Models as representations of problem situations

Within RME, models are seen as representations of problem situations, which necessarily reflect essential aspects of mathematical concepts and structures that are relevant for the problem situation, but that can have

different manifestations. This means that the term ‘model’ is not taken in a very literal way. Materials, visual sketches, paradigmatic situations, schemes, diagrams and even symbols can serve as models (see Treffers and Goffree, 1985; Treffers 1987, 1991; Gravemeijer 1994a). For instance, an example of a paradigmatic situation that can function as a model, is repeated subtraction. Within the learning strand on long division, this procedure – elicited, for instance, by the transit of a large number of supporters by coach (see Gravemeijer 1982; Treffers, 1991) – both legitimizes and gives access to the formal long division algorithm. As an example of a way of notation the arrow language can be mentioned. The initial way of describing the changes in the number of passengers on a bus ends up being used to describe all kind of numerical changes later on (see Van den Brink, 1984).

For being suitable to give the intended support to learning processes, models must have at least two important characteristics. On the one hand they have to be rooted in realistic, imaginable contexts and on the other hand they have to be sufficiently flexible to be applied also on a more advanced, or more general level. This implies that a model should support progression in vertical mathematizing without blocking the way back to the sources from which a strategy originates – which is similar to the Vygotskian notion of scaffolding (Vygotsky, 1978). In other words, the students should always be able to revert to a lower level. It is this two-way character of models that makes them so powerful. Another requirement for models to be viable is that they – in alignment with the RME view of students as active participants of the teaching-learning process – can be re-invented by the students on their own. To realize this, the models should ‘behave’ in a natural, self-evident way. They should fit with the students’ informal strategies – as if they could have been invented by them – and should be easily adapted to new situations.

Coming to the point of why models can contribute to level raising, the work of Streefland comes into the picture. About fifteen years ago, Streefland (1985a) elucidated in a Dutch article how models can fulfill the bridging function between the informal and the formal level: by shifting from a ‘model of’ to a ‘model for’. In brief, this means that in the beginning of a particular learning process a model is constituted in very close connection to the problem situation at hand, and that later on the context specific model is generalized over situations and becomes then a model that can be used to organize related and new problem situations and to

reason mathematically. In that second stage, the strategies that are applied to solve a problem are no longer related to that specific situation, but reflect a more general point of view. In the mental shift from ‘after-image’ to ‘preimage’ the awareness of the problem situation and the increase in level of understanding become manifest.4 The change of perspective involves both insight into the broader applicability of the constructed model, and reflection on what was done before (Streefland, 1985a; see also 1992, 1993, 1996). Especially in the areas of fractions, ratio and percentage Streefland enriched the didactics of mathematics education with models that have this shifting quality.

A first example is connected to his design research on fractions within the context of a pizza restaurant (Streefland, 1988, 1991). In the trajectory he designed, the learning process starts with the ‘concrete’ model of the ‘seating arrangements’ to compare amounts of pizza, which model is evoked by the designed tasks that are presented to the students, and later schematized to the ‘seating arrangement tree’ and the ratio table by means of which formal fractions are compared and operations with fractions are carried out. In this process of schematization and generalization, again the roles of the designer and the teacher are very important. By designing a trajectory in which new problems prompt the students to arrive at adaptations of the initial ‘concrete’ model and by accentuating particular adaptations that the students come up with the process of model development is guided. The bar model that will be discussed later in this article is a second

example. In the development of teaching a unit on percentage in which this bar model is the backbone for progress, Leen Streefland and I worked very closely together.

Although we owe the concept of the shifts in models to Streefland, he did not do his work in isolation. Again, the role Freudenthal played should not be underestimated. The distinction between the two meanings of ‘model’ was already an issue in his writings in the 1970s, when he wrote: “Models of something are after-images of a piece of given reality; models for something are pre-images for a piece of to be created reality” (Freudenthal, 1975, p. 66). In connection with these two functions of models he distinguished also ‘descriptive models’ and ‘normative models’ (Freudenthal, 1978). However, the difference with Streefland is

that Freudenthal was thinking about models at a much more general didactical level – such as models for lessons, curriculum plans, goal descriptions, innovation strategies, interaction methods, and evaluation procedures – and not on the micro-didactical level that Streefland had in mind. By applying Freudenthal’s thinking within a micro-didactic context he revealed the level raising mechanisms of models and the didactical use of this power. His idea of ‘model of’ and ‘model for’ undoubtedly turned out to be an eye-opener for many (see e.g., Treffers, 1991; Gravemeijer, 1994a, 1994b, 1997, 1999; Van den Heuvel-Panhuizen, 1995, 2001; Gravemeijer and Doorman, 1999; Yackel et al., 2001, Van Amerom, 2002). It is a simple, immediately recognizable and applicable idea, in which the essence of learning processes, namely raising the level of knowledge, is given a didactical entrance. For this reason it has been followed up in thinking about the didactics of mathematics education both within and without the RME community.

In particular, Gravemeijer (1994a, 1994b, 1997, 1999) worked out this idea. He showed that the shift in models can also be connected to the process of mathematical growth in a more general way. The distinction between ‘model of’ and ‘model for’ led him to split up the intermediate level, between the situational level and the formal level of solving problems and mathematical understanding, into a referential and a general level. In addition to this, Gravemeijer emphasized the connection between the use of models and the re-invention principle of RME. Because of the shift in model – that causes the formal level of mathematics to become linked to informal strategies – the top-down element that characterized the use of models within the structuralist and cognitive approaches to mathematics education could be converted to a bottom-up process.

Although the bottom-up process implies that the models are invented by the students themselves, the students should be provided with a learning environment – the whole of problems, activities, and contexts, placed within scenarios or trajectories, together with the stimulating and accentuating role of the teacher – to make this happen. As said earlier, within  RME, re-invention is taken to be guided re-invention. However, an essential facet of this process is that the students should have the feeling of having the lead in it. The emergence of models and their further evolution must occur in a natural manner.

The previous requirement puts a large onus on the development of educational materials. Education developers have to look for problem situations that are suitable for model building and fit within a scenario or

trajectory that elicits the further evolution of the model, to let it grow into a didactic model that opens up the path to higher levels of understanding for the students. It should be clear that this puts certain demands on such a problem situation. A key requirement is that the problem situation can be easily schematized. Another demand is that, from the point of view of the students, there should be a necessity for model building. This aspect requires that the problem has to include model-eliciting activities, such as for instance, planning and executing solutions steps, generating explanations, identifying similarities and differences, and making predictions. Although these criteria already give a good indication of what is necessary to have a model emerge, the most important is that the problem situations and activities bring the students to identify mathematical structures and concepts.

To discover which problems and activities can do this, ‘phenomenological didactical analyses’, as Freudenthal (1978, 1983) called them, are needed. These analyses are focused on how mathematical knowledge and concepts can manifest themselves to students and how they can be constituted. Part of this analysis is done by means of thought experiments and inter colleague deliberation – including discussions with teachers – in which both knowledge about students and ideas about the desired mathematical concepts function as a guiding pre-image. The more important part of the analysis, however, is done while working with students and analyzing students’ work. In this way what is important for constituting the model and hence what has to be ‘put’ in the problem situation can be found, so that situation-specific solutions can be elicited, which can be schematized, and which will have vertical perspective.