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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