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The Van Hiele model describes levels of thinking through which students develop when reasoning about geometry. It recognises five progressive levels of geometric thinking that learners pass through. According to the model, the student, supported by suitable instructional experiences, goes through these levels in a hierarchical order, starting with recognition of figures as a whole (level 1), progressing to the understanding of the properties of figures and informal reasoning about these figures and their properties (levels 2 and 3), and ultimately reaching a formal deductive and rigorous learning of formal geometry proof-writing (levels 4 and 5) (Van Hiele, 1986; Fuys, et. al., 1988). The following is a description of the Van Hiele levels:

Level 1: Recognition

Students at this level, which is also called visualisation, have the ability to learn the names of figures and view the figures as a whole according to their appearance only. For example, if they are asked to explain why a particular quadrilateral is a rectangle, the typical response would be, “because it looks like a rectangle” (Shaughnessy & Burger, 1985, p. 420). The student in this case associates a rectangle with other objects that are shaped like a rectangle, such as a door, window, and so on, from his previous encounters. He is able to name the figure

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but properties of the figure do not feature at all in his ability to recognise the figure. At this level, the student can, for example, differentiate between a rectangle and a square. However, he does not do this because of the knowledge of the properties of these two figures but because a square and a rectangle “seem to be different”. Because of the inability to understand definitions (which are based on the properties of the figures), the student at this level is not able to, for example, recognise a square as a special case of a rectangle.

Level 2: Analysis

At this level, also known as the descriptive level, students do not view a figure only as a whole, but rather by its features as well. They can identify the properties of figures, for example, they can recognise and describe a square as a figure that has all its sides equal. However, they see these properties discretely, without the ability to understand that there exists a relationship among these properties or that there are closely related classes of figures (Van Hiele, 1986; Usiskin, 1982). For example, the student might be able to give a list of the properties of a square without relating these to each other, such as mentioned “it has four equal sides and four corners”, even though the four corners are implied in the four sides. Therefore, the student does not yet have insight into which properties are essential or which ones are sufficient to define a figure.

Level 3: Ordering

Students at this level, which is also called informal deduction, can recognise the relationship between different properties of a figure as well as between the properties of different figures. They use the properties that they already know to describe shapes and reason about relationships between shapes. A student can now see how one figure could be described in different ways if it shares the same properties as another figure. For example, the student can reason that, since all the properties of a rectangle are also properties of a square, then a square is a rectangle, but a rectangle is not necessarily a square. Therefore, a basic deduction can be made. However, students at this level do not understand the role and significance of proof or formal deduction. (Mayberry, 1983; Pusey, 2003).

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Level 4: Formal deduction

At this level, the student understands the significance of deduction and can understand the logical development of a proof as well as appreciate the role of postulates, theorems and proof (Usiskin, 1982). They can produce a logically sound argument and conclusion. Definitions involving essential and sufficient conditions are now understood. Students can define things and limit them with a minimum amount of information, compared to the “laundry list” associated with the second level (Pusey, 2003). For example, it is sufficient for a figure to have four sides if it is to be identified as a quadrilateral but it is necessary for all the sides to be equal in length for it to be a square or a rhombus and it is necessary that all four angles be right angles for it to be a square.

Level 5: Rigour

The fifth and final level is described by students being able to go between geometries (Euclidean vs non-Euclidean) and think outside of one axiomatic system (Pusey, 2003). The essence of geometry and the necessity for rigour is appreciated. Pupils can accept logically correct proofs even if the concepts are counter to general intuition. Geometry is seen in the abstract with a high degree of rigour, even without concrete examples (Khembo, 2011). This level is the least developed in the original works of Van Hiele and has received little attention from researchers (Crowley, 1987). Furthermore, since the majority of high school geometry courses are taught at level 3, it is not surprising that most research has also concentrated on lower levels.

The Van Hiele levels have been modified since their original inception (Pegg & Davey, 1991). The original Van Hiele levels were numbered from 0 to 4 and Van Hiele claimed that all students were at least at level 0 (Senk, 1989). However, in his research he was working with secondary students who did not operate at a level below 0. With more research involving elementary students, researchers saw it necessary to classify geometric thinking that was below the Van Hiele introductory first level (level 0). Rather than categorise these students as being below level 0, Van Hiele and some other researchers introduced a system that renumbered the levels 1 to 5 to provide for this (Hoffer, 1981, Van Hiele, 1999; Atebe, 2008). Therefore, a student not at the first level, that is, previously below 0, was now assigned level 0, known as the pre-recognition level (Pusey, 2003).

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While students at the visualisation level (level 1) can describe shapes based on their appearance, students at the pre-recognition level (level 0) recognise only a few of the shape’s visual features. For example, even though they might be able to distinguish between a rectangle and a circle, which are rectilinear and curvilinear shapes respectively (Atebe, 2008), they might not be able to distinguish between a circle and an oval (both curvilinear). Students at the pre- recognition level, therefore, can reason about objects that are “the same shape” by only focusing on certain visual or tactile properties (Mansi, 2003). In this study, all references made to research studies that used the level 0 to 4 numbering system have been modified to the level 1 to 5 numbering system.