Reviewed by Evangelista Lus W.P. (Impome 2011)
Prof. Koeno Gravemeijer is professor of mathematics and science education in University of Eindhoven, Netherland. Earlier he was affiliated with the Freudenthal Institute of Utrecht University. He has written many articles and has done many researches related to curriculum development, instructional design, domain-specific instruction theories includes the theory for realistic mathematics education/RME (http://www.isls.org/icls2010/keynote_2.html). One of his writing is “Realistic mathematics education theory as a guideline for problem-centered, interactive mathematics education” (2010) which can be read in book titled “A decade of PMRI in Indonesia” edited by R. Sembiring, K. Hoogland and M. Dolk. and it will be reviewed in this essay.
Based on fact that there has been changeover in mathematics education from thinking about instruction as the ‘transmission of knowledge’ towards ‘learning as the construction knowledge’, and as the result there has been a move towards problem centered, interactive mathematics, which requires a change in how instruction is conceptualized. Those give its own challenge for the teacher that is to guide the construction of the students in indirect manner which can be helped using RME theory. So that, in his article, Gravemeijer discusses three main points: problem-centered mathematics; fostering reinvention; and RME theory (includes the role of concrete material, context problem and the cultivation of mathematical interest).
1. Problem-centered mathematics
Gravemeijer explain that problem-centered, interactive, classroom has different didactical contract than traditional classroom which social norm culture in which teacher is always right. Students who are learning in classroom culture in which teacher is always right will not easily start to think on their own (Desforges & Cockburn in Gravemeijer, 2010) while in the problem-centered classroom, students have to think for themselves, not for the teacher expectation. They have to explain and justify their solution, try to understanding other students’ reasoning, ask question about the explanation in which they do not understand, and challenge arguments they do not agree with. And to make students particular with that norm, teacher should not just tell them verbally or make a rule of that, but teacher have to shape it through experiences like by offering challenging or intriguing open task, arranging them to work in group, give inspiring content and using concrete material.
2. Fostering reinvention
Motivating and involving students in learning process is not enough. Mathematics education also has to reach the conventional mathematics goal through helping the students to build on their own thinking while constructing more sophisticated mathematics while the teacher only can get involved in what students construct. Gravemeijer wrote that to reconcile the constructivist stance we can use a ‘hypothetical learning trajectory/HLT’ which is combination of instructional and mental activities, and their relation with the instructional goal (Simon, 1995 in Gravemeijer, 2010). HLT have to fit with long-term learning strands which is uneasy and complicated to design. Due of that, Gravemeijer argue that teachers should be offered or given support such as by a local instruction theory (a theory about possible learning process for a given topic) which is included in a domain-specific instruction theory such as RME theory.
3. RME theory
According to Gravemaijer, RME is a domain-specific theory which offers guidelines for instruction that aims at supporting students in constructing, or reinventing mathematics in problem-centered interactive instruction. Further, in RME, the teachers have to play an active role in facilitating productive class discussion and in selecting and framing mathematical issue as the discussion topic. The word ‘realistic’ has a meaning as a real in sense to be meaningful for students. This mean instructional starting point or problems that give to the students have to be experientially real for the students and make sense for them. For young students this may be connected to the daily life while for other students this may be involving number. So, student’ reality is different from teacher’.
a. Concrete material
Concrete material or manipulative is used as bridge for the students to jump the gap between their informal knowledge and the more formal abstract knowledge. The idea of using manipulative is to make abstract mathematical knowledge become concrete and tangible for students. However this does not mean that teacher can tell students what they are to ‘see’ when students do not see it. Based on Gravemeijer, manipulative is used in RME for scaffolding students own thinking. Model that be used are to represent the students’ own thinking and have to be designed so they can effectively support a transition towards more sophisticated mathematics.
b. Context problem
As explained before, RME theory take situation that is real and meaningful to children/students as a start up point. However we or teacher should still concern about the goal that should be achieved. There are two goals that we may discern, first is to offer a motive to the students and the second is to offer footholds for a solution strategy. Offer motive to the students means create a situation or problem in which it makes sense to the students and motivate student to solve it. While the second one means by using context problems can support students to construct their own strategy through their own thinking. In the other words, contextual problems may support students in construing informal solution that have potential to contribute in constructing more sophisticated mathematics solution.
c. Cultivation of mathematical interest
One thing that also becomes concern of RME is to teach students thinking mathematically. To reach that goal and to do reinvention we need combination of horizontal mathematization which concerns the transformation of contextual problem into mathematical problem and vertical mathematization when it applies in mathematical matter. Gravemeijer explained that if we want the students to enhance their mathematical skills and insights, they have to vertically mathematize their own mathematical activity. Teacher should initiate and help students to do vertical mathematization by giving question to enhance students thinking and appreciate or show genuine interest in what students come up with. Through that way, students will become more interest in thinking mathematically.
In conclusion, reforming mathematics education depends on the ability of the teacher to create a problem, to engage with students in interactive instruction, and depends on instructional design which can help students to construct their own thinking and thinking mathematically. To be able to do that, teacher can use RME theory as a guideline.