Ch. 2, page 1
Symmetry
Symmetry
Research on Young Students’ Understanding and Using Symmetry
Children from a very early age experience symmetry because it is an aspect of our
bodies, of nature, and of many person-made constructions. Booth (1994) also showed
that young students develop in their art from placing paint within and next to other lines
and dobs of paint (called a topological stage), to patterning and using symmetry. They
naturally develop a desire to paint in patterns and will also follow a shape design such
as painting in triangles or squares. Students may move away from the initial symmetry
but still fill all the spaces and balance their artworks.
Figure 1. Reflection in a lake.
Studies carried out in the United Kingdom, the United States of America (USA),
and Australia have found that seventh grade children have difficulty with symmetry
(Kouba, Brown, Carpenter, Lindquist, Silver, & Swafford, 1988; Owens, 1997a). Two
questions on symmetry involving mirror reflections, were poorly answered by Year 6
students in New South Wales (NSW) on Basic Skills Tests (Australian Council for
Educational Research, 1989-1991): 69% were correct on a question involving a grid
and a vertical reflection line, but only 20% coloured in parts of a reflected face
correctly. By comparison, over 80% of students in Year 3 and Year 6 were correct on
questions involving folding (Owens, 1997a).
Experience and understanding of the concept of symmetry assists recognition of
transformed shapes. Perham (1978) found that first graders' scores on subtests on the
recognition of shapes and left-right orientation were relatively high but were low on
subtests on perspective, figure-folding, and reasoning. After instruction the
experimental group only improved significantly on the perspective subtest. Moyer
(1978) found that explicit knowledge of the physical motion associated with a
transformation did not necessarily help the child's ability to perform the transformation
task.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Symmetry
Ch. 2, page 2
Features of Spatial Tasks and Their Effect on Performance
When Thomas (1978) found students in grades 1 and 3 were less able than grade 6
to choose a point on the side of a triangle which was rotated or flipped, an analysis of
the errors of the third- and sixth-graders revealed that they considered the vertices as
well as the sides of the triangle. This result suggests that conservation may not have
been the most important determinant of the results of this study but some other factor
such as the strategy used to make the decision or some features of the task.
The fact that task features are significant factors was confirmed by the results of the
alphabet task for which students in all grades (up to 11) found it was very difficult to
visualise the rotation of letters which had rotational symmetry (the S and N) and the
horizontal reflection of the non-symmetric J. The half-turn clockwise also yielded
greater differences between the grades than the two reflections or the counter-clockwise
rotation. Vurpillot (1976) explained that the use of a horizontal reference line in spatial
perception tasks encourages subjective preference for distinguishing a "top" and a
"bottom" of a shape while a vertical reference line encourages preference for
homogeneity of perception favouring recognition of symmetry.
The need to consider variations in the type of transformation as well as the type of
figure involved was taken up by Schultz (1978). She varied the type of transformation
(rotation, reflection, translation), the mode (horizontal or diagonal), the lengths between
positions before and after the transformation (long, short, overlap), the size of the
configuration (large or small), and the type (meaningful, that is the sailing-boat
configuration, or not meaningful). The configurations were made of three coloured
parts. She found that: (a) lack of familiarity and unexpected sizes of shapes interfered
with comprehension but not as much as type of transformation and features of the
transformation itself; (b) "meaningful configurations apparently facilitated the
operational comprehension of a task" (p. 205) and large shapes were preferred; (c)
translations were far more "do-able" than reflections and rotations by 7, 8, and 9 yearold children; (d) rotations and diagonal reflections increased error rate or were found to
be not "do-able"; (e) diagonal translations often resulted in re-orientation of the shape in
the same direction; and (f) the distance of a displacement was a significant variable.
However, the study did not give the significance of the differences in the percentages in
categories.
Further support for the effect of orientation of a figure on "do-ability" is given by
Perham (1978) who found that horizontal and vertical displacements in translation and
rotation tasks were significantly easier than diagonal-displacement tasks for first
graders. His study suggests that the orientation of a figure is more important than the
type of transformation, and that standard spatial abilities subtests were either too easy or
too difficult to be effective research instruments.
Instead of determining the kinds of transformations that students could carry out by
giving them test items in which students had to recognise transformed shapes,
Mansfield and Scott's (1990) study observed 23 pre-school to Year 1 children selecting
shapes to cover other shapes which were either marked with suitable divisions or were
not. (For example a square could be covered by two right-angled isosceles triangles or
two rectangles.) Although older children in this study tended to be able to solve more
problems than younger students, this was mainly the result of their persistence rather
than their more efficient or varied strategies. Covering shapes which did not have
divisions was more difficult for children than covering those with divisions. Children
who succeeded on the tasks tended to recognise shapes which would not lead to a
solution, and displayed a willingness to re-position pieces. Although eight of the ten
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Ch. 2, page 3
Symmetry
top-scoring children used rotation, only two of the ten lowest-scoring children used
rotation. Some children turned pieces over. Inefficient strategies used by the children
included picking up and discarding pieces without having rotated them. During the two
interviews which they were given children tended to use similar strategies, so it is
possible that children, in the second interview, were reluctant to discard inefficient
methods because they knew that the strategies they used would eventually lead to the
problems being solved (Mansfield & Scott, 1990).
Figure 2. Can the left triangle be a tile to cover the right triangle?
Key Ideas in Symmetry
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There are several key ideas on symmetry. These include:
There are properties associated with symmetries just as with other spatial concepts.
In reflection symmetry, corresponding points are the same perpendicular distance
from the line of symmetry
Seeing and using symmetry in vertical lines is easier than in horizontal which is
easier than in diagonal
Rotational and slide symmetry also exists. Slide symmetry is usually seen as
horizontal repetition and commonly used. Because of the properties, tessellations are
readily made.
In rotational symmetry we refer to order being the number of turns to return to the
original position.
Experience assists visual recognition of symmetries.
Lessons can develop from recognising symmetry by folding in early childhood to
using analysis and mirrors for reflection symmetry (much harder).
Visual arts can be used to illustrate mathematical concepts like symmetry,
asymmetry.
There are many cultural groups that use different designs based on reflections and
slides in a 4 square pattern.
Teaching Objectives for Symmetry through Motion
Students need to recognise shapes in different orientations and to develop the skill
of appreciating what an object or group of objects might look like from another
perspective. These changes in perspective and orientation are related to motion.
Motions with manipulatives (e.g. cardboard cutouts and tiles) that represent twodimensional shapes include flips, slides, turns, and folds. These motions assist students
to develop concepts such as (a) reflection symmetry (flips in horizontal, vertical, and
diagonal lines or folding), (b) area (slide repetitions associated with covering of areas),
and (c) rotational symmetries.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Ch. 2, page 4
Symmetry
horizontal line
vertical line
oblique line
folding
Figure 3. Reflection symmetries in horizontal, vertical, and diagonal lines, folding.
turn
Slide
slide
Figure 4. Slide symmetry (here with a rotation), covering areas.
Movement is imaged by students as they make associations between shapes. For
example, they can image one triangular shape moving to become another triangular
shape as a point slides along a taut string. They might see how a triangular shape can
become a quadrilateral by bending one side into two or how a square can be pushed
over to make a rhombus.
Learning Tasks for Readers
Symmetry Activities 1
experiencing
* Take a rectangular sheet of paper. How many lines of
symmetry does it have?
* How many ways can you fold it in half? Discuss with your
class.
* Take a four-square rectangle and extend one side by a square.
Now find as many places to put another square to make a
symmetrical design. Where are the lines of symmetry in each
case.
Figure 3. Creating symmetrical designs.
There are four positions. The rectangle has only 2 lines of
symmetry (the diagonals do divide it in half but they are not lines
of symmetry). The two designs with a diagonal line of symmetry
are popular in traditional designs such as with First Nations in the
Americas as they are easily woven into cloth. They are attractive,
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Ch. 2, page 5
Symmetry
one looks like a fish which may also be important in culture. Most
people find the diagonal symmetries harder to see.
One position that does not make line symmetry gives an example
of rotational symmetry The rotational design has order 2 because
there are 2 turns to return it to its original position.
The dime puzzles. Two of the cards are illustrated below. Copy
and cut out the pieces and try to make symmetrical shapes.
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Figure 4. Dime puzzle cards.
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Put the mira on an end point of the line design left below and
make a diversity of designs by rotating the mira around the
point.
Figure 5. For making symmetry designs
Use a red mira to draw a perpendicular bisector of an interval
(terminating line) above and bisector of an angle. Use the mira to
find the number of symmetries for a range of shapes.
Figure 6. Interval and angle to bisect.
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Make a simple design with 4 sticks, predict what it will look like
if reflected in (a) one, (b) two, (c) vertical lines; (d) a vertical
then horizontal line, (e) vertical, horizontal, vertical, (f) diagonal
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Kay Owens
Ch. 2, page 6
Symmetry
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Why do you find some reflection difficult to see?
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Look at the activities on symmetry in the Syllabus. Do they
indicate a development? Do they cover rotational symmetry?
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Why is symmetry so important (a) in the building industry, (b)
in nature?
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Give the lessons at the end of the chapter. The pentomino
activity and the pattern block symmetry are particularly
worthwhile.
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Develop an early lesson based on folding and consider how to
encourage an understanding of symmetry that is to move the
lesson from activity to abstracting ideas.
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Develop a later lesson on rotational symmetry.
connecting ideas
summarise
and record
Symmetry of Three-Dimensions
With three-dimensional objects we should distinguish between the symmetry of
faces and symmetry of the object. Symmetry of an object can be in a plane e.g. a
rectangular prism. A cylinder has plane symmetry but also line symmetry. Find other
3D geometric shapes and environmental objects with plane symmetry and line
symmetry.
If you devise a mathematics trail and ask students to look for symmetries you
need them to focus on 2D faces or confusion can arise.
Symmetry in Designs
Designs using symmetry have been well used in traditional cultures. A number of
these are illustrated. They are usually combined with tessellations. Carvings are often
symmetrical or deliberately not quite symmetrical. When carver plan, they measure the
distance from the central line of symmetry to say the edge of the eye in order for the
eyes to be symmetrical. They might use a piece of cane or string.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Ch. 2, page 7
Symmetry
Papuan Gulf
Sepik
Trobriand
Huon Gulf
Figure 7. Traditional carvings from Papua New Guinea
Figure 8. Papua New Guinea mask, Sepik, tortoise shell, clay, shells, cassowary
feathers; stick used to mark symmetrical point; pot symmetry.
Designs are a feature of Papua New Guinea bilums or string bags. Each row is made
stitch by stitch with the length of string pulled through in a double figure of eight. This
complex stitch is then developed into a repeated, symmetrical design by careful
counting. Designs can be varied and extended by increasing the numbers appropriately.
The maker also keeps a keen eye on the length of each shape, especially diamond
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Symmetry
Ch. 2, page 8
shapes, so that their length and breadth are equal despite the different height and width
of stitches. New designs are being made regularly and these may spread quickly around
the regions.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Ch. 2, page 9
Symmetry
Figure 9. Bilums and bilum making in Papua New Guinea.
Learning Tasks for Readers
Symmetry Activities 2
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experiencing
Look at the pattern made with 4 squares from a combination of
reflections and slides or glides, or turns. Make these using
plastic acetate sheets and make some more designs. These are
commonly used in traditional cloth-making or printing.
Look at the symmetries in traditional cloths and other artifacts like
this Malay kite.
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On grid paper prepare a symmetrical design that can be woven.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Ch. 2, page 10
Symmetry
Note the numbers that are to be remembered. Try it out with
paper for weaving. Alternatively, set up a small hand weaving
frame and weave a design by counting the number of over and
under weft strings.
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Discuss the links between creative arts (music and visual arts)
and their links with symmetry. Consider how you might design a
lesson involving these two KLAs.
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Read PAMphlets such as Artistic Shapes for further ideas.
connecting ideas
summarise
and record
Find examples of woven patterns by Indigenous people of the
Americas, for example from Peru. Find designs in European
crafts.
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Write down an explanation of symmetry for Stage 1, another for
Stage 2, and another for Stage 3.
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Why is symmetry so important for:
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nature
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build environments
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deciding properties of shapes, for example, height line of
isosceles triangles
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens