This paper is the reflection of the article of Koeno Gravemeijer that titles How Concrete is Concrete? Koeno Gravemeijer is a realistic mathematics education expert from Freudenthal Institute, Utrecht University, Nederland.
Mathematics is abstract, so most of the students think that mathematics is not easy. For making the learning of mathematics easier, the teachers introduce tactile or visual models. They think it is able to make the abstract mathematics concrete. However, Gravemeijer will show that our common way of making things concrete for the students does not work because we help students to make a connection with what we know, not what they know.
The first example that will be shown by Gravemeijer is about how the student of grade 6 solved the comparing fraction problem. The context is about a bakery that would cut banquet bars (a sort of large cookies) to order. They were given paper strips of a given length to enact the cutting process. Then they were asked to cut the banquet bars into eight equal pieces, or six, or ten, and so forth. After they asked to use similar strips to compare 1/3 and 2/6, they solved this problem by comparing the lengths of pieces produced in the two different divisions-either dividing by three or dividing by six. However, students concluded that 1/3 was not equal to 2/6.
Gravemeijer analized the answer of the students then he found that the way students cut the strips was not very precise. It made the answer of the student is like that. It means that the tactile representations do not support an insightful solution. Furthermore, in this case, we may denote the two ways in which ‘concrete’ can be understood as either “material concrete” or as “common sense concrete”. The students’ point of view is sometimes different with teacher’s point of view. In this case, the teacher see the relation between 1/3 and 2/6 in the paper cutting, but the students just see the blocks with various sizes.
So, how is the better one? We prefer to help students to reason how 1/6 relates to 1/3. First, a bar is divided into three parts and each piece into halves next, to get six equal pieces. Then, they will understand that 1/6 is a half of 1/3 and they able to concluse that 2/6 equal to 1/3.
The second example is about addition tasks for the student of grade 1. She is Auburn. When Auburn counting on 16 + 9, she got 25. She also know that if she had 16 cookies and another 9 added, she would have 25 cookies. But, when she filled the worksheet, she wrote 15. When she was interviewed why the answers are different, she said that she didn’t know. Lets look at a part of that worksheet below.
If we analize the worksheet, we able to know why Auburn’s answer is 15. She just imitated the example from the worksheet. She thought that the mathematics of the worksheets seem different with the world of everyday-life experience. She will not be inclined to use everyday-life knowledge to make sense of ‘school math’ problems. Gravemeijer argue that the knowledge gap between teachers and students is too big to make this work. And manipulatives cannot bridge this gap.
So, what is the better? We could use arithmetic rack to fasilitate the students for communicating and scaffolding.
In conclution, we may try to follow Freudenthal’s adagio that ‘mathematics should start and stay within common sense’ by trying to foster the growth of what is common sense for the students.