Model is a tool used for mathematician to reveal their ideas and thinking, and also to help them as tools to solve questions or puzzle. In Realistic Mathematics Education (RME), we know well about the using of context and model. There are two different used of model in RME, model of situation and model for students’ thinking. In model of situation, students try to describe what they find in the context given, just exactly similar with the problem found. Usually, students tend to draw the situation in their paper. In the other hand, model for the picture are becoming more abstract. It is followed by their understanding more toward the problem. Students make a drawing not exactly the real situation but more close to the abstract calculation. I would like to give you an example; Corry just celebrated her 8th birthday. She plans to give 5 heart stickers to her best friends, for which each 5 stickers will be put on an envelope. Now, she is struggling to know, how many heart stickers that she should buy if she had 20 heart stickers. For more clear information, I would like to invite you to this following picture:
In the model of, students still wonder and draw exactly how many candy that Corry has and the unitize it into five. In the Model for, they need not to draw each of hearts stickers in their paper since they already understand that each envelope has 5 heart stickers. In this phase, students understand the concept of number behind the envelopes picture represented about, thus they need not to draw a truly picture of hearts. Constructing their thinking is not an easy effort, since every student is different. Different contexts can also develop different models that is related to the strategies used. Sometimes we find that, the model are used by the students is not the model that we intend them to reveal. Now, how if the students even cannot reveal their ideas and thinking into a model? What should we do?
In the 6th meeting of RME course, we were posed by Rindu’s question (one of my colleagues who also learn RME in Utrecht University). As the assignment, we have to read an article and then answer the question in the tittle based on Koeno Gravemiejer thinking and perspective. The following paragraph was my question.
After reading the text, what do you think Gravemeijer will answer to the question “What should teacher do if their students cannot reveal their ideas in to a model.
In RME, a teacher can influence students’ inventing activity in indirect manner (Gravemeijer, 2004). Those we have to be able to plan instructional activities based on their point of view, that is considering their thinking through what strategy that might be used and what kind of big idea that might be born. As anticipating of these variant ideas, thus teacher has to think every possible conjecture. Sometimes, a learning process cannot reveal the goal, such as they cannot reveal their idea in the model intended to their work. This problem can be investigated from two different of view; student point of view, and the design itself. As Gravemeijer advices in article, we have to think and see from student point of view, in which is perhaps there are some aspects that are being ignored while designing the instructional activities. Thus, In hypothetical learning trajectory a teacher is required to revise the learning trajectory based on her/his finding in learning process. This revising process is known as mathematical teaching cycle (Simon, 1995). We can revise our design and let the students experience the new design until they can follow the learning trajectory designed. If they still cannot show their idea and strategy in a model or even cannot reach the learning goal, we can re-revise it. Furthermore, since teaching is a learning process, we have to serve them scaffolding in order to help them reveal the model. Support, support, and keep support them. Posing some question to stimulate their thinking could be a brilliant way. Remember! Never give a direct answer to them, you will never help them then, just give the clues and let they develop their thinking and their own ideas.
I would like to say that I am kind of lucky person that have seen and observed some learning process videos in RME recorded in researches conducted by De Hertog, Dolk, Fosnot, Van Galen, et al. the masters of RME. Having a block session of lectures with Dolk and Abbels is kind of amazing thing for me, the budak kampong. I hope you enjoy reading it. 🙂
Gravemeijer, K. (2004). Creating opportunities for students to reinvent mathematics. ICMI 2004 (pp. 1-17). State College, PA, USA: ICMI.
Dolk, M., & Fostnot, C. (2001). Developing Mathematical Models. In Young Mathematician At Work (pp. 73-91).
Fosnot, C., & Dolk, M. (2001). Developing Mathematical Model. In Young Mathematician at Work; Constructing Multiplication and Division (pp. 73-89).