Horizontal and Vertical Mathematization

Treffers (1987) describes progressive mathematization in terms of two different types of mathematical activity--horizontal mathematizing and vertical mathematizing. Horizontal mathematizing is described as "transforming a problem field into a mathematical problem" (p.247). This definition of horizontal mathematizing suggests that what constitutes a problem field is non-mathematical (i.e., some context related to a real-world situation). This is consistent with the way in which the NSF Working Group on Assessment in Calculus interprets mathematizing (see Schoenfeld, 1997, p. 31). 

We interpret horizontal mathematization more broadly to include problem situations that are already a mathematical problem. In our view, what constitutes a problem field or problem situation depends on the background and experience of the individual. Thus, what constitutes a problem situation for a learner in a real analysis course is potentially different from that of a learner in a high school algebra course and does not necessarily involve real-world applications.

In broadening what is meant by a problem situation to include mathematical contexts, we take horizontal mathematizing to mean formulating a problem situation in such a way that it is amenable to further analysis. This might include, but is not limited to, activities such as experimenting, classifying, conjecturing, and structuring. The activities that are grounded in and build on these activities, which might include activities such as reasoning about abstract structures, generalizing, and formalizing, are thought of as vertical mathematizing. Students' new resulting mathematical reality may then be the context for further horizontal mathematizing. Thus, the term "progressive mathematizing" has both short-term and long-term connotations. In the short-term, progressive mathematizing refers to a shift or movement from horizontal activities to vertical activities. This shift is not necessarily linear, as vertical activities often "fold back" (Pirie & Kieren, 1994) to horizontal activities. Over the long-term, progressive mathematizing refers to the fact that students' mathematical realities resulting from previous mathematization can be the context for additional horizontal mathematizing. In the remainder of this paper, we use the short-term connotation of the phrase progressive mathematizing. The separation of mathematical activity into horizontal and vertical aspects is somewhat artificial, as in reality the two activities are closely related. However, for the purposes of sorting out what might be considered advanced mathematical activity, we find this distinction useful. 

In  particular, we view vertical mathematization as one way to characterize advanced mathematical activity. Note that advanced mathematical activity is framed in relation to students' previous activity. Thus, in the examples that follow, we do not limit our discussion to vertical mathematizing, but rather elaborate both horizontal and vertical aspects to mathematizing. For it is when the two aspects are set in relief against each other that advanced mathematical activity is illuminated. In our work in undergraduate mathematics education, we have emphasized mathematizing in the sense described above and therefore we draw on examples from these research efforts to illustrate and clarify aspects of advanced mathematical activity. In the next section, we bring to the fore three different aspects of horizontal and vertical mathematizing within the activities of symbolizing, algorithmitizing, and defining. These examples of progressive mathematization are only a start, and are in no way meant to capture all aspects of mathematical activity.

Vertical mathematizing as an example of advanced mathematical activity whereas horizontal mathematizing describes building blocks necessary for advanced mathematical activity that are not themselves “advanced.” We find that the distinction about what constitutes advanced mathematical thinking is not necessarily reflected in grade or content level, but rather in the nature of the mathematical activities in which students participate. Our development of the notions of horizontal and vertical mathematizing reflects our interests in understanding and characterizing the nature of these activities and helps illuminate aspects of advanced mathematical thinking.