Sue Forsythe - How Children Use Symmetry When Visualising 2D Shapes

"The learning of geometry is my particular area of interest", says Sue Forsythe of the School of Education, "because of its visual nature and the way it allows learners to use their intuition to access mathematical ideas".

About My Research

My research focuses on how 12-13 year old students learn and reason about 2 dimensional shapes when working with a dynamic figure which has been constructed using a geometry software program.

Dynamic Quadrilateral

The Dynamic Figure

Geometry is both a practical and an abstract art and many students become confused between the diagram of a shape (its representation) and the abstract theoretical object. Hence I am exploring the potential of the dynamic figure on the computer screen to act as a mediator between the representation and the theoretical object. I am particularly interested to explore whether working with the dynamic figure can help students’ understanding to develop to a more sophisticated level.

Research Approach

Pairs of students work with the dynamic figure in the geometry software program which has been uploaded onto one computer. I record the on screen activity and the dialogue between myself and the students in each session which lasts about 50 minutes.

The figure has two perpendicular fixed length lines which I have called bars (see illustration). The figure can be changed into many different shapes (triangles and quadrilaterals) by dragging the bars inside the figure. Notice here that when I refer to the figure it is the generic object made in the geometry software program. When I refer to a shape it is a specific shape which is generated by dragging the bars inside the figure.

I am interested to see whether the students notice similarities between the shapes when they drag the bars in certain ways. For example when one bar is dragged so that it always crosses the other bar at its mid point then kites, rhombus, isosceles triangles and arrowheads can be generated. They have a common property (one bar bisects the other) and could be said to form a ‘dragging family’ of shapes. This notion of a family of shapes which includes special cases is quite a sophisticated idea for this age group.

Research Findings

Two of the most important themes that emerge from the data are the roles of orientation and symmetry.

Orientation

When children learn about shapes in primary school they often tacitly observe the typical orientation in which shapes are presented. For example a square tends to be drawn with its sides vertical and horizontal and if it is oriented differently, say with its sides at a 45 degree angle, it is commonly called a diamond. Orientation clearly matters to humans although in mathematics the orientation of a shape is not one of its properties. In the sessions the secondary school age students noticed the orientation of any shape they generated. For example they refer to upside down isosceles triangles or, when the bars in themselves were presented at an angle to the vertical, one girl referred to ‘an angled kite’.

Symmetry

This has emerged as an important aspect of how the students observed the shapes they generated from the figure. The students were observed to drag one of the bars inside the figure so that near symmetry was preserved. It is difficult to hold a computer mouse and drag keeping accurate symmetry but the students kept  the figure fairly close to being symmetrical whilst dragging one bar through the middle of the figure. When I looked at the recordings of the shape with the side and angle measurements displayed on the screen, expected equal sides and angles (if symmetry were to be true) were often within 0.2 centimetres and 2 degrees apart. This indicates fairly accurate symmetry and the intention to drag maintaining symmetry which is what I have named this style of dragging.

The students’ concept of symmetry was clearly very strongly held which suggests that it is an innate aspect of human cognition, a finding backed up by much research into human cognition. The students were able to use symmetry to identify equal sides and angles within any shapes they made, even ones with which they were less familiar such as arrowheads. It was common for students to test for symmetry by imagining folding the shape in half, which they had experienced as a practical activity at school. Obviously they could not fold the computer screen but were able to mentally carry out this operation, a very useful strategy and one which shows the power of the original folding activity to become part of a students’ cognitive tool box.

Shape OneShape Two
Shape Three Shape Four

Some symmetrical shapes which can be generated by dragging the dynamic figure

The findings have implications for the way in which shapes and their properties are taught in schools. Working with the dynamic figure and dragging the bars to generate different shapes gives students the opportunity to explore geometrical shapes in a different way than they could using pencil and paper. It encourages them to take a more holistic view of the shape and to notice properties of the diagonals. The notion of symmetry also appears to be a significant aspect of how we view shapes and it may be more intuitive for children to explore the symmetry of shapes and use this to derive the other properties of shapes.

About Sue Forsythe

Sue Forsythe is a research student working towards completion of her doctoral degree in the School of Education.

Sue is supervised by Professor Janet Ainley and Dr Alison Fox.

School of Education
University of Leicester
21 University Road
Leicester
LE1 7RF

Sue will present her work at the Festival of Postgraduate Research 27 June 2013 - see Sue's Festival poster.

The Festival is open to all members of the University community and the public - book your place here.

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