Teaching about Angles and Triangles for 3rd Grade Students
Using Origami
Galit Ashkenazi-Golan & Vered Gabai
Faculty of Science Education, Seminar Hakibutzim, Israel
galit.ashkenazi@gmail.com & veredami@smile.net.il
Abstract
Sorting angles and triangles is an important part of the basic geometry education.
Using Van Hiele (1999) levels of geometry thinking, classifying aids the students
during the descriptive level, when figures are bearers of their properties. When
searching for creative ways to teach basic geometry concepts, Origami is a
natural candidate. The Japanese art of paper folding gives rise to
experimentation with shapes and angles. During the creation of a final product,
those shapes are classified, marked, and then, sometimes, folded again, to
create new shapes and angles. In this research, a 3rd grade class was randomly
divided into two groups: one studied about shapes and angles from the textbook
(control group), and the other studied shapes and angles solely through origami
(experiment group). The achievements of the experiment group were no lower
than those of the control group, and for some topics significantly higher, while
teacher and students reported the origami classes to be enjoyable.
Theoretical background
According to the Geometry Standard of the NCTM, four main areas should be
addressed: properties of shapes, location and spatial relationships,
transformations and symmetry, and visualization (Clements & Sarama, 2000).
Among the geometry topics taught in Elementary school are the angles and the
triangles. The concept of “angle” proves to be difficult to teach and hard to
understand. There are several definitions of angles, and there are aspects of the
“dynamic” versus “static” definitions (White & Mitchelmore, 2003). In the research
of Keiser, Klee and Fitch, when 6th grade students were asked to define angles,
they found lack of proper geometric vocabulary, and “We also learned that the
constancy of the angle measure as an angle is enlarged or reduced is an idea
that a few of our students really understand” (Keiser, Klee, & Fitch, K. (2003)).
In the second grade, the students are familiar with the concept of “right angle” in
an intuitive manner, as a result of discussing the properties of the square the
rectangle, and out of observations, without a formal definition. The students
mainly compare the different angles to a right angle – an acute angle is smaller
than a right one, and an obtuse angle is larger than it (Rozen, 2006).
Van Hiele’s theory and the NCTM standards support active learning, using variety
of manipulatives. According to Koester, instructions should include activities that
begin with playing and exploring, activities that gradually introduce the
mathematical concept and language to the students. (Koester, 2003).
In the 80’s Origami has been put into use in teaching the subject of Geometry
(Yitzhaki, 2010). In the base of the Origami lie seven axioms which describe the
process of the folding of the paper according to the following basic concepts:
point, folding and plane. With the help of these concepts it is possible to solve
different Math and Geometry problems (Alperin & Lang, 2006).
Researches that tested the effectiveness of Origami in the teaching of Geometry
show either that it does not fall from traditional methods of teaching (Boakes,
2009) or that it is significantly increasing achievements (Arıcı & Aslan-Tutak,
2013).
We were interested in testing the effect of teaching 3rd grade students the
concept of angles and triangles using origami.
The purpose of the research:
The research compared two different teaching methods of the following subjects:
1. Angle – it’s definition and marking, identification of a right angle and its
comparison to an acute angle, obtuse angel and straight angle.
2. Triangle – categorization by its angle type (an acute triangle, a right triangle
and an obtuse triangle) and categorization by its edge type (a scalene
triangle, an isosceles triangle and an equilateral triangle).
The research was conducted in two classes of third graders in Israel. In the test
group class, the subjects were taught using the art of paper folding – Origami and
with no use of textbooks. In the control group class, the subjects were taught
using a standard textbook.
Method of research
The research included a math questioner meant to test the students level of
knowledge in the topic of angles (an acute angle, a straight angle and an obtuse
angle), the topic of triangles and its categorization according their angles (acute
triangle, straight triangle, obtuse triangle) and the topic of sides (scalene triangle,
isosceles triangle, Equilateral triangle). The questioners were given to the
students twice: before the process of formal teaching in a class setting took
place, when answering of the questioners was based on their knowledge from
previous years, and twice at the end of the teaching process, after the subjects of
both classes (the test group and the control group) learned the above subjects
with the same teacher.
An Example of the Activities
The activities for this research were designed by Mrs. Miri Golan from the Israeli
Origami Center.
The students were taught simple paper-folding designs, such as the one
presented in the following page.
Front
1. Fold Horizontal and Vertical.
Then unfold.
2. Fold bottom edge to the center.
Flip
Back
4. Fold top flaps
to center vertical
line.
5. Fold top flaps down to
bottom top without folding
the triangles on the side.
Done!
3. Fold bottom edge flaps to the
center vertical line.
Front
Back
6. Fold edge triangles
along their inner edge
Flip
8. Bend
down top
tip.
7. Fold top
corners down
to edge made
by the top flap.
fold
During the paper-folding instructions, the teacher used the shapes that were
folded to note and mark the angles and sides created. For example, in the heartshape folding above, at the beginning (step 1) we have a square – four right
angles (and four equal sides); at step 5 we have a different square; at step 6 two
obtuse angles were created, etc.
Results
The questionnaire that was used was validated by Dr. Steinberg. The
questionnaire included questions in the following topics:
The questionnaire was answered before and after the intervention.
Table 1 and Figure 1 sum the results of the improvements in each topic in the
experiment and in the control group:
Experiment
Control Group
Group (N=23)
(N=23)
Topic
Mean of
Mean of
Improvement (SD) Improvement (SD)
1. Sorting angles between rays (by 4.74 (2.89)
3.78 (3.24)
size)
2. Naming the type of the angles 2.48 (1.64)
1.74 (1.66)
between rays (acute, straight or
obtuse)
3. Marking (internal) angles on a 5.57 (3.70)
5.96 (3.30)
polygon
4. Naming the type of internal 3.48 (2.17)
3.26 (2.22)
angles in a polygon (acute,
straight or obtuse)
5. Drawing triangles by angles and 0.61 (1.72)
0.52 (1.80)
relative length of sides
6. Sorting triangles by relative 1.57 (1.92)
0.74 (2.34)
length of sides
7. Sorting triangles by angles
0.74 (0.91)
0.78 (1.83)
8. Sum of angles in a triangle
0.13 (0.37)
0.04 (0.55)
Table 1: achievement improvements in the experiment and in the control group
Figure 1: achievement improvements in the experiment and in the control group
Discussion
Although none of the differences in the achievement improvement above are
statistically significant, the experiment group improved more than the control
group in six out of eight topics, which is a statistically significant result. Using
origami, the students improve their achievements in more topics than when using
the textbook.
During the intervention, the teacher reported that the students loved the lessons
using origami, and often approached her in the school yard to ask when the next
lesson will be.
In light of those reports and the improvement of the achievements, there seems
to be no contradiction between having fun during geometry lessons, and learning,
maybe it is the other way around.
References
Alperin, R. C., & Lang, R. J. (2006). One-, two, and multi-fold origami
axioms.Origami, 4, 371-393.
Arıcı, S., & Aslan-Tutak, F. (2013). The effect of Origami-based instruction on
spatial
visualization,
geometry
achievement,
and
geometric
reasoning.International
Journal
of
Science
and
Mathematics
Education, 13(1), 179-200.
Boakes, N. J. (2009). Origami Instruction in the Middle School Mathematics
Classroom: Its Impact on Spatial Visualization and Geometry Knowledge of
Students. RMLE Online: Research in Middle Level Education, 32(7), 1-12.
Keiser, J. M., Klee, A., & Fitch, K. (2003). An Assessment of Students'
Understanding of Angle. Mathematics Teaching in the Middle School, 9(2),
116-119.
Koester, B. A. (2003). Prisms and pyramids: Constructing three-dimensional
models to build understanding. Teaching children mathematics, 9(8), 436442.
Rozen, G. (2006). Geometry from a Different Angle. Mispar Hazak , 17, 45-47 [in
Hebrew]
White, P., & Mitchelmore, M. (2003). Teaching angles by abstraction from
physical activities with concrete materials. International Group for the
Psychology of Mathematics Education.