Mathematics as mathematizing

One of the basic concepts of RME is Freudenthal’s (1971) idea of mathematics as a human activity. As has been said before, for him mathematics was not the body of mathematical knowledge, but the activity of solving problems and looking for problems, and, more generally, the activity of organizing matter from reality or mathematical matter – which he called ‘mathematization’ (Freudenthal, 1968). In very clear terms he clarified what mathematics is about: “There is no mathematics without mathematizing” (Freudenthal, 1973, p. 134).

This activity-based interpretation of mathematics had also important consequences for how mathematics education was conceptualized. More precisely, it affected both the goals of mathematics education and the teaching methods. According to Freudenthal, mathematics can best be learned by doing (ibid., 1968, 1971, 1973) and mathematizing is the core goal of mathematics education:

What humans have to learn is not mathematics as a closed system, but rather as an activity, the process of mathematizing reality and if possible even that of mathematizing mathematics. (Freudenthal, 1968, p. 7)

Although Freudenthal in his early writings unmistakably referred to two kinds of mathematizing, and he made it clear that he did not want to limit mathematizing to an activity on the bottom level, where it is applied to organize unmathematical matter in a mathematical way, his primary focus was on mathematizing reality in the common sense meaning of the world out there. He was against cutting off mathematics from real-world situations and teaching ready-made axiomatics (Freudenthal, 1973).

It was Treffers (1978, 1987) who placed the two ways of mathematizing in a new perspective, which caused Freudenthal to change his mind as well. Treffers formulated the idea of two ways of mathematizing in an educational context. He distinguished ‘horizontal’ and ‘vertical’ mathematizing. Generally speaking the meaning of these two forms of mathematizing is the following. In the case of horizontal mathematizing, mathematical tools are brought forward and used to organize and solve a problem situated in daily life. Vertical mathematizing, on the contrary, stands for all kinds of re-organizations and operations done by the students within the mathematical system itself. In his last book Freudenthal (1991) adopted Treffers’ distinction of these two ways of mathematizing, and expressed their meanings as follows: to mathematize horizontally means to go from the world of life to the world of symbols; and to mathematize vertically means to move within the world of symbols. The latter implies, for instance, making shortcuts and discovering connections between concepts and strategies and making use of these findings. Freudenthal emphasized, however, that the differences between these two worlds are far from clear cut, and that, in his view, the worlds are not, in fact, separate. Moreover, he found the two forms of mathematizing to be of equal value, and stressed the fact that both activities could take place on all levels of mathematical activity. In other words, even on the level of counting activities, for example, both forms may occur.

Although Freudenthal introduced some important nuances in the formulation of the two ways of mathematizing, these do not affect the core of Treffer’s classification or its significance. Furthermore, it was Treffers’ merit that he made it clear that RME clearly differentiates itself, through this focus on two ways of mathematizing, from other (then prevailing) approaches to mathematics education. According to Treffers (1978, 1987, 1991) an empiricist approach only focuses on horizontal mathematizing, while a structuralist approach confines oneself to vertical mathematizing, and in a mechanistic approach both forms are missing. As Treffers and Goffree (1985) stressed, the kind of mathematizing on which one is focused in mathematics education has important consequences for the role of models in the different approaches to mathematics education, and also for the kind of models that are used.

Another characteristic of RME that is closely related to mathematizing is what could be called the ‘level principle’ of RME. Students pass through different levels of understanding on which mathematizing can take place: from devising informal context-connected solutions to reaching some level of schematization, and finally having insight into the general principles behind a problem and being able to see the overall picture. Essential for this level theory of learning – which Freudenthal derived from the observations and ideas of the Van Hieles (see, for instance, Freudenthal 1973, 1991) – is that the activity of mathematizing on a lower level can be the subject of inquiry on a higher level. This means that the organizing activities that have been carried out initially in an informal way, later, as a result of reflection, become more formal.

This level theory of learning is also reflected in ‘progressive mathematization’ that is considered as the most general characteristic of RME and where models – interpreted broadly – are seen as vehicles to elicit and support this progress (Treffers and Goffree, 1985; Treffers, 1987; Gravemeijer, 1994a; Van den Heuvel-Panhuizen, 1995, 2002). Models are attributed the role of bridging the gap between the informal understanding connected to  the ‘real’ and imagined reality on the one side, and the understanding of formal systems on the other.