Some background
Nearly every teacher would agree with the statement that young
children learn through the psychomotor and the affective (modern terminology
for affective read emotional intelligence) just as they do through
the cognitive.
We believe however that there is, on many occasions in public schools, too
much emphasis on the development of cognitive faculties of young children.
In many cases to the exclusion of considering that hemispheric development,
namely the development of the affective and the cognitive in a balanced and
harmonious way, should occur.
On the other hand, those who use the approach we advocate attempt to balance
left and right brain learning. This should happen on a continuum and the objective
must to be implement methodology to achieve this throughout grade school.
What does this mean? Let us take the example of young children. For them, learning
cognitively should not be just about learning factual material or absorbing
information. It is seeing that both the content and the methodology used appeals
and relates to both left and right hand brain activities.
As far as these children are concerned, something like the following sequence
applies. It means making sure the young children experience material initially
through the psychomotor and emotional intelligence before coming to cognitive
learning.
Although this lesson is about teaching symmetry to 9/10 year olds, the process
(as far as grade school is concerned) begins in grade 1. For example, in the
early grades one objective is for children to move gently and gradually from
a psychomotor activity to symbolic form. Part of this process will occur through
the drawing of different forms. When this occurs in the manner described, namely
gently and gradually, balanced and harmonious hemispheric development occurs.
The result of this is that children are able to adjust to the world we live
in more easily and in a less stressful manner than would otherwise be the case.
Start of the process
The process of seeing that balanced and harmonious hemispheric development
occurs can, of course, be taken right back to birth. An example is where growth
patterns occur naturally and with little teaching from adults. If we go right
back to the first year of growth we see that babies instinctively learn to
live in a multidimensional world, a world where the law of gravity rules supreme
and where they learn to adapt their own body and nature to such laws.
Soon small children have learned to move upright in a variety of ways
and they have also learned both inner and outer experiences of what
can be termed "inner
symmetrical experience". An obvious inner experience is when they recognize
right and left and whether to write with the left or right hand.
By the time they reach grade school children should have experienced a variety
of symmetrical forms that exist in the outer world. As long as they are aware
of such symmetry there is no reason why, once they enter grade school, they
can learn to develop both their awareness of symmetrical forms and the properties
of such forms in a more structured way.
Through the early grades children learn, through a study of themselves and
the outside world, that symmetry (starting with simple study of themselves)
is a natural part of the world and their experiences of it. How all this is
taught however must be left to another time.
We can say that children should learn about forms of symmetry existing in the
outer world (although not perfectly symmetrical leaves would be an obvious
example for young children). This is an easy example where it is not difficult
to plan a structure that recognizes and relates to children's mindset.
Talking generally and theoretically, intellectual understanding can be added
to experiential learning that has occurred through activities that appealed
to psychomotor and emotional intelligence. Somewhat simplified, the sequence
for learning is psychomotor, emotional intelligence followed by intellectual
experience.
Another facet, again describing a theoretical description but giving a specific
topic area, would be to take different axes. For example, one basic sequence
is to work first with the vertical axis, then with the horizontal axis. At
an appropriate time, content will also include crossing the axes.
Hopefully, from the above the reader will obtain a feeling for, and an understanding
of, the type of content and methodology implemented through the approach we
recommend. We will now turn to briefly describing the teaching of symmetry
up to grade 4 followed by what we are trying to achieve in grade 4. This latter
will include covering the contents taught in this lesson.
Development up to grade 4
Children from an early age develop the concept of symmetry from their
experience of themselves and the outside world. How do children
do this? An obvious example to start with is to relate content
to their own physical being. For example, this can be achieved
throughout their early childhood by singing and repeating songs
and rhymes (the song "I have two hands, I have
two feet etc.)
Thus, they experience matching parts and the symmetry that exists with
their own body in a way that is appropriate to their mindset. During
this process they are learning experientially in a way that engages
their emotional intelligence. This would not be the stage in which
to introduce a structure which engages the intellect through, say,
formal exercises or, for example, learning the vocabulary of symmetry.
Grade 4
When should such formal content be taught? Grade 4 would be the
appropriate time to introduce such content (for some general background
concerning developmental models see previous lesson at: members.cox.net/waldorfedu2/newsletters/Fractions1.html, "Age
and Stage of Growth" and "Children of Nine/Ten").At
the age of around 9 a formal and structured approach to subject matter
occurs. Perhaps it should be added that it would still be relevant
to start a lesson in grade 4 with reference to the symmetry that exists
within each one of us. This time, however, appropriate vocabulary can
be introduced. For example, "line of symmetry" or "access
of symmetry". In other words, where is the line of symmetry
in our physical being?We know that drawing and completing symmetrical
forms develops eye-hand coordination and a feeling for form, harmony
and geometrical relationships. Also many drawings enable students
to express inner experience with outer form. All these are now placed
in a more formal structure than previously. They can involve the
student in observation and the ability to reproduce outer forms with
accuracy according to individual ability. They can also draw many
examples of symmetrical forms, enabling them to express their imaginative,
artistic and creative aptitude and ability.
Students at this stage should be able to recognize, identify and
create symmetrical figures and be able to classify them according
to their axes of symmetry. Simple examples would be identifying figures
(whether created and drawn from their imagination or examples from
real life) according to their axes.
Contents of lesson "Symmetry"
The contents of the lesson "Symmetry" fits in with
both national and California standards. I would hope that this may
indicate how a methodology, based on the integration of left and
right hemispheric spheres, can be used in a manner appropriate for
public schools. I also appreciate that many public school teachers
will find many things in this section that they teach and may well
give similar reasons to the ones we have described for doing so.
Index
Teacher
Notes
Introduction
Assignment 1 - Symmetry
Assignment 2 - Lines of Symmetry
Assignment 3 - Symmetrical
Drawing
Assignment 4 - Snowflakes
We believe that as the lesson plan is simple and straightforward no further elaboration
(except for one explanation) is needed in this section. However, if this is not
the case we welcome feedback.
Teachers will recognize that the assignments are carefully scaffolded
leading the student step by step through the concepts. The following
is the sequence:
a) awareness of the symmetry that is found in the physical body;
b) simple line symmetry;
c) figures with multiple lines of symmetry;
d) a hands-on activity providing experience of rotational symmetry;
e) formalizing experiences of rotational symmetry;
f) identifying and naming rotational symmetry.
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Teacher
Notes
Purpose/Objectives
For a variety of reasons, which are dealt
with in the newsletter, students need to learn
about symmetry. The lesson below describes one
way in which symmetry can be taught in grade
4. I appreciate that some of you will not be
conversant with national or state standards.
However, we know that some educators working
in universities or for state departments of education
are reviewing our material and making known our
web sites to serving and student teachers.
Many working in such environments will be working or conversant with
national standards. For these people the following may be of interest.
This lesson meets the following standards in the National Standards
for Mathematics (October 1998 Draft): Principles and Standards for
School Mathematics.
National Standards
Standard 3: Geometry and Spatial Sense
Mathematics instructional programs should include attention to geometry
and spatial sense so that all students recognize the usefulness of
transformations and symmetry in analyzing mathematical situations (our
note: this is the objective that applies to symmetry and is one of
four stated objectives).
Focus Area for Grades 3-5
All students should identify and describe line and rotational symmetry
in various two dimensional shapes (our note: this is the objective
that applies to symmetry and is one of four stated objectives).
Introduction
The lesson begins with a basic introduction to symmetry and leads students
through to more challenging concepts. The lesson is designed for grades
3-5, with the earlier work providing a review of concepts that students
will have met in earlier grades. The final lesson on snowflakes introduces
students to rotational symmetry.We attempt, wherever possible, to develop
literacy through different content areas including mathematics. In
this lesson students should learn the following vocabulary - the methodology
by which this occurs we leave, on this occasion, to the teacher.
Vocabulary to be introduced
a) clockwise, anticlockwise;
b) rotate, rotation;
c) mirror image;
d) vertical, horizontal;
e) line of symmetry, axis of symmetry;
f) hexagonal;
g) stellar. |
Introduction
One way in which to introduce children to the concept of symmetry is
for them to experience it first within their own
body
Example 1
Discuss with students the symmetry that exists in the human body. Help
students to recognize where the line of symmetry is in human body.
Identify where there is no line of symmetry in the human body.
Example 2
Have a student lay down on a large sheet of paper while you draw around
his/her outline.
Ask this student (once upright!) to draw lines of symmetry on his/her
outline. For example, down the center of the face and down the center
of the body.
Discuss the qualities in our bodies that make up the concept of symmetry.
Discuss lines that divide the drawing into two parts that are not lines
of symmetry.
Example 3
Draw a line and stand on one side of it.
Invite a student to stand alongside you on the other side.
Create a simple "pattern" by taking two strides, one away from
the line and one towards it.
Invite the student to repeat this pattern on his/her side of the line.
Explain that between you, you are creating a symmetrical pattern.
Continue with other students, making the pattern gradually more complicated.
Suggest that students work with a partner taking it in turns to create
and complete patterns. |
Assignment
1 - Symmetry
For these activities you will need:paper, pencil, colored pencil, scissors,
ruler, small mirror |
Activity 1
- Fold a piece of paper in half.
- Draw a shape on the folded edge.
- Cut out your shape.
- Open out your shape.
- You will use this shape in Activity
3.
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Activity
2
- Take a piece of paper and fold it twice.
- Draw a shape on the folded edge
(not the open edge).
- Cut out your shape.
- Open out your shape.
- You will use this shape in Activity
3.
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Activity 3
Place your mirror along the fold of your first cut out
shape so that half the shape is reflected in the mirror.Is
the half reflected in the mirror identical to the half that
is behind the mirror? If your answer is yes, then this is a
mirror image and the drawing is symmetrical.Repeat this exercise
with the second cut out shape. You will see that you can carry
out this exercise twice, holding the mirror vertically and
then horizontally.This shape has two lines of symmetry.Examine
the following figure. It has has four lines of symmetry.
- If a shape can be divided by a line into
two identical parts it has line symmetry. The two parts
of the shape are mirror images.
- We call the dividing line, the
axis of symmetry or the line of symmetry.
- Some shapes have more than one
line of symmetry.
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A
pressed leaf or flower will give you an example
of symmetry. |
Activity 4
- Look around your classroom.
- Look around your school.
- Look around as you travel to
and from school.
- Look in magazines and newspapers.
- What examples of symmetry do
you see?
- Collect examples of symmetry
from your environment. Your examples may include,
drawings, pictures, written explanation, actual
objects.s
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Assignment
2 - Lines of Symmetry
Work in groups of 2-4 students to carry out this activity.
- Some letters have horizontal symmetry. e.g. B
- Some letters have vertical symmetry. e.g. A
- Some letters have both horizontal and vertical
symmetry. e.g. H
- Some letters have no line symmetry. e.g. F
- Look carefully at the letters
below.
- Discuss with your group
the symmetry of each letter.
- What type of symmetry
does each letter have?
- When you have reached
a decision draw in the line or lines
of symmetry.
- Write a description
beneath each letter.
- Work on one letter at
a time.
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C
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D
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E
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K
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M
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N
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P
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S
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X
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 Assignment
4 - Snowflakes
Snowflakes are hexagonal stellar ice crystals; they are made up of
six sided patterns.People who have studied snowflakes by magnifying
them and photographing them have not been able to find any two identically
shaped snowflakes. It appears that every snowflake is different!! (Although
it should be mentioned that in 1988, researchers did find one pair
of snowflakes that had no discernible differences!)
Let's construct some snowflakes
You will need: a square of white paper, 8" x 8" works well.
a larger piece of white paper, a pencil, a pin, scissors
This is what you do. 
- Place your snowflake on a piece of paper.
- Carefully draw around the outside, creating
an outline of the snowflake.
- Put a pin through the center of the snowflake
and the sheet of paper so that the snowflake
is free to turn but attached to the underlying
paper at the center.
- Mark one point out of the six of your snowflake
so that you can always recognize it.
- Turn your snowflake clockwise so that the marked
point moves to the next point in your outline
drawing.
- Examine the snowflake and the underlying drawing.
- Do they still fit exactly?
- Move your snowflake again in a clockwise direction
so that the marked point moves to the next point
in your outline drawing.
- Examine the snowflake and the underlying drawing.
- Do they still fit exactly?
- Continue until the marked point has been moved
to every point of the underlying drawing.
- Did the snowflake and the drawing match perfectly
after every move?
This is called rotational symmetry.Now,
write a short description of rotational symmetry starting
with, "Rotational symmetry is when ....." Top
(Index)
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