## How do children learn Geometry? (Sam Curran)

Piaget (1953; 1960; 1967) suggests that a child’s initial geometrical discoveries are topological: that they can recognise the boundary aspect of space and distinguish between open and closed figures from the age of 3. Piaget (1953) suggest this development seems to be formulated during the latter sub-stages (tertiary, circular reactions, curiously and novelty) of the formative sensorimotor stage when a child interacts with the world around them and begins to explore the properties of new objects. Bruner (1961), a critic of Piagetian theory, reaffirms this by proposing that children learn by exploring their surroundings and physical environment.

However, Bruner (1961) places more importance on social learning than Piaget and implies that children learn in a social constructivist model from More Knowledgeable Others (MKOs) and their peers (Vygotsky 1962; 78). Perhaps this is particularly evident in an interactive classroom environment. This seems to be an underlying criticism of Piaget’s method: that he fails to recognise the social aspect of learning. Donaldson (1979) goes further in her criticisms of Piagetian theory by stating that his experiments weren’t appropriate and that children didn’t understand what the tasks required to them to do. Hughes (1986) supports this and also states that due to the arrangements of the task children were limited to egocentricity and couldn’t see another viewpoint. However, Glasserfeld (1995) refuted these criticisms and attributes the children’s lack of understanding to the conceptual difference of the mistranslated Piaget text. Regardless of the agreement of the various cognition theories, there seems to be some truth that children learn Geometry in a social aspect. This may be particularly evident when a child starts formal education and learns from teacher exposition and their peers. A child’s formative study of Geometry in school is quite heavily centred towards Euclidean Geometry which consists of understanding the patterns and properties of shape (DfE, 2011). Both Piaget (1967) and Bruner (1961) allude to the concept of prototypical images, where an image of a shape is constructed in a child’s mind and stored for later use, although both describe it in different ways. This learning style could perhaps be fostered if there was a numerous amount of teacher exposition and board work in lessons. Piaget (1953) theorised that children have symbolic schemata which are mental pictures or images or what they have experienced in lessons. Bruner (1966) described this method of remembering images as iconic. Van Hiele (1985) seems to further support these by identifying this as level 1, visualisation, of his model of geometrical reasoning where children have a simple understanding of basic geometrical figures like shapes, but no knowledge of their interrelating properties.

However, it could be argued that students have some knowledge of the connectionist properties of Euclidean Geometry; in a Piagetian sense by linking it back to previous schemata and also by using visual prototypes to identify other shapes in the Van Hiele model. This is often done in comparison with everyday objects; for example, a rectangle is formed in the mind because it looks like a box. It could be conjectured that this formative geometric reasoning is normally only applied to Euclidean spaces (shapes or figures which a defined by a set of axioms or postulates) in the school environment but can be applied to other spaces outside of this.

At primary school level, the teaching strategies used are often a mixture of inductive (practical investigations and kinaesthetic activities) and deductive (formal teaching and exposition) which constitute the first stage of Geometry teaching (Mathematical Association, 1938, cited in French, 2004, p.54). There does seem to be evidence that children are more influenced by deductive teaching particularly in their approach to prototypical images. Kerslake (1979) investigated whether primary-school-aged children could recognise angles and shapes of different orientations and found that most only correctly identified the ‘typical’ image (the orientation of angle/shape that was normally drawn by their class teacher) whereas ‘unfamiliar’ images were not recognised. This is perhaps explained by the fact that younger children often believe a rule based on one example, normally from the class teacher and are unable to extrapolate this to other shapes (Burger, 1986).

This research seems to validate Piaget’s assertion that at this stage children view figures holistically without any realisation of their properties. However, children who have natural ability in Geometry, normally those with a pre-disposition to the right cerebral hemisphere (OECD, 2002), may not be categorised by this developmental model, although current research has concerns over the validity of this model (Carter, 2004). Another possible flaw in Piaget’s theory is that the levels are classified by stages where subjects are at an approximate age: progression in Geometry is not necessarily linear. The seemingly more flexible Van Hiele Model (1985) may be a better indicator of assessing a child’s progress in Geometry as it is based on instruction and experience rather than age.

Upon learning about angles and lines, children may be able to make better links between shapes. Van Hiele (1985) termed this level as analysis where pupils could understand the properties of shape but not yet link them. Piaget (1967) and Bruner (1961) both support this in their respective pre-operational stage and symbolic models, although Bruner recognises that these seemingly autonomous mental structures can be blended together and related depending on the age and experience of the child. This model is in sharp contrast to Piaget’s age-centric theory; Bruner philosophises that a child can learn any task given the right teaching. Again, perhaps due to the abundance of command, teacher-led strategies (Mosston, 1966) pupils may develop an inflexible arbitrary knowledge of Geometry which can be a barrier to progression (Hewitt, 1999). Some students particularly struggle with angle and cannot quantify or grasp the ‘big idea’ that it is a measure of turn (Ofsted 2008; 2012).

Strategies to overcome this could include having more varied examples of a shape’s orientation and for the teacher to develop a connectionist orientation (Askew et al., 1997) and make more links between topics. Teachers could also try to promote higher-order thinking (Bloom, 1956, cited in Krathwohl, 2002, p.123) by encouraging children to analyse what they have learnt, although the process of carefully examining what they have been taught may be too complex for children to grasp at high school level.

Once children have grasped the basic notions of angle and shapes, they can begin to make links between them. At the Piagetian concrete operations stage, a child can think logically and solve problems which are heavily generalised and require an inductive manner of thinking. This is perhaps comparable to GCSE exams where children are generally able to competently solve ‘method’ questions but sometimes struggle with ‘worded’ or ‘applied’ problems (Ofsted, 2012). This obstacle to progression in Geometry at this level is perhaps explained by the heavy influence of Euclidean Geometry on the Primary and Secondary Mathematics curriculum. This could possibly be rectified by introducing non-Euclidean geometries such as eliptic geometry. However, as the child is probably not likely to have mastered 3-Dimensional thinking and may not have encountered plane geometry, this may not be advisable.

Roughly when a child starts secondary school, they enter the Van Hiele abstraction stage where they can compare shapes and their properties.The progression to the child thinking in a slightly more abstract manner and the knowledge of the properties of 2-D shapes may help a child to understand plane geometry and that of 3-D polyhedra and platonic solids such as cubes and tetrahedrons. A knowledge of Euclidean Geometry and a developing knowledge of spatial awareness (through studying topics like tessellations) is conducive to understanding affine geometry, the study of parallel lines which is introduced in Mathematics at KS3 Level (DfE, 2013).

The progressive and interrelating nature of Geometry seems to be further represented by the fact that knowledge of affine geometry can help with understanding of the beginnings of co-ordinate and graphical geometry in Secondary School in modules like transformations. It may also be beneficial to mastering vector geometry both at secondary school (translations) and A-Level (magnitude, direction, scalar product and equation of a vector).

However not all geometrical learning is linear and discrete; it can be discontinuous as pupils develop at different rates. Many pupils develop misconceptions on plane geometry and require technology support packages such as LOGO and Cabri which can be transferred to paper so pupils can learn in an enactive manner (Bruner, 1961; French, 2004; Johnston-Wilder and Mason, 2005). A possible criticism of the Piagetian and Van Hiele models is that they are heavily generalised and do not account for variations in ability. Furthermore, Piaget’s model is domain specific and surmises that cognitive development is homogenous across all fields which may not be true; as the distinction between pure and applied mathematicians implies. It seems that a clear understanding of all the fields of Geometry is needed before a child can develop deductive logic and understand formal Euclidean proofs such as proving there are 180 degrees in a triangle and congruency. Piaget argues that children don’t enter the formal operational stage until they are 14 and that they are not cannot learn formal proofs before this period. Van Hiele describes similar properties in his penultimate geometric level deduction although he does not specify which age pupils reach this level. This seems to be supported by the curriculum as proofs are not normally covered until Year 10 although some students study it in Year 9 in accelerated study programs (DfE, 2011). However, in many cases pupils do not reach this level due to the perceived ‘test-teaching’ nature of proofs in school; formulas and rules are often learnt by rote with no understanding of the mathematics behind them: for example the hypotenuse of a right-angled triangle used in Pythagoras’ Theorem is typically learnt as an arbitrary concept ‘c’ rather than a necessary fact that it is ‘always opposite the right angle’.

A supposition could be presumed that all students need a good comprehension of Algebra to comprehend more sophisticated Geometry topics. This seems to be evidenced by curriculum content; GCSE and A Level Mathematics contain more Algebraic Geometry and a reduced amount of Euclidean Geometry. Indeed, Geometry in A Level Mathematics is almost exclusively made up of Co-ordinate and Differential Geometry (Calculus) and some Trigonometry with barely any pure, Euclidean Geometry. Van Hiele expressed an additional level, rigour, although this seems unlikely to be achieved unless a student studied Mathematics at degree level and mastered all the axiomatic structures and can confidently deal with Non-Euclidean Geometry. In conclusion, a broad (and not necessarily linear) generalisation of the way in which children learn geometrical concepts seems to concur with the fields of Geometry in reverse chronological order in which they were discovered. The Euclidean axiomatic system (and formal proofs) was the first area of Geometry to be discovered over 2,000 years ago (Heath, 1967) whereas projective geometry was invented around the 17th century and finally topological geometry was formulated in approximately the 19th century (Piaget, 1953). In learning Geometry, pupils seem to develop from pure and synthetic geometry (Euclidean) but need to have an understanding of Algebra to attain more sophisticated levels of analytic (Algebraic Geometry). It could be suggested that there is a finite level of a geometrical reasoning that a student and that their understanding of geometry will eventually plateau.

The literature suggests issues as follows which will be addressed in this research study:

• Are the Van Hiele Levels of Geometric reasoning an accurate way of assessing children’s geometry skills?

• How could an awareness of the Van Hiele and other theories change teaching strategies so the children progress as much as possible in Geometry?