Streefland (1991) developed realistic mathematics lessons (based on fractions in elementary school) using the three levels construction principle:
(1) the local, or classroom level;
(2) the global, or course level; and
(3) the theoretical level.
(1) Classroom level
In this level, lessons are designed based on all the characteristics of RME and focus on the construction through horizontal mathematization (an example of the lesson is provided in the appendix). First, an open material is introduced into the learning situation and opportunity for carrying out free productions is provided. Then, characteristics of RME are applied to the lesson by:
(1) situating the intended material in reality, which serves as source and as area of application, starting from meaningful contexts having the potential to produce mathematical material; involves;
(2) intertwining with other strands; such as fractions and proportions; and
(3) producing tools in the form of symbols, diagrams and situation or context models during the learning process through collective effort.
(4) Finally, learning through constructions is carried out by arrangements of the students activities, so they can interact with each other, discuss, negotiate, and collaborate. And this is where the educational principle of interaction is applied. By this mean, the students contribution to their own learning path can be guaranteed. The students can be encouraged to follow this kind of constructional activity by giving them an assignment which leads to free productions.
(2) Course level
The material constructed at the classroom level is now used according to its mathematical and didactical essence in order to realize the general outline of the course. This means the measures taken to achieve contributions to the learning process at the local level must be continued at the general level.
(3) Theoretical level
All activities which took place in the both preceding levels such as design and development, didactical deliberation, and trying out in the classroom form the source of theoretical production, the generative material for this level. Constructing a theory in the form of a local theory for a specific area of learning. By using development research method, the local theory is revised and tested again in the other cyclic developments.
In order to design RME lessons, the components of a lesson plan will be identified and connected to realistic mathematics education. Those components are goals, content, methodology, and assessment.
(1) Goals
De Lange(1995) characterized three levels of goals in mathematics education: lower level, middle level, and higher order level. In the traditional program the goals were more or less clear. For example students should be able to solve a linear equation using a specific method. However, most of the goals of the traditional program are now classified as lower level goals that are based on formula skills, simple algorithms and definitions. In the realistic mathematics education goals are classified as 'middle' and 'higher' level goals. At the middle level, connections are made between the different tools of the lower level and concepts are integrated; it may not be clear in which strand we are operating, but simple problems have to be solved without unique strategies. This means that for both the teacher and the students the intended goals are not always immediately clear. Moreover, the new goals also emphasize the reasoning skills, communication and the development of critical attitude. These are popularly called 'high order' thinking skills. To conclude, in order to redesign a lesson based on the realistic approach it should contains these two types of goals.
(2) Materials
De Lange (1996) pointed out that materials are associated with real-life activities where domain specific, situational knowledge and strategies are used within the context of the situation. A variety of contextual problems is integrated in the curriculum right from the start. In a general way , RME developers need to find contextual problems that allow for a wide variety of solution procedures, preferably those which, considered together, already indicate a possible learning process through a process of progressive mathematization.
(3) Activities
The role of the RME teacher in the classroom are (de Lange, 1996;Gravenmeijer, 1994): a facilitator, an organizer, a guide, and an evaluator. Based on the process of progressive mathematization, generally one can conclude that the role of teacher on the steps of the teaching-learning process based on realistic approach are: Give the students a contextual problem that relate to the topic as the starting point. During interaction activity, give the students a clue, for instance, by drawing a table on the board, guide the students individually or in a small group in case they need help; Stimulate the students to compare their solutions in a class discussion. The discussion refers to the interpretation of the situation sketched in the contextual problem and also focus on the adequacy and the efficiency of various solution procedures.
Let the students find their own solution. It means the students is free to make discoveries at their own level, to build on their own experiential knowledge, and perform shortcuts at their own pace. Give another problem in the same context. On the other hand, the role of students in RME are mostly they work individually or in a group, they should be more self-reliant, they can not turn to the teacher for validation of their answers or for
directions for a standard solution procedure, and they are asked to produce free production or contribution.
(4) Assessment
In the development research on assessment based on the RME viewpoints carried out so far, has already produced some keys of how assessment can be improved, especially written assessment (Van den Heuvel-Panhuizen, 1996). In addition, doing assessment during the lesson, teachers may ask students to write an essay, to do experiment, collect data, and to design exercises that can be used in a test, or to design a test for other students in the classroom. Assessment can be continued by giving the students some problems as homework. But, in order to relate with the national standardized test, the assessment procedures should reflect the goals of the curriculum. Regarding to assessment in RME, De Lange(1995) formulated the following five principles of assessment as a guide in doing assessment:
- The primary purpose of testing is to improve learning and teaching. It means assessment should measure the students during the teaching-learning process in addition to end of unit or course.
- Methods of assessment should enable the students to demonstrate what they know rather that what they do not know. It can be conducted by having the problems that have multiple solution with multiple strategies.
- Assessment should operationalize all of the goals mathematics education, lower, middle, and higher order thinking level.
- The quality of mathematics assessment is not determined by its accessibility to objective scoring. In this case, objective test and mechanical test should be reduced by providing the students with the tests in which we really can see whether they are understand the problems.
- The assessment tools should be practical, available to the applications in school cultures, and accessibility to outside resources.
- situating the intended material in reality, starting from meaningful contexts having the potential to produce mathematical material; involves;
- intertwining lines of learning with other strands; and
- producing tools in the form of symbols, diagrams and situation or context models during the learning process through collective effort;
- in the activity part of the lesson plan, the students are arranged so they can interact with each others, discussion, negotiations, and collaborations. In this situation they have opportunity to work with or doing mathematics, communicate about mathematics; and
- assessment materials should be developed in the form of open question which leads the students to free productions. The assessment should be given to the students either during or after the instruction process, or as the homework.
Finally, based on this model, an example of mathematics lesson plan is designed .
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