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HOW CONCRETE IS CONCRETE?


Koeno Gravemeijer
 
One of Mathematics’ characteristics is having an abstract object which might cause a lot of difficulties for students to access. Therefore, considering this issue, it is important to make the abstract mathematics to be concrete. To make something concrete in mathematics education, we are inclined introduce, ‘manipulative’, in the form of tactile objects or visual representations, to help students make connections between what we know and what they know.
Making things concrete, may be elaborated as either making concrete what we know (material concrete) or hooking up with the students already know (common sense concrete). In this manner, we can start with what is common sense for the students. From this point onwards, we may try to follow Freudenthal’s guideline that ‘mathematics should start and stay within common sense’, by trying to foster the growth of what is common sense for the students.
The large difference between the knowledge of the teachers and the experiential knowledge of the student cause a mismatch. Many things may seem logical for teacher or adults that are so self-evident for the students. And the only way to bridge the gap between teacher’s mathematics world and students’ everyday life world is by trying to connect to what the students know, and helping the students to construct mathematics in a bottom-up manner.
In relation to this case, it can be very valuable to try to imagine the students’ point of view, and try to concern the way the students see the situation. Since when we ignore the students’ point of view, we would run the risk of disconnecting mathematics the students learn from their common sense. As a result, they may start to treat mathematics school and everyday-life reality as two disjunct worlds.
The author illustrate this with an interview a first-grader, named Auburn, conducted by Cobb (1989). Auburn solved the first task, addition, ’16 + 9’, by counting on, and she arrived at the answer, ’16 + 9 = 25’. Later, when she filled out a worksheet that contains the same task in the following manner;
                  
                                           
Surprisingly, Auburn has two different answer on one question. In interview section, she said that if we had 16 cookies and another 9 added,then if we count them altogether we would get 25. She performed it well by using common sense concrete.
Noticing this case, for Auburn, doing mathematics in the worksheet seems belong to a different world which disconnected from the world of everyday-life experience. As we notice, Auburn will not be inclined to use everyday-life knowledge to make sense of ‘school math’ problems.
Such manipulative -tactile and visual models- approach will not be used to make the students ‘see’ the abstract mathematics, instead, material and visual representations as manipulative may be used by the students as means of scaffolding and communicating their own ideas.In this article, author takes the so called arithmetic rack as an example (Treffers, 1990). The structure of the colored beads on the rack can support the students’ arithmetical reasoning. When adding 7 and 8, for instance. On their prerequisite knowledge students may already know that 5 + 5 = 10, 7 = 5 + 2 and 8 = 5 + 3, and visualize that on the arithmetic rack. 
                       
As a next step is asking the students how they might use the rack and asking them to invent ways of symbolizing to describe their reasoning.
Through prerequisite knowledge, students may come up with the idea;
                                
Over time, the students may become so proficient that they will not need visual scaffolding anymore.
Though manipulatives (tactile and visual models) have some risks, it can support learning processes starting with situations that are concrete in the sense o familiar to the students. It can be used as means of scaffolding and communicating students’ ideas.

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