Conception of Learner and Learning

According to van Hiele (cf. de Lange, 1996) the process of learning proceeds through  three levels:

• A student reaches the first level of thinking as soon as he or she can manipulate the known characteristics of a pattern that are familiar to him or her.
• As soon as he or she learns to manipulate the interrelatedness of the characteristics he or she will have reached the second level.
• He or she will reach the third level of thinking when he or she starts manipulating the intrinsic characteristics of relations. Traditional instruction is inclined to start at the second or third level. According to de

Lange (1996) this should not be the case if we start in the real world. The researchers at the Freudenthal Institute notice that the significance of the level theory of van Hiele does not exist in its theoretical use, but it does in its practical implications. Firstly, mathematics has to start on a level at which the concepts used have a high degree of familiarity for the students, and secondly its aim has to be the creation of a relational framework (Gravemeijer, 1994). Although RME and constructivism are not the same, to some extent they have some compatible characteristics, one of which is the similarity of the conceptions of learning and learners in both theories. As are the case to constructivism, the following conceptions are relevant to RME (Anderson et al., 1994; Louck-Horsley, et al., 1998; van den Berg, 1996):

• Each learner brings his or her preconceptions to the educational experience.These preconceptions are highly influential on subsequent learning. Learnerspossess a diverse set of alternative conceptions about mathematical ideas thatinfluence their future learning;
• Each learner actively constructs meaning. Learners acquire new knowledge by constructing it for themselves;
• Each learner is ready to share his or her personal meaning with others, and based on this negotiation process, reconceptualizes the initial knowledge structures. The construction of knowledge is a process of change that includes addition, creation, modification, refinement, restructuring, and rejection;
• Each learner takes responsibility for his or her learning. The new knowledge learners construct for themselves has its origin in a diverse set of experiences;
• Each learner is convinced that success in learning with understanding is possible. In other words, all students regardless of race, culture, and gender are capable of understanding and doing mathematics.

The conception of learning in RME is in line with the conception of learners. The starting point in the learning process of the realistic approach is emphasized on the conception that the students are familiar with. Each learner has a preconception or a set of alternative conceptions about mathematical ideas. After a student is involved meaningfully in a learning process, the student develops the conceptions to a higher level. In this step, the student actively acquires new knowledge. The construction of knowledge is a process of change that proceeds slowly from the first to second and then to the third. In this process the student is responsible for his own learning.

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