Treffers (1987, 1991) differentiates two types of mathematization, namely vertical and horizontal, which is depicted by Gravemeijer (1994) as reinvention process (Figure 2).
Figure 1 Horizontal and Vertical Mathematization (Gravemeijer, 1994)
In horizontal mathematization, students start from contextual problems. They try to describe the problems using their own language and symbols, and solve the problems. In this process each student could use his/her own strategies that might different from others. In vertical mathematisation we may start from contextual problems, but in the long run students develop certain procedure which can be used to solve similar problem directly without utilize context. Gravemeijer (1994) calls this as mathematisation of mathematical problems to distinguish with horizontal mathematisation which is mathematisation of contextual problems. RME might be distinguished from other theories of mathematics instruction such as mechanistic, empiristic and structuralistic, based on the existence of horizontal and vertical mathematisation components (Treffers, 1991).
In horizontal mathematization, students start from contextual problems. They try to describe the problems using their own language and symbols, and solve the problems. In this process each student could use his/her own strategies that might different from others. In vertical mathematisation we may start from contextual problems, but in the long run students develop certain procedure which can be used to solve similar problem directly without utilize context. Gravemeijer (1994) calls this as mathematisation of mathematical problems to distinguish with horizontal mathematisation which is mathematisation of contextual problems. RME might be distinguished from other theories of mathematics instruction such as mechanistic, empiristic and structuralistic, based on the existence of horizontal and vertical mathematisation components (Treffers, 1991).
Figure 2 Mathematisation and direction (Treffers, 1991)
The two mathematisation components do not exist in mechanistic mathematics education. Mechanistic approach is algorithmic in nature. It tends to use method of telling and drill to exercise mathematical formulas and procedures. In empiristic theory of mathematics education horizontal mathematisation is clearly realized in informal procedure as a basis for learning. However, without support of models, schemas and the like, it is difficult to arrive at formal level. In structuralistic education, operation, mathematical patterns, and the like are concretized by using tools and media of instruction as representation of mathematical ideas and concepts. Vertical mathematisation is developed by using those structured materials. However, application of mathematics could no be reached, except students have understood how to use learned procedures. As consequence students would not be able to develop further their natural and informal procedures.
The two mathematisation components do not exist in mechanistic mathematics education. Mechanistic approach is algorithmic in nature. It tends to use method of telling and drill to exercise mathematical formulas and procedures. In empiristic theory of mathematics education horizontal mathematisation is clearly realized in informal procedure as a basis for learning. However, without support of models, schemas and the like, it is difficult to arrive at formal level. In structuralistic education, operation, mathematical patterns, and the like are concretized by using tools and media of instruction as representation of mathematical ideas and concepts. Vertical mathematisation is developed by using those structured materials. However, application of mathematics could no be reached, except students have understood how to use learned procedures. As consequence students would not be able to develop further their natural and informal procedures.
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