1. Concept forming: in the first phase of a course they allow the students a natural and motivating access to mathematics.
2. Model forming: they supply a firm hold for learning the formal operations, procedures, notations, rules, and they do so together with other models, which have an important functions as supports for thinking.
3. Applicability: they uncover reality as source and domain of application.
4. Exercise of specific abilities in applied situations.
There are three different uses of context were distinguished:
- Third order context use: The most significant and characteristic for the conceptual mathematization process is the use of the context to introduce and develop a mathematical model or concept.
- Second order context use: Of course the context may play a less important role. The role of the context in mathematization is less essential, but still very important: a real world problem is presented to the student, and the student is expected to find the relevant mathematics, to organize and structure and solve the problem. In this kind of context, the real worl is essential and mathematics is the tool to organize reality.
- First order context use:We speak of first order context use if the mathematical operation are embedded in context. A simple transition from the problem to a mathematical problem is sufficient. These kind of problems are often found in traditional schoolbook. The context in this situation is only used to ‘camouflage’ the mathematical problem.
A quite trivial aspect is that the context should be motivating. However, students have a personal idea about what is motivating. So in general it is quite difficult to say what makes a motivating context. In 1981, after some three years observations, de Lange et al made the following two remarks concerning the motivating aspects of contexts:
- For younger students artificial contexts are acceptable and under certain cicumstances – motivating. For older students contexts must be more realistic to be acceptable.
- To make sure that the context is motivating to all, or to as many students as possible, one should offer a whole range of different contexts.
As we have indicated before, a realistic context does not always mean that there has to be a direct connection to the physical world. The Hewet team also accepts mathematical activities derived from topics that owe their existence to a story, cartoon or any creation of an ‘imagined’ reality. A ‘somewhat’ realistic context seems a minimal condition for students of this age. One may get away with using comic that supposedly is very popular in the Netherlands but even in this case students sometimes complain. But maybe surprisingly, in the intuitive explorative phase students are forgiving when the real context in not so real.
The real world of one child may differ considerably from the real world of another. Or, as Thomas called it in Symbolic Interactionism: “the definitions of the situation” may differ. Although Thomas never formulated precisely the meaning of this key term, the conception of definition of the situation provided a simple and powerful rationale for the significance of the subjective aspect in social life and thus provided symbolic interaction with the prime methodological rule which seems important in the philosophy behind mathematics: “if men define situations as real, they are real in their consequences”.
Our observation lead to the following recommendations, given the present state of teacher training and traditions in teaching:
- One should avoid, especially in the phase of conceptual mathematization contexts that are emotionally disturbing (defense, war, illness, ethic affairs).
- One should avoid artificial contexts.
- One should avoid too neutral contexts, which might fail to motivate unless the mathematical content is stimulating enough.
- One should not expect from the students too much background information; most, if not all, information should be contained in the text.
One should choose the context and edit the exercise in a away that stimulates interindividual action (in order to promote sociocognitive conflicts).