3.2 The theories underpinning this study
3.2.1 Van Hiele theory of geometric thinking
The van Hiele theory originated from companion dissertations (that is, they worked in a similar area of geometry research but focusing on its different aspects) which were completed simultaneously at the University of Utrecht, The Netherlands, in 1957. Pierre van Hiele devoted his lifetime clarifying, amending, and advancing the theory after Dina died shortly after completing her dissertation. Their work has come to be known as the van Hiele theory, and has helped shape and direct much of the research investigations associated with geometry around the world.
The roots of their theory are found in the theories of Piaget (1978). However, since the van Hiele dissertations and early articles were in Dutch, their findings were not widely disseminated outside Holland until a paper presented in 1957 by Pierre van Hiele to the mathematics education conference brought the theory to the attention of the mathematics education community. The Soviet Union (Russia) educators and psychologists found the paper to be of particular interest and undertook major revisions of their geometry curriculum based on this theory.
Contrary to the claims of Piaget, Inhelder, and Szeminska’s (1960) theory, the van Hiele theory suggests that learners progress through levels on the basis of their experiences rather than age, and as such it is imperative that teachers provided experiences and tasks so that learners could develop along this continuum (level 1 to level 4) (Breyfogle & Lynch, 2010). Further, though Piaget et al.’s (1960) theory attempts to explain why learners find geometry difficult, what sets the van Hiele theory apart is its strength in the suggestion of phases of alleviating the problem. Piaget et al.’s (1960) do not go that far. The second strength of the van Hiele theory is that, unlike Piaget el al’s theory which applies to several areas of mathematics, it was developed specifically for geometry.
The theoretical frameworks The theories underpinning this study
The work of the van Hieles has been presented in Fuys, Geddes, and Tischler’s (1984) English Translation of Selected Writings of Dina van Hiele-Geldof and Pierre M. van Hiele as part of the research project investigating the van Hiele theory on how people learn geometry. They and subsequent researchers have demonstrated that the van Hiele theory can help improve geometric understanding (Vojkuvkova, 2012; Pusey, 2003) and as such ‘has become the most influential factor in the American geometry curriculum’ (Van de Walle, Karp, & Bay-Williams, 2010, p.
309), including studies from a South African perspective (for example, Atebe & Schäfer, 2011; de Villiers & Dhlamini, 2013; Luneta, 2015; Siyepu & Mtonjeni, 2014; van der Sandt, 2007; van Niekerk, 2010). That is, the van Hiele has had tremendous influence on geometry education reform in the last half of the twentieth century (Ndlovu, 2013).
The van Hiele theory of learning geometry and de Villiers’ (1990) model for the functions of proof influenced my thinking about of the functions of proof in mathematics. The discussion of the theory is modelled around the van Hiele theory’s three aspects: characteristics of the levels;
properties of the levels; and phases describing steps to help learners progress from one level to the next. This study is confined to the first two aspects; a discussion of the third aspect is beyond the framework of the present study. One remark is worth making, nonetheless. For learners to make progress from one level to the next, the learning process should move through five phases which are not strictly sequential: information, guided orientation, explicitation, free orientation, and integration.
3.2.1.1 General characteristics of the van Hiele theory
The general characteristics of the van Hiele theory are that it is sequential and each level builds on the thinking strategies developed in the previous one. The levels are hierarchical in that advancement to the next level is a function of mastering the thinking strategies of the preceding level(s). Ideas and concepts that are only implied at one level become the objects of study at another level and so become explicit. Each level has its own language and symbols. Therefore, learners working at different levels cannot understand each other’s explanations even though they may be describing the same shape or idea. Also important is that teaching needs to match learners’
thinking and language. So, if the learner were at different levels, learning cannot take place and as
The theoretical frameworks The theories underpinning this study
a consequence, progress would be stunted. As already mentioned, progress from one level to the next is more dependent on and can be accelerated by instruction and experiences than age.
3.2.1.2 The five levels of the van Hiele theory
Pierre van Hiele, reporting on the studies that he together with his wife conducted, identified five levels through which learners develop their thinking in geometry. Although originally the van Hieles numbered the levels from 0 to 4, I adopted the American numbering scheme and labelled the levels from 1 to 5. Here I took the liberty to provide an overview of the Levels in the van Hiele (1986, pp. 39-47) theory. In this study, I focused on both general and behavioural terms of the levels but described Level 4 in some detail as it pertained more to this study.
The theory describes the basic Level 1 (visualisation/recognition) as one in which learners recognise shapes on the basis of their physical global, holistic characteristics, like size or position, and therefore formulate their ideas based on visual perception (Usiskin, 1982). At this level, learners need to learn the vocabulary of geometric shapes by comparing the shapes to known prototypes to be able to identify, reproduce, and name a shape as a whole, but not in any orientation (Feza & Webb, 2005). At Level 2 (analysis/description), learners describe the properties of geometric shapes through investigations and practical methods and acquire the appropriate technical terms to make generalisations for classes of figures. Level 3 (informal deduction/ordering) entails learners making sense of definitions although these may be expressed in minimum terms (Lim, 1992). For example, they begin to understand what is meant by the term
“proof” in mathematical sense. They also understand the interrelationships between the properties of shapes and see that new results can be obtained by making short chains of deductive arguments based on properties learned from concrete experiences but they may not be able to derive such proofs themselves (Senk, 1989). It is important that at this stage (Level 3) learners are provided with the opportunities to explore, feel and see, build, take shapes apart, and make observations about shapes they created with drawings, models, and computers (Van de Walle, Karp, & Bay- Williams, 2007). These activities involve constructing, visualising, comparing, transforming, and classifying geometric figures.
The theoretical frameworks The theories underpinning this study
Level 4 entails understanding of the functions of proof, definitions, axioms, and theorems and making longer chains of deductive arguments (proof) (de Villiers, 1997). The thinking is concerned with logical deduction of new results from axioms, definitions, with theorems and their converses, and the necessary and sufficient conditions in proofs (Crowley, 1987). For instance, on the strength of knowing that, “given parallel lines cut by a transversal, alternate angles are equal and that angles on a straight line are supplementary”, a learner can deduce that the interior angles of a triangle are supplementary. As already mentioned, this study was designed to focus on the deductive level which requires learners to be able understand and use the ideas of the Euclidean geometry system. More particularly important, the focus was on one aspect of Level 4 where learners are supposed to understand and hold appropriate understanding of the functions of proof in mathematics. As a consequence, this project sought to explore and understand whether Grade 11 learners’ level of geometric thought through measurement of their understanding of the functions of proof. Generally, the van Hiele theory is premised on the understanding that successful construction of proof depends on experiences in thinking at lower levels and specifically an appreciation of its functions in mathematics.
In Level 5 (abstract/rigour), the highest level of the van Hiele theory of development, learners manipulate geometric axioms, definitions, and theorems to compare and establish non- Euclidean geometries. As the label of the level suggests, non-Euclidean geometry is less intuitive and the Euclidean system of axioms that high school learners are accustomed to, are modified. It is worth noting that the first four Levels are the ones mostly pertaining to school geometry and Level 5 is meant for tertiary level geometry courses. Non-Euclidean geometries can also be identified in examples like spherical, elliptical, and hyperbolic geometries. More recently, there has been growing interest in transformation, fractal, turtle, analytical and vector geometries.
3.2.1.3 Critics of the van Hiele theory
I have cited the main ideas emphasised in the theory and illustrated how the main aspects of the theory is related to the research problem. Though I gave an exposition of the theory, to offer a balanced argument, I introduce into the discussion the main proponents and critics of the theory.
On the one hand, the van Hiele theory is regarded as one of the best framework known for teaching
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and learning geometry to date (Wu & Ma, 2006). Whilst van Hiele's model of geometric thinking is of undoubted value in geometric education, I was mindful that there have been some criticism associated with the application of some of its notions of levels and its hierarchical aspect. On the other hand, some studies that have raised questions about some of the characteristics of the theory.
For instance, Burger and Shaughnessy (1986) argue that the theory fails to detect the discontinuity between levels and found instead that the levels were dynamic and of a ‘more continuous in nature than their discrete description would lead one to believe’ (p. 45).
Although agreeing with the assertion of the van Hieles that each level has its own language, the study by Fuys, Geddes, and Tischer (1988) also found that learners’ progress was marked by oscillation between levels in different geometric content. Also, Gutierrez, Jaime and Fortuny (1991) found that students can develop more than one level at the same time. As Pegg and Davey (1998) argue, van Hiele’s broad propositions ‘are not as black and white as they are often portrayed to be’ (p. 114). Regarding the levels even van Hiele (1986) himself expresses doubt about the existence or testability of levels higher than the fourth and considered them as of no practical value.
I concur with Clements and Battista (1992) suggestion that a pre-recognition level at the lower end of the levels (Level 0) needs to be added to accommodate learners who cannot even identify shapes. Mason (1998) points out that although in terms of the van Hiele theory, a learner cannot achieve one level of understanding without having mastered all the previous levels, research studies in the US and other countries have found that the levels are not sequential as claimed; some mathematically talented learners appear to skip levels, suggesting that they may have developed logical reasoning skills in ways other than through geometry instruction.
For this study, the only drawback related to the learning phases which are meant to move a learner from one level to the next in the van Hiele theory. Although appreciation of the systematisation function of proof is supposed to be achieved at Level 3 (informal deduction/ordering), the need to develop an understanding of the functions of proof is only explicitly introduced at Level 4. I am inclined to suggest that within Level 4, two sublevels,
“Functional understanding of proof” as well as “Argumentation” be introduced prior to construction of proof. This view is consistent with de Villiers and Njisane’s (1987) suggestion that
The theoretical frameworks The theories underpinning this study
the theory needs refinement with regard to the levels at which deduction takes place. In addition, empirical research suggests that functions such as explanation, discovery, and verification can be meaningful to learners at Levels 1 and 2 when introduced gradually. For example, de Villiers (1996) points out that the function of communication pervades geometry education. De Villiers (2004) argues that a prolonged delay renders later introduction of proof as a meaningful activity even more difficult and may also make learners become accustomed to seeing proof as just a means to verify. Despite the criticisms of the theory, it is nonetheless supported by other experts in geometry education as key in understanding learner thinking.