Geometry
is an important parts of mathematics curriculum (Mistretta, 2000). In Indonesia,
Geometry for the secondary school is taught since the second semester of junior
high school. The first standard competency of geometry topic is understands the
concept of triangular and quadrilateral then determine the measurement. One of
the basic competences that should be accomplished for that standard competency
is identifying the characteristic of persegi panjang, persegi, trapesium, jajar
genjang, belah ketupat dan layang-layang. Those are the terminology that is
used in Indonesia to differentiate the shape which is included to the
quadrilateral (segi empat).
Besides,
in other countries geometry is also taught in the secondary school. For example
in Singapore, we know from their O Level Secondary Mathematics Syllabus that
they have the subtopic of Angles, Triangles and Polygon in the topic of
geometry and measurement. One of the contents is about classifying special
quadrilaterals on the basis of their properties. I believe that the purpose of
both of countries to teach the students those of contents are more or less
same. However, I think there is a little bit difference between those contents
that are taught. It is shown by the way each country state the content of the topic.
Singapore curriculum states that they will learn about classifying the special
quadrilaterals based on the properties. It shows that the classification will
be taking place in the learning process to determine some kinds of special
quadrilaterals. It is quite different from the process that will be done in the
learning process in Indonesia school.
Indonesian
curriculum states it directly what are shapes that will be learned to be
indentified based on the topic of quadrilateral (segi empat). They are persegi
panjang, persegi, trapesium, jajar genjang, belah ketupat and layang-layang. It
shows that students do not classify some kind of special quadrilateral because the
classification of the shape is already decided from the beginning through the given
names of the shape. Hence, the students are only identifying the characteristic
of each shape without identifying such a special properties of quadrilaterals.
For example, students only learn about what the characteristic of persegi
(square) are or what the characteristic of the persegi panjang (rectangle) are.
However, I doubt the students will understand that those shapes are the special
kind of quadrilateral and also acknowledge the relationship between properties
of those shapes. It is because of the shapes have been classified from the
beginning which is not the result of students’ identification.
According
to Malloy (1999) based on the van Hiele theory of geometric thinking levels,
the students who are entering the middle school are generally in the level of concrete,
analysis or informal deduction levels. Nevertheless, Schultz (1988) in Mistretta
(2000) revealed that the high school students should be in the level of
informal deduction level of thinking. Based on Fuys, Geddes, and Tischler (1988), O'Daffer and
Clemens (1991) in Malloy (1999), Lawrei
and peg (1997) in Mistretta (2000), Naylor (2000) propose that in informal
deduction level, the students should be able to formulate generalizations or
definitions by ordering the previous properties, attributes and rules logically
in order to convince the truth of the generalizations through the logical
arguments, it begins by recognizing the relationship among properties.
Van
Hiele (1999) and Naylor (2000) believes that every student has their own
thinking level and it does not depend on the age of the student, otherwise it
is more dependent on the students’ experiences in getting the instructions that
could foster the development. Since, the fact is there are various conditions
of the students’ thinking level, it’s the teacher’s role to facilitate students
in enhancing the geometric thinking skills. That role can be done by providing
the lesson that is accessible for all level and it is promoting the development
for every level.
In
the case of quadrilateral, the teacher should be able to teach the special
quadrilateral by covering all of students’ thinking level. That approach is purposed
to help the student works on their own thinking level but also it should help
the students along to the next level (Naylor, 2000). Since the high school
students should be in the informal deduction (Schultz, 1988 in Mistretta, 2000),
then the teacher should promote all of students to attempt that level
achievement. Thus, it is very important for teachers to give the students
experiences that move them from the concrete and analysis level to the informal
deduction level (Craine and Rubenstain, 1993).
One of the recommended instructions that can be used for teaching
special quadrilateral is by using the quadrilateral hierarchy. That is because
of the relationship among special properties or attributes in the quadrilateral
hierarchy could foster students’ thinking level of Van Hiele for being in the
informal deduction (Crowley, 1987 in Craine and Rubenstain, 1993). The
quadrilateral hierarchy provides a structured classification and definitions of
special quadrilateral that can lead the students to draw their own framework of
properties. The students are expected to identifying the special properties of
quadrilaterals and then classifying them into some categories without
forgetting the relationship between those special kinds of quadrilaterals. However,
the quadrilateral hierarchy that is discussing is like the picture below.
Furthermore
the teacher could design the learning process that encountering the students to
build their own definitions about those special kinds of quadrilaterals. The example
that is proposed by Craine and Rubenstain (1993) which is teacher asks the
students to observe the properties of special quadrilaterals which are helped
by the questions of the teachers. For example the question is asking the
students to observe which quadrilaterals having the properties of “two pairs of
parallel sides”? The next process is
giving the students summarization of the information that students should know
through a simple diagram that contains quadrilaterals hierarchy. Lastly, give
the students a statement that should be proven by generate the properties that
they have identified previously. “These observations help them internalize
the properties and retain them for future references” (Craine and
Rubenstain, 1993).
However,
in Indonesia the special quadrilaterals are taught without giving the
understanding that they are interrelated each other. Students are learning each
special quadrilateral independently so that they are only facilitated to be in
the analysis geometric thinking level which is lower than the informal
deduction level. It is because the curriculum lead the teacher to do the
learning process by facilitating the students to build the definitions only
through identifying the properties without attempting to generate them to build
the relation.
Regarding
to enhance the geometric thinking level of students, the quadrilateral
hierarchy should be taught in Indonesia. Nevertheless, the challenges are about
the concept of special quadrilaterals in Indonesian Curriculum that has been
applied by the teachers a long time ago. The first thing that should be done is
developing awareness of the teachers toward five Geometric Thinking levels from
Van Hiele and internalizing the belief that a high school student has to be in
the 3rd level which is informal deduction rather than in the 2nd
level which is analysis.
References :
Mistretta, R.M. (2000). Enhancing
Geometric Reasoning. ProQuest Education Journal. pg. 365
Naylor, M.
(2000). The levels of Geometric Reasoning. ProQuest
Education Journal. pg. 30
Craine T. V.,
Rubenstein, R. N. (1993). A quadrilateral hierarchy to facilitate learning in
geometry. ProQuest Education Journal. pg. 30
