List of Figures
Discussion of results
Research Study- ‘How do children learn shape?’
1. Van Hiele Levels of Geometric Reasoning
2. Full Explanations of Data Collection
3. Full explanations of tasks
4. Questionnaire given to teachers
5. Ethical Checklist
6. Blank Consent Form/Written Information Sheet
7. Raw data from investigation
8. Golden Ratio SPSS Data
9. Female KS3 pupil’s Triangle of
10. Rotated Rectangle
11. Male and Female KS4 pupils’
12. Male KS1 pupil’s similar isosceles
13. Female KS1 pupil’s congruent scalene Triangles
14. Selected Primary School Teachers’ Rectangles
15. Secondary School Teachers’
16. KS2 pupils’ Rectangles
17. KS5 pupils’ Rectangles
I am indebted to many people in helping me to compose this dissertation, of which there are a few significant individuals I want to identify for special commendation here.
Firstly, to my lecturers Ian Wood and Fiona Lawton who have provided me with invaluable support in both the formulation and production of my research study. Both have furthered me professionally; I also collaborated with Ian over an article which has since been published in the Association of Teaching Mathematics (ATM) Journal.
I would especially like to thank my former A Level Mathematics Teacher Elizabeth Best who has been an inspirational mentor, who further sparked my interest in Mathematics and made me decide to go into a career in education. She also gave me the idea to create a book on Mathematical exercises to use within the classroom which I am hoping to publish.
Thirdly, I would also like to everyone on the Secondary Mathematics QTS degree course who have always believed and supported in me; in particular Cameron Swindells, Jonathan Carver, Josh Philppotts and Becky Hamilton.
Finally and most significantly, I would like to thank my immediate family. My Mother for fostering my interest in education and always believing in and encouraging me to keep going through adversity and my Father for maturing me. Also, I would like to thank my Sister for helping raise me as a child and being the best possible sibling. They have been great friends and have always been there for me.
Is the Van Hiele Model useful in Determining how Children learn Geometry?
University of Cumbria
Copyright © 2013 Samuel James Curran
The aim of this study is to investigate how children learn Geometry (at all levels of compulsory education) in Mathematics. This study was chosen because of my difficulties in the area and the possible under-representation of Geometry in the Mathematics Curriculum. Five tasks were given to two students for each Key Stage 1-5 inclusive. These were then analysed using the Van Hiele model of Geometric reasoning; which was used to make an assessment of children’s geometrical ability. The study also draws on theoretical frameworks from eminent researchers like Vygotsky, Piaget and Bruner as well as engaging fully with current educational literature and research. A questionnaire on Geometry was also completed by a variety of primary, secondary and A-level mathematics teachers. It was found that geometrical ability increases with age (although young children can display sophisticated knowledge of shape) and that students mainly drew shapes of a non-prototypical orientation. This has increased my subject knowledge and enhanced my classroom practice and also may have the implication of changing other practitioners’ teaching strategies.
List of Figures
Figure 1- Relationships between different types of P14 Quadrilaterals
Figure 2- Inductive and Deductive Research P22 Methods
Figure 3- Pentagon and Hexagon Triangle P56 Relationship
Figure 4- Cyclic Quadrilateral Circle Theorem P58 Diagram
Figure 5- Gender data of schools involved in the P60 Study
The Department for Business, Innovation and Skills (DBIS) (2012) report that one in four adults have Mathematics skills which are deficient to those of primary school age and less than 60% of all pupils in the 2012 GCSE Mathematics cohort achieved a C or above (DfE, 2013 a). This arguably indicates that numeracy skills are not ideal in the current environment; so it may be beneficial to gain a detailed knowledge of how pupils learn so standards and attainment can be increased.
Mathematics could be perceived as ‘pure’- with more emphasis being given to historically eminent topics like Number and Algebra, as opposed to ‘applied’ modules like Geometry. Johnston-Wilder and Mason (2005) suggest that Geometry is given less teaching time in the classroom than other disciplines. There is also tangible evidence to suggest that this is inherent at all levels of mathematical study: Geometry forms none of the syllabus for the ITT Numeracy QTS Skills Tests (DfE, 2013 b) and is only present in 3 of the 9 attainment descriptors (heavily in Trigonometry but only lightly in Vectors and Logarithmic Functions) at A-Level (CIE, 2013). Furthermore, it has less teaching time than the strands of Number, Algebra and Handling Data in the National Curriculum of both Primary and Secondary School Mathematics (DfE 2011 a; 2013 c). This perceived underrepresentation does not appear to be amended in proposed curriculum reforms; Geometry forms less than a quarter of the amalgamated attainment descriptors in the draft of the 2014 Secondary Mathematics curriculum (DfE, 2013 d).
This was validated in my own experiences in learning Mathematics at school. I have few recollections of studying shape topics; many of my lessons were orientated on Number and Algebra. This study is of particular meaning to me as I experienced many difficulties in learning shape at school and developed an emotional and mental block on it which still persists to this day.
Senechal (1990) states that shape is a vital and key component in learning Mathematics and, if properly developed, can aid cross-curricular links to Science and more creative subjects. Cuoco, Goldenberg and Mark (2012) propose that thinking geometrically seems to provide an alternate perspective on life, investigations and problem-solving. Used in conjunction with a solid understanding of the ‘core’ concepts of Mathematics like Number and Algebra, this could provide a young person with real benefits in life.
For all of these reasons, I decided to conduct a research study investigating how children learn shape. By undertaking this study, I hope to increase my subject content knowledge as well as enhance and aid my classroom instruction. It could also possibly influence other practitioners’ teaching strategies and help an innumerable amount of pupils.
This literature review will focus on the eminent and recent literature which examines the usefulness of the Van Hiele Model in assessing children’s geometrical ability and theories of how children learn Geometry.
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) suggests 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, p.21) reaffirms this by proposing that children learn by exploring their surroundings and physical environment.
It could be argued that Bruner (1961, p.23) however places more importance on social learning than Piaget. Vygotsky (1962; 78) implies that children learn in a social constructivist model from More Knowledgeable Others (MKOs) and their peers in a classroom environment. Chazan and Lehrer (2012) suggest this is particularly evident in an interactive classroom setting. This seems to be an underlying criticism of Piaget’s theory of cognitive development: that he fails to recognise the social aspect of learning. Donaldson (1979) goes further in her criticisms of the findings arising from Piaget’s experiments by stating that his experiments were not appropriate and that children did not 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 could not see another viewpoint. However, Glaserfeld (1995) refutes 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 manner, at least partially. DfE (2012) identify that social learning is particularly prevalent when a child starts formal education and learns from teacher exposition and interactions with their peers. However, DfE (2009) suggest that the role of More Knowledgeable Others (such as teachers) are more important in a child’s geometrical development than their fellow pupils, particularly in practical ‘hands -on’ topics like measures and mensuration.
DfE (2011 c) states that the curriculum content of Geometry in KS1 and 2 is Euclidean Geometry, as children begin to understand the patterns and properties of 2-D shapes. 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. 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.
Based on research carried out on students in their own mathematics classes as part of composing their doctoral dissertations in 1957 in Utrecht, Netherlands, husband and wife Pierre and Dina Van Hiele (1985) devised a model of geometric levels that children progress through (See Appendix 1, p.53-59 for a more detailed version of the model):
- Level 1 (Visualisation) - Children have knowledge of basic shapes but no comprehension of their properties. They cannot link or compare shapes.
- Level 2 (Analysis) - Children understand the properties of shapes but do not use them in a comparison of shapes.
- Level 3 (Abstraction) - Children can make links between shapes based on their properties and can understand some very simple proofs, although they may struggle on more formal examples.
- Level 4 (Deduction) - Children have a good knowledge of Geometry and can use and apply some formal proofs. Children are likely to reach this level at the end of secondary school.
- Level 5 (Rigor) - This is a level of geometric understanding is equivalent to that of a Mathematician and is unlikely to be reached in compulsory education. People at this level have a deep knowledge of formal proofs and can work confidently in most areas of Geometry.
Van Hiele (1985) states a child’s initial study of Geometry in KS1 is the visualisation level (Level 1 in his model), where children can name 2-D and 3-D shapes and recognise them in the real world but possess no knowledge of their interrelating properties.
However, DfE (2011 a) state that children are taught to make connections between shapes from KS1. This seems to imply that children will have some knowledge of the common 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 such as comparing the number of equal sides. Mitchelmore and Outhred (2004, p.467) observed that this is often done in comparison with everyday objects; for example, a rectangle is formed in the mind because it looks like a box. Carraher, Nunes and Schliemann (1993) characterise this as a child making a link between the dichotomy of school and street mathematics. 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 could be applied to objects in everyday life. Furthermore, the Van Hiele model (1985) does not acknowledge that children learn in a number of different ways; Baume and Fleming (2006, p.5) suggest children mostly learn through multiple representations of a problem. This and the success of using multisensory approaches in teaching may influence the rate of progression in children’s geometrical knowledge.
French (2004) states that 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 (Ofsted, 2012 a). There does seem to be evidence that children are influenced by deductive teaching, particularly in their approach to prototypical images.
Kerslake (1979, p.34) investigated whether primary-school-aged children could recognise angles and shapes of different orientations; she found that most pupils only correctly identified the ‘typical’ image (the orientation of angle/shape that was normally drawn by their class teacher) whereas ‘atypical’ images were not recognised. Burger (1986, p.41) rationalises this as younger children often believing a rule based on one example, normally from the class teacher, and being unable to extrapolate this to other shapes.
This research seems to validate Piaget’s assertion that, at this stage, children view figures holistically without any realisation of their properties. However, Sperry’s (1961, p.1750) hemispheric dominance theory suggests that children who have a natural ability in Geometry, normally those with a predisposition to the right cerebral hemisphere, may not be categorised by this developmental model. Carter (2004) disagrees with Sperry’s theory and questions the validity of it. Although there may be some disagreement with the various cognition theories, it could possibly be assumed that only some children can make connections between shape at this stage of learning.
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).
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. Ofsted (2012) compare this to the situation in GCSE exams where children are generally able to competently solve ‘method’ questions but sometimes struggle with ‘worded’ or ‘applied’ problems.
Roughly when a child starts secondary school, they enter the Van Hiele abstraction stage where they can compare shapes and make connections between them such as in the diagram below:
illustration not visible in this excerpt
Figure 1- Relationships between different types of quadrilaterals (Haylock, 2010)
The progression to a child thinking in a slightly more abstract manner and 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. DfE (2013 d) highlight that 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.
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 C o-ordinate Geometry in Secondary School in modules like transformations (DfE, 2013 d). It may also be beneficial to study Vector Geometry, both at secondary school (translations) and A-Level (magnitude, direction, scalar product and equation of a vector).
However, Haggerty (2001) asserts not all geometrical learning is linear and discrete; it can be discontinuous as pupils develop at different rates. 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 Organisation for Economic Co-operation and Development’s (OECD) (2008) 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. Piaget (1953) argues that children do not enter the formal operational stage until they are 14 and that they cannot learn formal proofs before this period. Van Hiele (1985) 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 the DfE (2011 b) states that proofs are not usually covered until Year 10 although some students study it in Year 9 in accelerated study programmes.
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. Parliament (2012) and Ofqual (2012) perceive this to be a weakness of the course, which could stop pupils reaching the final Van Hiele (1985) level of rigour, where a pupil has mastered all the axiomatic structures of Geometry and can confidently deal with Non-Euclidean Geometries.
In learning Geometry, pupils seem to develop from pure and synthetic Geometry (Euclidean) but need to have an understanding of Algebra to understand more sophisticated levels of analytic (Algebraic Geometry). There may be a finite level of geometrical reasoning that a student can reach and that their understanding of Geometry will eventually plateau.
Research carried out by Senk (1989, p.308) and Gutiérrez and Jaime (1998, p.37) describes the Van Hiele model positively and highlights its impact on the American Mathematics Curricula. However, Burger (1986, p.41) highlights a deficiency of the model as the levels of knowledge within it are discrete, not continuous, and in some cases overlap, with children sometimes displaying reasoning at numerous levels simultaneously. Conversely, this may actually be an asset of the theory: it could be more widely interpreted and thus may be applicable to more research studies.
The literature seems to suggest that determining whether the Van Hiele Model is appropriate in assessing children’s geometrical abilities is something which needs to be examined. The potential impact of knowledge of the Van Hiele model may have on teaching and learning also seems to be a relevant issue to be considered.
Throughout my teaching practice and career I have always tried to be a reflective practitioner and recognise what needs to be changed about my own and possibly whole school practice. Hubbard and Power (1999) and Bell (2005) argue that developing this thoughtful style may allow me to assess pupils more in-depth in this investigation to facilitate more accurate analysis and demonstrate good practice in conducting my research study.
My model of research is not essentially interactive as it is being individually conducted by me. Vygotsky (1978) argues this modality of inquiry may deprive me of the possible collaborative benefits of a social constructivist model of research. According to Freire’s (1982, p.30) theory, my style of action research is still participatory as I am trying to enforce change using a reflective approach, but only on an individual level. Dadds (1998; 2009) argues that trying to enforce change is a key attribute of practitioner research which is something I am trying to do in my study. Ollerton (2004) highlights this as the key distinction between a reflective practitioner and a practitioner researcher, actually doing something specific about the issue. However, I may be able to amalgamate the most desirable assets of both roles in my study by changing things but also being reflective in my practice.
I am using what I term a ‘peflective’ paradigm in my approach to this research study (Curran, 2013). ‘Peflective’ signifies for me that I am taking a positivist viewpoint with a reflective element.
The style of action research implemented in my study is very reflective and cyclical by identifying a relevant theory, collecting data with my tasks and reflecting and reacting to the research by changing mine and possibly influencing other teachers’ strategies in teaching shape. However, McNiff and Whitehead (2002) highlight that there may only be limitations to what I can change, something which indicates I may need to keep improving and refining my practice.
I am using a positivist paradigm in my approach to this research study. One theory of how children learn Geometry, Van Hiele’s (1985) model of geometric reasoning, is used to construct my research study. I am taking a ‘realist’ view of the classroom environment as I am summative assessing children’s geometrical ability in my research study by analysing their scores using the Van Hiele model (1985) to produce quantitative data. Variables are to be very tightly controlled in this test, and I believe that I can control all of them adequately: the tests will take place in non-mathematical rooms at each school and each child will have access to the same equipment, instruction and resources. Hudson and Ozanne (1989, p.2) recognise that, although a positivist approach is logical and may produce objective data, it may not address the underlying cause and reasons for the data occurrence which an ‘idealist’ ontology might yield.
Carson et al. (2001) state that a researcher using a positivist ontology stays emotionally detached of the setting and research process. This is something I would like to create in my study. However, I will have to work hard to ensure this neutrality given that the research is being conducted in my former educational establishments, which may induce understandable emotional attachments.
The epistemological viewpoint I have taken is also positivist as I believe in quantifying intelligence through tests, measurement and observation.
Conversely, Stenhouse (1974) is a proponent of the interpretivist approach which he feels has rigour as the power of research lies with the teacher. However, the potential weakness of a positivist methodology may be negated by the logical positivism that is used- the assessment made of the pupils is being made using a fairly reliable theoretical framework. Howson and Urbach (1993) advocate the credentials of logical empiricism, something which I have used as tasks 4 and 5 rely on the scientific verification of prototypical images which seems a reliable framework on which to base my conclusions on.
I have used a mixed methods paradigm in my collection of data. Johnson et al. (2007) define this as collecting a mixture of qualitative and quantitative data and using both viewpoints to justify my conclusion which is what I have tried to implement in this study. This is exemplified by the duality of my approach in analysing task 4 where participants are asked to draw a rectangle that looks visually appealing. Although the analysis is partially positivist in conducting a statistical test (non-parametric one sample t-test) and using known theoretical research (about the golden ratio), it is also interpretivist as the test used is inferential so a supposition about the data can be assumed. Furthermore, the results will be related to the literature review and my own observations to see how useful the Van Hiele Model is in assessing how pupils learn Geometry.
I have taken a deductive approach in writing my literature review (see below diagram) as I examined theoretical approaches in order to structure my approach but have implemented an inductive method of data collection as results are collected and then related to practice. The fusion of these 2 approaches may be complementary as it could allow me to gain a deep knowledge of what I have researched and enact what I have learned in my classroom practice (Weick, 1979).
illustration not visible in this excerpt
Figure 2- Inductive and Deductive research methods (Newton and Rudestam, 2012)
In conclusion, the methodology I have used seems sound as it tries to complement a positivist research paradigm with a mixed methods paradigm in data collection.
Data Collection (See Appendix 2, p.60-61 for more detailed information)
The tests for my study consisted of 10 pupils completing 5 geometrical tasks (see Appendix 3, p.62-70 for a detailed explanation on them) and 37 teachers also completed a questionnaire (See Appendix 4, p.71-72) to ascertain their views on Geometry which was then compared with the data from the pupils. Likert (1932) proposed a standard 5-point answer scale for questionnaires. However, I chose a 4-point scale by eliminating the ‘Neither Agree Nor Disagree’ option so I could eliminate neutrality and gain a stronger polarity of opinion.
The data were collected at a primary and secondary school and a sixth form college in the same town all within a two mile radius in the North East of England.
A teacher-selected, systematic sample was used in the collection of the data. 2 pupils were tested from each Key Stage 1-5 in a systematic approach by taking the same number of pupils from each stage.
The sample size of 10 students may not be entirely statistically reliable. Bartlett, Kotrlik, and Higgins (2001) articulate larger sample sizes as being generally accepted to have increased precision and statistical power, whereas reduced samples tend to have decreased confidence intervals and a greater susceptibility to outliers. This seems to be evidenced by the dubiousness of whether the results of this study would be replicated in a larger investigation.
On the other hand, Haeussler, Paul and Wood (2013) advocate the advantages of a small sample size being expedient and necessary as it allows data to be collected and analysed efficiently although they recognise the potential limitations in accuracy a small sample size could have. The cognitive differences in the age of the subjects involved in the study seem to validate this: developmental differences seem far more prominent amongst children than adults (Piaget, 1952; 1953).
All tests are analysed using the Van Hiele (1985) model of Geometrical reasoning (See Appendix 1, p.53-59 for a more detailed version).
Tests (See Appendix 3, p.62-70 for more detailed explanations of tests)
1. Estimating length of line
2. Estimating the size of an angle
3. Draw a right angle
4. Draw 4 different types of triangle
5. Draw a rectangle that looks nice
Van Hiele Level: 2 Van Hiele Level: 2 Van Hiele Level: 2 Van Hiele Level: 2 Van Hiele Level: 3
Throughout the research study, an ethical approach was followed at all times (See Appendix 5, p.73-74). Particularly due to the young age of the participants in the study, full school and parental consent was sought and obtained (See Appendix 6, p.75-76) and a transparent and safeguarding approach was followed at all timesschools were fully involved in the testing procedure; all appropriate protocol was followed and participation was entirely voluntary and the children and school had the right to withdraw at any time. Permission also needed to be gained for the possible publication of the study.
The ethnicity of most of the pupils at the schools surveyed was White British.
However, all students were selected free of bias and no discrimination was made at all, particularly for cultural factors such as religion and race. Furthermore, due to the importance of the study, it was ensured that the benefits were reciprocal and that the research was challenging. The schools will have a more detailed knowledge about how their pupils learn Geometry and the study will influence the author’s and other practitioners’ teaching strategies when teaching shape. In addition, an original approach has been followed as the study is completely of my own design and examines a field which has not been extensively researched as other areas of Mathematics.