According to van Hiele (de Lange, 1996) learning process is a process through three
levels:
levels:
- A student reaches the first level of thinking if she/he could manipulate familiar characteristic from recognized patterns.
- A student reaches the second level of thinking if she/he could manipulate the relation of those characteristic.
- A student reaches the third level of thinking if she/he could manipulate intrinsic characteristics of relations.
Traditional instruction has been modeled from the second to the third level. According to de Lange (1996) this never happen if we start from real world. Researchers in the Freudenthal Institute notice that van Hiele theory is important not because of its theoretical application but its practical implication. First, mathematics might be started from the level in which concepts being used have a high degree of familiarity for students, and second the goals of mathematics instruction is to create a relational framework (Gravemeijer, 1994).
So, mathematics learning process should be emphasized to concepts in which students familiar with. Each student has a set of knowledge that she/he has as a result of her/his interaction with environment or from previous learning process. After students are involve in meaningful learning process, they develop their knowledge to the higher level. In this process students are actively involve in gaining new knowledge. The construction of knowledge is a modification process that move slowly from first level to second level and third level. In this process students are responsible to their own learning activities.
The thinking framework of mathematical ideas go through three phases, namely exploration, identification, and application phases which stimulate new exploration phase that could be used to develop new mathematical concept. Those phases are cyclic processes without objective that was determined beforehand. There is no intention to come to certain goal. Learning progress highly depends on students’ ability to acquire new knowledge. Gravemeijer (1994) argues that there is no place for pre-program teaching and learning process, since all processes depend on students individual contribution and should be approved by teacher and students. Based on this argument Clarke, Clarke, and Sullivan (1996) contend that RME put its theory as process of inquiry as well as socio-constructivism of learning.
Cobb (1994) contends that RME theory is compatible to domain specific instructional theory which is rely on real world application and modeling. Alignment between constructivism and RME is mainly on similar characteristics on mathematics and mathematics learning. Both theories use premise that mathematics is human creative activity and mathematics learning took place when students are able to develop effective way of solving problems (de Lange, 1996; Steefland, 1991; Treffers, 1987). Using Cobb’s description, de Lange (1996) states RME tenets:
So, mathematics learning process should be emphasized to concepts in which students familiar with. Each student has a set of knowledge that she/he has as a result of her/his interaction with environment or from previous learning process. After students are involve in meaningful learning process, they develop their knowledge to the higher level. In this process students are actively involve in gaining new knowledge. The construction of knowledge is a modification process that move slowly from first level to second level and third level. In this process students are responsible to their own learning activities.
The thinking framework of mathematical ideas go through three phases, namely exploration, identification, and application phases which stimulate new exploration phase that could be used to develop new mathematical concept. Those phases are cyclic processes without objective that was determined beforehand. There is no intention to come to certain goal. Learning progress highly depends on students’ ability to acquire new knowledge. Gravemeijer (1994) argues that there is no place for pre-program teaching and learning process, since all processes depend on students individual contribution and should be approved by teacher and students. Based on this argument Clarke, Clarke, and Sullivan (1996) contend that RME put its theory as process of inquiry as well as socio-constructivism of learning.
Cobb (1994) contends that RME theory is compatible to domain specific instructional theory which is rely on real world application and modeling. Alignment between constructivism and RME is mainly on similar characteristics on mathematics and mathematics learning. Both theories use premise that mathematics is human creative activity and mathematics learning took place when students are able to develop effective way of solving problems (de Lange, 1996; Steefland, 1991; Treffers, 1987). Using Cobb’s description, de Lange (1996) states RME tenets:
- Use of contextual problems (contextual problems as applications and as starting points from which the intended mathematics can come out).
- Use of models or bridging by vertical instruments (broad attention is paid to development models, schemas and symbolization rather than being offered the rule or formal mathematics right away).
- Use of students' contribution (large contributions to the course are coming from student's own constructions, which lead them from their own informal to the more standard formal methods).
- Interactivity (explicit negotiation, intervention, discussion, cooperation and evaluation among pupils and teachers are essential elements in a constructive learning process in which the student's informal strategies are used as a lever to attain the formal ones).
- Intertwining of learning strands (the holistic approach implies that learning strands can not be dealt with as separate entities; instead an intertwining of learning strands is exploited in problem solving).
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