verbalisation, and formalisation (French & Finlay, 2000). Learners’ representations of situations are based on their understanding of physical patterns, existing conceptual representations and symbolic patterns. Learners use these patterns to recognise situations which can be modelled and connect these to conceptual representations through symbolic representations. Models are therefore focused on describing patterns and other mathematical representations so that learners can build an understanding of the system that is modelled. “Mathematical models are conceptual systems that are: (a) expressed for some specific purpose and (b) using some representational media” (Lesh & Lehrer, 2003, pp. 111-112).
Lesh and Lehrer’s (2003) developed a modelling cycle to show the process a learner goes through when they develop models. Figure 2.2 shows how the model is represented by some representational media and it is described or explained. Both the describing and representing is based on the purpose of the model. This purpose of the model is the “end-in-view” (p. 112). Lesh and Lehrer (2003) state that the modelling cycle can be revisited and revised more than once, and through this revisiting and revising they predict the learner to show different ways of thinking regarding the symbols the learner chose to use, the process he followed, and the way he got to the solution. Learners make sense of new information by relating it to previous understanding and experience. Olivier (1992) explains that learning depends on the schema formed in a previous learning experience. Learners make sense of new information by relating it to earlier learning. It is therefore easier for a learner to add to his existing schema, than to change it.
Describe or Explain
Figure 2.2: A modelling cycle (Lesh & Lehrer, 2003, p.112)
Olivier (1992, p. 200) further elaborates on “overgeneralization”; when a learner applies the same operation to a new situation, because he was successful in previous instance. A state of equilibrium occurs when an individual experiences mismatches between internal conceptual understanding and the understanding of the modelling situation (Izsák, 2004). As the informal understanding of models evolves through multiple conceptual systems, levels of understanding become mature and mathematical thinking can now be organised around abstractions rather than experiences (Lesh, Doerr, Carmona & Hjalmarson, 2003).