Problem Solving, constructivism and realistic mathematics education
3.4 Constructivism in mathematical modeling in schools
There is very little doubt that children enter mathematics classes with views and ideas of certain mathematical occurrences that they experience in real life. They, for example, may be aware that water reservoirs are generally constructed at the top of the highest hill in a
village because the pressure required to feed the water to large distances must be great.
They may be aware that in order to establish the height of a mountain, it is basically impossible to start from the top of the mountain with a tape measure and work downwards towards the foot of the mountain. They may not know how the height is calculated, but they could have a sense that mathematics would playa role. Often, mathematics educators make the flawed assumption that learners are empty vessels into which knowledge must be poured. The constructivist view is completely opposite and is based on the theory that learning is an active process and learners construct their own meaning. This therefore implies that learners themselves are responsible for their own learning.
In the constructivist theory it is accepted that "the learners have their own ideas, that these persist despite teaching and that they develop in a way characteristic of the person and the way they experience things, leads inevitably to the idea that, in learning, people construct their own meaning" (Brookes, 1994 : 12)
In developing new knowledge, learners often use their pre-existing knowledge. The following, according to Scott (1987 : 7), are key points when considering the construction of meaning by learners.
1. That which is already in the learner's mind, matters. This point simply reiterates the fact that the learner's pre-existing knowledge is important. When modeling, a learner would have to develop strategies based on previous knowledge. If the learner has never worked with equations previously, it would be difficult to model a real life phenomenon based on algebra (consider the equations used in example 2 in 2.4.2).
2. Individuals construct their own meaning. Each learner may be at a distinctly different
due to environmental, societal or cultural differences or mental abilities. Talking to a learner from a deep rural area about a reservoir may be difficult if they have no fresh water source in that village. In the modeling experiment in this research, learners talked of shopping complexes in rural areas where no roads existed. Their knowledge of rural environments was minimal. Thus, modeling may be affected by the different meanings that are constructed by different individuals.
3. The construction of meaning is a continuous and active one. Often learners generate ideas, test or evaluate them and then review and reaffirm these ideas or hypotheses.
That is exactly what the process of modeling expects. But all the ideas and hypotheses can only be derived from the previous knowledge the learner has. For example, if learners are asked to find a point that is equidistant from two other points, their responses will generally be that the midpoint between the two given points will be equidistant. This comes from their experience of working with midpoints. As this researcher determined, learners were able to find other points that were equidistant as well, not because they worked with it before but because they constructed their own meaning. In this research, they found many points that were equidistant from two points, but it was a revelation to them that these points would lie on the perpendicular bisector.
From a constructivist perspective, educators must therefore carefully consider the learners' prior knowledge when developing modeling activities. For learners to restructure their ideas and knowledge they need to have some already determined starting point. Expecting learners, for example, to differentiate an equation in order to find its gradient at a point would not be advisable for a Grade 10 learner, but may be quite appropriate for a Grade 12 learner. With the constructivist perspective that the learner is responsible for his or her own
learning, the role of the educator now becomes more modified. Modeling activities allows the learner to interact with the problem and his or her prior knowledge to construct new meaning for himlherself. Modeling activities takes cognizance of the fact that the educator does not just transfer knowledge, but acts as a facilitator for the learner to construct knowledge.
So according to Constructivism learners change and develop the meaning of that which they experience. Modeling activities, especially using dynamic geometry software, such as Sketchpad, offer learners the experience of working with many examples within a few minutes. Learners can "see" the results as they interact with the software. This enables the learner to go from their level of understanding to, either a related level of understanding or a completely new level of understanding. These software programs provide the learners with immediate feedback as they test their ideas. According to Ranson and Martin (1996 : 9), "reasoning and testing ideas in this way reveals the indispensable mutuality or sociability of learning". Learners can easily determine a correspondence between what they know and the new knowledge they 'see' unfolding as they work through the exercise.
Often there may be a conflict with old knowledge and the new knowledge they are discovering. Cognitive restructuring of knowledge takes place, where the new knowledge is assimilated using existing schemas that were already established. This is illustrated in the discussion of the empirical findings in Chapter 4.
According to constructivism we quite frequently learn through the process of trial and error (Zecha, 1995 : 82). When faced with a particular problem, the learner engages in different methods to solve the problem. True and useful statements are retained whilst false or
untrue statements are discarded. Problems do arise when the learner cannot sufficiently recognize which is useful and which is not. New methods are attempted with the accumulation of the useful knowledge and conjectures are made. Eventually, suitable solutions are attained, which can be justified. This process is remarkably analogous to that of mathematical modeling.
Closely linked to the socio-constructivist theory is the Problem-centered learning (PCL) approach, developed in South Africa in the mid 1980's by researchers at the University of Stellenbosch. The PCL approach is based on a socio-constructivist theory of the nature of knowledge and learning and hinges on the following (Olivier, Murray and Human, 1992 : 33):
• The learner is active in the process of acquiring knowledge.
• In acquiring this knowledge, the learner makes use of past experiences and existing knowledge.
• Learning is a social process and the learner acquires new knowledge through interaction with other learners and educators.
PCL does not simply focus on the acquisition of knowledge but also attempts to increase the learners' ability to understand and become good problem solvers.
Olivier, Murray and Human (1992 : 33) stated that ''young children enter school with a wide repertoire of informal mathematical problem-solving strategies that reflect and are based on their understanding of the problem situation and on their existing concepts".
They further stated that "instead of ignoring or even actively suppressing children's informal knowledge, and imposing formal arithmetic on children, instruction should recognize, encourage and build on the base of children's informal knowledge".
Research conducted by the Research Unit for Mathematics Education at the University of Stellenbosch (Olivier, Murray and Human, 1992 : 33) showed that the "majority of children invent powerful non-standard algorithms alongside school-taught standard algorithms; that they prefer to use their own algorithms when allowed to do so and that their success rate when using their own algorithms is significantly higher than the success rate of children who use the standard algorithms or when they themselves use standard algorithms". This research clearly contributes to the general constructivist theory of children creating their own knowledge from their own experiences and not from the experiences of the educator or textbook author.