Modelling and modelling perspective

Researchers and educators of mathematics have to follow certain guidelines in order to develop an MEA. These guidelines are referred to as the six principles of design, known as the modelling and modelling perspective (Chamberlin, 2004; Lesh et al., 2000). The modelling and modelling perspective are based on six principles that arose out of the work of a number of researchers and educators, but were subsequently refined by Lesh et al. (2000). According to Lesh, Amit and Schorr (1997, p. 2), these principles have been violated by mathematics problems seen in every major mathematics textbook and test, “therefore, in some sense, they are quite radical”. Furthermore, “The principles ensure that each MEA will have the intended curricular

39 and learning characteristics” (Chamberlin & Moon, 2005, p. 39). There are six principles.

2.3.1.1 Reality principle

The task focuses on problems with which learners are confronted in their lives where they are encouraged to make sense of a situation, based on extensions of their own personal knowledge and experiences. The activity must aid the learner to be able to interpret the problem given. Possani et al. (2010) emphasize that the context that motivates the MEAs must be motivational and realistic enough for the learners. At the same time, the mathematical elements embedded in the problem must not be compromised, and the specific mathematical concepts embedded in the context must be clearly outlined. Other researchers refer to the ‘reality principle’ as the meaningful principle, which is meant to increase the learners’ interest and stimulate the kind of activities in which mathematicians engage when solving problems (Chamberlin & Moon, 2005). They stress that the more realistic the problem, the more potential exists for creative solutions based on the learners’ familiarity with the problem.

2.3.1.2 Model-construction principle

The task involves the construction of a model where the learners will construct, explain, manipulate and predict a structurally significant system. MEAs must be designed to elicit creative behaviours and high-level thinking, especially at the level of synthesis (Chamberlin & Moon, 2005). The MEAs must be able to push the learner to explicitly describe and explain a given situation mathematically. The problem-setting must be authentic enough to “need mathematical concepts in the construction of a model” (Possani et al., 2010, p. 2128).

2.3.1.3 Self-evaluation principle

The design of the task makes it easier for the learners to assess the use of their responses and of those of others, and also be able to judge whether their responses are adequate. The activity must contain a criterion that enables learners themselves to revise and test their current way of thinking mathematically. The learners must be able to recognise the appropriateness and use of their model without input from the educator (Chamberlin & Moon, 2005).

40 2.3.1.4 Model-documentation principle

The mathematical concepts embedded in the task have to enable the learners to reveal how they understand the situation mathematically, and to reveal their mathematization processes as they work on the task. It ensures that while working on the activity the learners create some form of documentation to reveal their thinking of the problem situation. Lesh et al. (2000) refer to this principle as the ‘taught revealing activity’, in that it is able to reveal how the learners are thinking as they work on an MEA which can be documented by the educator. The model-documentation principle helps the educator who implements MEAs to focus on the thinking processes of the learners during problem-solving, as well as on their final model (Chamberlin & Moon, 2005).

2.3.1.5 Model-generalisation principle

This feature requires the learners to be able to produce sharable and re-usable solutions so that mathematical model(s) could be transferred and used in other real- life situations. If the model can be transferred to other parallel real-life situations requiring a similar model, then the learners’ responses are deemed to be successful (Chamberlin & Moon, 2005).

2.3.1.6 Simple prototype principle

The task must be designed to elicit the creation of a model while still being as simple as possible. The activity must be as simple as possible, whilst at the same time being mathematically significant. According to Chamberlin and Moon (2005), the principle requires the model created by the learners to be easily interpreted by other learners. They emphasize the difference between this model-generalization principle and the simple prototype principle in that in the simple prototype principle the learners may use the prototype in a similar situation but not in a parallel situation.

Lesh et al. (1997, pp. 2-3) also articulated a number of questions to be asked regarding each of the principles as an MEA is being designed.

(a) Reality principle

Could this really happen in a real-life situation? Will the learners be encouraged to make sense of the situation based on extensions of their own personal knowledge and

41 experiences? Will the learners’ ideas be taken seriously, or will they be forced to conform to the educator’s notion of the ‘correct’ way to think about the problem situation?

(b) Model-construction principle

Does the task create the need for a model to be constructed, or modified, or extended, or refined? Does the task involve constructing, explaining, manipulating, predicting, or controlling a structurally significant system? Is attention focused on underlying patterns and regularities rather than on surface-level characteristics?

(c) Self-evaluation principle

Are the criteria clear for assessing the usefulness of alternative responses? Will the learners be able to judge for themselves when their responses are good enough? For what purposes are the results required? By whom are they required? When?

(d) Model-documentation principle

Will the responses require the learners to explicitly reveal how they are thinking about the situation (‘givens’, goals, possible solution paths)? What kind of system (mathematical objects, relations, operations, patterns, regularities) are they thinking about?

(e) Model-generalization principle

Does the model that is constructed apply to only a particular situation, or can it be applied to a broader range of situations?

(f) Simple prototype principle

Is the situation as simple as possible while still creating the need for a significant model? Will the solution provide a useful prototype (or metaphor) for interpreting a variety of other structurally similar situations?

42 The study drew on these principles in the design of an effective MEA that served as a medium of instruction in developing learners’ conceptual understanding in algebra at the level of Grade 6.

2.3.1.7 Characteristics of the MEA in the lens of the modelling and modelling perspective

Chamberlin and Moon (2005) explain that MEAs that are designed on the bases of the six principles of the modelling and modelling perspectives must have the following characteristics:

i. Inter-disciplinary: This enables educators to integrate other disciplines. In addition to mathematics literacy, which is the main goal of MEA, MEAs have a context related to social studies, science, physical education, etc. When the learners use knowledge from various subjects when solving MEAs it increases their ability to reason creatively.

ii. Well-structured problems: MEAs are well-structured problems in the sense that all the necessary information to solve them is within the problems or is readily available to the learner. The learner does not have to do any research in order to solve the problem.

iii. Realistic problems: MEAs must be realistic problems that are relevant in the lives of the learners (Lesh et al., 2000). According to Cooper and Harries (2003), realistic problems in MEAs are likely to promote learning mathematics with understanding compared to problems without context. iv. Meta-cognitive coaching: MEAs are administered successfully when the

educator acts as a meta-cognitive coach when the learners are solving MEAs, and pose questions to the learners rather than answering them. v. Explication of learner thinking: MEAs provide the opportunity for

educators to explore the learners’ thinking as they work on the MEAs which can give much insight when the curriculum is being revised.