The Value of Symmetry in the Primary Classroom
Symmetry can be found all around us. From the flowers and leaves of nature to the wheels and windows of the industrial. Because of this, children have an obvious interest in symmetry, particularly in its aesthetic and well-proportioned qualities. The NSW Board of Studies (2000, p. 5) states that children need to be exposed to learning experiences that relate to their everyday lives. Symmetry is an area in which this has a number of possibilities. Symmetry can also provide rich hands-on activities that a number of literature and research has stated that is crucial to children’s development of mathematical concepts, particularly in spatial abilities. Symmetry is also important, as it is an area that can be an effective foundation for other mathematical concepts and other key learning areas.
This paper will address the significance of teaching symmetry in the primary classroom through an exploration of research and theory. It will also outline practical strategies for teachers to teach symmetry effectively.
The basic facts of Symmetry
The motions of symmetry are taught in the space and geometry strand, within the sub-strand of two-dimensional space of the NSW K-6 Mathematics Syllabus (BOS, 2002, p. 125). Space and geometry is the study of spatial forms and important and critical skills for students to acquire are those of recognising, visualizing and drawing shapes and describing the features and properties of three-dimensional objects and two-dimensional shapes in static and dynamic situations (BOS, 2002, p. 117). It involves students learning and extending their spatial abilities and perceptions. (Douglas, p. 19) Spatial sense includes two main spatial abilities, spatial orientation and spatial visualisation and imagery.
Two types of symmetry are taught in the primary classroom, line symmetry and rotational symmetry. Line symmetry is when one half of a shape is a reflection of the other half (Turner, 2003, p.17).
When a shape looks the same in different positions as it is turned at a fixed point, it has rotational symmetry (Turner, 2003, p. 22). The number of matches the shape makes when you rotate it around once is called the order (Maths Is Fun, 2005, para. 2).
An object has point symmetry if it matches itself when rotated a half turn (180) about a point and if an object has point symmetry than it has rotational symmetry of order 2 (Sanok, 1978, p. 51).
Line symmetry is first introduced in stage one and rotational symmetry by stage three (BOS, 2002, p. 125 &128).
The importance of hand-on activities
Educational and psychological research has shown that children build ideas about shapes from action, rather than merely passively viewing (Douglas, 1998, p. 5). Children need to explore parts fully, including their parts, properties and transformations (Douglas, 1998, p. 5). The Board of Studies states that the manipulation of a variety of real objects and shapes is crucial to the development of appropriate levels of imagery, language and representation (BOS, 2002, p. 117). When working mathematically students apply strategies they use computer software to draw shapes of their own and mark any lines of symmetry.
Children’s ideas develop from intuitions grounded in action, building, drawing, moving and perceiving (Douglas, 1998, p. 6). For line symmetry, a design or shape can be folded in half to match the other side and in rotational symmetry, students can trace the design and by turning it recognise where the deign matches the original at different parts of the turn. It is important for children to be exposed to concrete manipulations of symmetry before having to interpret other representations (Bobis, Lowrie & Mulligan, 2004, p. 101). Using play-doh to cut and test, and create shapes that reflect line symmetry is also another concrete manipulation that can be employed in the classroom. Other hands-on activities include; using mirrors to explore line symmetry, creating shapes on geo-boards to explore symmetry, using pattern blocks to create shapes and as mentioned in the mathematics syllabus, creating and manipulating shapes on computer programs such as paint.
Symmetries connection to the real world
A guiding principal of the NSW primary syllabi is that children’s learning is enhanced when they see connections among their learning experiences and relate them to their everyday experience (BOS, 2000, p. 5). Many spatial relationships are commonly found in a child’s day to day life and it is recommended that these experiences be linked to spatial and geometric understandings of mathematics (Bobis, Lowrie & Mulligan, 2004, p. 87). It is therefore important that teachers provide a number of activities that develop students’ understandings of symmetry and learning experiences that have relevance to their everyday life. For example, teachers need to get students to look at nature and build up an appreciation for symmetry. Students can explore the symmetry found in flowers, tree leaves, animals and so on.
Karen Hancock (2007, p.24), a mathematics teacher devised a lesson on identifying objects with rotational symmetry in the staff car park. Pictures and sketches were made by students of a variety of objects with different orders of rotation. Teachers can create activities that similarly explore objects found in the school grounds. This means that students will be more engaged as the learning has relevance to their everyday experiences. Student’s could also explore the symmetry found in the letters of the alphabet, in particular the letters in their names. Investigating familiar objects for symmetry emphasises the relation of geometry to the student’s world (Morris, 1977, p. 57).
Symmetries relationship to other learning areas
Research has found that early work in symmetry can provide foundations for extensions in several directions in mathematics, including transformational geometry, congruence and groups (Morris, 1977 p. 60) Therefore, the study of symmetry should be made part of a continuous line of geometrical experiences, not an isolated activity. Work with mirrors can lead to the rigid motion of slide, or translation (sliding) and from there to tessellations (Morris, 1977, p. 60). Creating shapes on computer software too, can further help students to explore transformations. When students explore symmetry on two-dimensional shapes, they are also informally investigating shapes and their properties. Groups can be created when students place shapes into categories such as number of lines of symmetry or point symmetry. Further, through students’ manipulations on a geo-board, concepts of area can be explored with the same shapes. Lastly, with the play doh activity mentioned earlier, and with the use of pattern blocks, children could place shapes on top of each other to test congruence.
Students may also develop and apply understandings of symmetry across curriculum areas of art, craft, technology and science (Bobis, Lowrie & Mulligan, 2004, p. 88).
For instance in art, a number of symmetrical ideas may be applied to create aesthetically pleasing designs or artworks of human faces, whereby the student completes the reflection of the face using art forms. Symmetry can also be explored in plants and animal shapes, integrating with a biology unit. It is important that teachers do not restrict learning, make connections and foster relationships between subjects or mathematical concepts.
Symmetry is so valuable as it can provide effective learning experiences in the primary classroom, through fostering a hands-on approach to learning, incorporating activities that relate to students and their everyday lives and being a stepping stone for integration into mathematical concepts and other key learning areas.
Must Have Resources for Teaching Symmetry
- play doh clay
- mirrors and mira mirrors
- pattern blocks
- digital camera
- drawing and painting computer software, including Paint Shop
Board of Studies NSW (2000). The Primary Curriculum, An Overview. NSW: Board of Studies NSW
Board of Studies NSW (2002). Mathematics K-6 Syllabus. NSW: Board of Studies NSW
Bobis, J, Lowrie, T & Mulligan, J. (2004). Mathematics for Children: Challenging Children to think Mathematically. Australia: Pearson Education Australia.
Clements, D. (1998). Geometric and Spatial Thinking in Young Children. USA: National Science Foundation.
Hancock, K. (2007). Symmetry in the Car Park. UK: The Association of Teachers of Mathematics
Maths Is Fun (2005). Rotational Symmetry. Retrieved August 17, 2008 from http://www.mathsisfun.com/geometry/symmetry-rotational.html
Morris, J. (1977). Investigating Symmetry in the Primary Grades. In J. Hill (Ed)., Geometry for Grades K-6 (pp. 55-62). USA: The National Council of Teachers of Mathematics Inc.
Sanok, G. (1978). Living in a World of Transformations. In J. Hill (Ed)., Geometry for Grades K-6 (pp. 50-55). USA: The National Council of Teachers of Mathematics Inc.
Turner, G. (2003). Targeting Maths, Dictionary. Sydney: Pascal Press