Problem Solving, constructivism and realistic mathematics education
3.2 Problem solving in a real world context
doing mathematics would be to solve real world problems. This is what Lester (1980 : 29) implied when he stated that "there is support for the notion that the ultimate aim of learning mathematics is to be able to solve problems". Although he does not specifically talk about solving problems in the real world, it is clear that the entire world is pervaded by all kinds of mathematics (financial and economic sectors, industry, computers and so on) and therefore one of the aims of doing mathematics is to solve problems in the real world.
When businessmen use graphs, spreadsheets, computer programs and so on to attempt to maximize profit and minimize cost, they use mathematical modeling tools to solve their everyday problems. So problem solving is part of daily life.
The Sixth Standard as determined by the Standards 2000 Writing Group (NCTM, 1998 : 76) states that "mathematics instructional programs should focus on solving problems as part of understanding mathematics so that all students -
• Build new mathematical knowledge through their work with problems;
• Develop a disposition to formulate, represent, abstract, generalize in situations within and outside mathematics;
• Apply a wide variety of strategies to solve problems and adapt the strategies to new situations;
• Monitor and reflect on their mathematical thinking in solving problems."
The second bullet represents the idea of also working in real world situations. It conveys the idea that "people who have developed a mathematical point of view of the world tend to act in ways that are mathematically productive" (NCTM, 1998 : 77) and this may be extended to the notion that mathematically productive people may be highly productive people in the work environment.
There is a similar strain of thought in Nunokawa's (1995 : 722) belief that problem solving and mathematical modeling are connected when he stated that "mathematical problem solving is the activity to connect the real world and the mathematical world'. Branca (1980 : 3) also cited ''problem solving as a goal (if not the goal) of mathematics learning" and he went on to say that problem solving involves "applying mathematics to the real world'.
Schoenfeld (1980 : 15) similarly stressed the importance of problem solving when he stated that "the problem solving process is one of the most important aspects of mathematics with which teachers should be concerned'.
This idea of solving problems isn't specific to mathematics. According to Polya (1980 : 1), problem solving is "human nature itself'. The Standards 2000 Writing Group (NCTM, 1998 : 134) also stated that ''problem solving is natural to small children". People have their days filled with obstacles that are not immediately attainable, and will therefore require skills to overcome these obstacles. In this process, human beings are problem solvers. As a problem solver one has to overcome obstacles by using information and techniques that one has already experienced or learned. If that is not possible, one then creates a new method. This is in fact similar to the process of modeling. The processes of mathematical modeling and problem solving have much in common. The main difference, according to Treilibs (1979) lies in "the objectives of each process: the first seeks to solve problems in order to make sense of a situation, the second sees problem solving as an end in itself'.
As a result of the widespread concern for learners' inability to solve problems, school mathematics curricula have been adapted to include problem-solving strategies. There has
been a perception that problem solving is difficult at school level and therefore it has been traditionally avoided. Amit, Kelly and Lesh (1994: 161) reported on research conducted on problem-solving. They stated that "based on research with hundreds of students in the preceding kinds of "real life" problem solving situations, the following results have
emerged consistently: (1) Even students whose prior experiences in school suggested that they are far "below average" in mathematical ability, their performance on such activities routinely shows that they are able to invent (or significantly extend or refine) mathematical models which provide the foundations for the small number of "big ideas" that lie at the heart of mathematics courses in which they enrolled, and (2) the mathematical models that they construct are often more complex and sophisticated than those that previous teaching and testing experiences suggested they were unable to be taught ... "
Bell (1979 : 49), identified four processes when solving problems as illustrated in Figure 18. The first is the assimilation phase. In this phase the actual goal is understood and the data available is collated. The second is the phase where ideas are looked at, taking into consideration the data available and the goal to be attained. Tpis is called the exploration phase. In the third phase, the point of illumination phase, a connection is made with the data and the goal. In other words the chain that connects the data with the goal is understood. In the final phase, the actual link between the data and the goal is verified by testing the final solution. This is the verification phase. If at this point the [mal solution is shown to be incorrect for certain aspects of the data then phase two will begin all over again.
It is in the assimilation stage that the problem solver understands the problem, taking into
information whilst irrelevant data will be discarded. It is also at this stage that the actual plan for the resolution of the problem takes place. Previous knowledge is recalled and a cursory relationship is established between what is known and what is expected.
Problem
verification
Solution
Assimilation
(understanding of various aspects of problem)
Model Exploration and illumination
Figure 18
As a result of the assimilation stage, the problem can be transformed into a mathematical form. Solving related problems, which are simpler, may do this. The idea here would be to ascertain patterns that would aid in solving the larger more complex problem. This then is mathematical modeling.
The model established (which may not necessarily be a completely new one) is evaluated.
If the verification process reveals that the solution was not suitable then the model is re- evaluated and reassessed (a different model may be used).