This got me thinking - what is the purpose of constructing nets? It is supposed to help children understand and visualise the attributes of a 3D shape. Constructing 3D shapes using pre-drawn nets reduces the amount of visualisation and mathematical thinking children need to do, and can easily become a waste of time. Perhaps a better idea would be to get the children to construct their own nets. This may take more time, and may require children to persist through difficulties and multiple attempts. However, if the children are provided with appropriate scaffolding, they should come away with a deeper appreciation of the attributes of 3D shapes, than if they simply constructed a shape using a pre-drawn net.
Secondly, we discussed naming of common 2D shapes. This took up the majority of our lesson. We discussed terms such as: quadrilateral, polygon, rhombus, rectangle, square, trapezium, parallelogram...we discussed how some shapes fit into multiple categories, e.g. a square can also be classified as a quadrilateral, polygon, rhombus, parallelogram, a rectangle and a trapezium (according to the Australian definition of the word). We concluded by saying that what matters most about the teaching and learning of shapes in the early childhood years is developing children's abilities to describe and recognise attributes of shapes. This skill is more important than ascribing the right names to the shapes.
Furthermore, one of the research findings of the study summarised in Hannibal (1999) showed that when the research participants were asked to identify which shapes are triangles from a number of stimuli presented, younger children in particular often drew upon "self-determined triangle-defining criteria" (p.355). It was also shown that this occurred more frequently when other shapes which were easily recognised to not be triangles such as a circle and a square were not present in the stimuli. According to Hannibal (1999) "teachers need to move beyond having children just label shapes to having them understand what defines a shape category" (p.356). In other words, it is important to help children identify the integral attributes of a shape and distinguish these from other non-intregral attributes such as size, ratio and orientation, in order to avoid the development misconceptions in regards to what constitutes shapes such as a triangle, a square, a rectangle, a pentagon, and so forth. Other implications for some teachers may include correcting some of their own misconceptions of what are integral and non-integral attributes of these shapes.
It has been suggested that the common usage of proto-typical shapes (in books, posters or worksheets on shapes) has contributed to the development of these misconceptions held by children (and perhaps even teachers). For example, if children are used to seeing equilateral triangles but are unfamiliar with seeing isosceles or scalene triangles, they are more likely to mistaken non-integral attributes (such as "three sides have to be equal") as integral attributes of a triangle. Therefore teachers should use more non-prototypical shapes in the classroom and including different examples of shapes, in order to show that size, orientation and ratio are not integral attributes.
Thirdly, in the workshop, we participated in an activity which focused on identifying and describing attributes of shapes. This involved creating different polygons out of pieces of paper. We were encouraged to form non-prototypical shapes. Each participant created as many shapes as they could in the given time, then we were asked to sort our shapes by one attribute. The photographs below show how my group sorted our shapes by the attributes of "number of sides".
Starting with shapes that have three sides, four sides, five sides, six sides and so forth...
Then we subcategorised each of our categories. For the shapes that have three sides we divided our shapes into triangles that contain a right-angle (right-angle triangles) and those that did not contain a right angle triangle. The pink line was used to show the subdivision.
We also subdivided our four sided shapes (quadrilaterals) into two groups. We created one group of quadrilaterals that have no right angles, a group of quadrilaterals that have four right angles. Then, we further subdivided the second group (of shapes with four right angles) into shapes that have four equal sides and those who do not.
We sub-divided the five-sided shapes (pentagons) into those that contained a reflex angle and those that do not.
With the activity, the main focus was on identifying and describing attributes of shapes, and justifying the reasoning behind how these shapes were sorted. Another research finding outlined in Hannibal (1999) suggested that when the children were more likely to make correct catagorisation decisions when asked to provide explainations of the reasoning for their decisions in the triangle sorting task. Justifying decisions requires one to reflect on one's actions, which also provides opportunities for self-correction.
From participating in this activity, I also see its potential to assist teachers in identifying misconceptions held by their students (for example if they experience difficulties in counting the number of sides of a shape) as well as develop children's conceptual understandings about shape that is foundational for future geometry learning. It is only by being able to identify and describe the attributes that they can develop a comprehensive understanding of why a square is also a rectangle - which is because that its attributes fulfils both the criterion of a rectangle (a quadrilateral with four right angles and therefore have two sets of parallel sides of equal length) and a square (a quadrilateral with four right angles and all four equal sides are of equal lengths).
