Using the Van Hiele Theory to Explain Pre-Service Teachers’ Understanding of Similarity in Euclidean Geometry

Abstract

Helping learners to develop a solid grasp of geometric concepts poses a challenge for teachers. Therefore, it is important that teachers have a sound understanding of the geometry they teach. The aim of this qualitative study was to explore pre-service teachers’ (PST’s) understanding of the concept of similarity in Euclidean geometry and to use van Hiele’s theory to explain misconceptions evidenced by the PSTs. Data in this study were collected from 34 first-year PSTs studying for a Bachelor of Education degree in high school mathematics. The authors analysed the written responses to a 13-item worksheet and also conducted interviews with seven of the participants. The analysis of the data was guided by van Hiele’s theory which was used to identify misconceptions amongst PST’s who had not yet developed the appropriate reasoning skills linked to particular van Hiele levels of geometric thought. It was found that these students used reasoning that is characteristic of the elementary levels to make judgments. Many PST’s faced challenges with similarity notation and the process of proving the similarity between two figures. This study recommends that PST’s should be given more opportunities to connect visual and analytic representations of similarity.

1. Introduction

There is a concern in South Africa about the effect of curriculum reform processes where the Euclidean Geometry strand was excluded from and then later included in the Further Education and Training (FET) band comprising Grades 10–12 [1,2]. From 2008 to 2013, Euclidean geometry was made an optional section of mathematics in the FET band because of the perception that it presented challenges to teachers and learners [1,2]. During this time, many learners chose not to study the strand—less than 4% of the Grade 12 mathematics learners in 2008 opted to do the geometry examination [3]. Some chose to opt out of geometry to have enough time to focus on compulsory papers to boost their overall marks [2,4,5]. Atebe and Schäfer [5] noted that teachers avoided the teaching of Euclidean geometry in schools because of low confidence. However, with the new Curriculum Assessment and Policy Statement (CAPS), Euclidean geometry was reinstated as a compulsory examinable section forming part of the second examination paper in mathematics from 2014 in grade 12 [6]. With this change, it became compulsory for teachers to teach Euclidean geometry regardless of their level of proficiency or familiarity with the content. Teachers who struggle with the content that they teach will not be able to support their learners in constructing an accurate representation of the concepts. Researchers to learn more about the difficulties that practising and preservice teachers experience in Euclidean geometry. The concept of similarity forms part of the Euclidean geometry strand in the South African mathematics FET curriculum. The concept is covered in an informal manner using a broad definition based on enlargements and reductions of figures in the early years. As the curriculum progresses, in the FET band, the similarity of triangles is treated formally by first starting with a non-minimal definition [7] that two triangles are similar if they have equal angles and if the lengths of their corresponding sides are in proportion. Learners then work with formal theorems to show that either of the two conditions is sufficient for the similarity of two triangles [6]. There are few studies which focus on similar formal treatments of similarity within a Euclidean geometry focus [7,8,9], and with this study, we hope to contribute to filling this gap. The following research questions were posed:

How do pre-service teachers perform tasks based on similarity?

How can misconceptions displayed by the PST’s be explained by van Hiele’s theory?

2. Literature Review

2.1. Geometry

Euclidean geometry involves working with the properties of shapes, such as lengths, angles, area and perimeter, to determine relationships and properties of shapes [10]. These properties of shapes play a significant role in science and technology applications, such as the construction industry, design, and architecture [11]. Euclidean geometry improves thinking and reasoning through the solution of riders and the writing of proofs to represent and make sense of the world [12]. However, many researchers [2,13,14] have expressed their concerns regarding underachievement in Euclidean geometry. According to Patkin and Lavenberg [15] (p. 14), “Euclidean geometry is experienced as the most challenging section of the curriculum”. Some studies have shown that pre-service teachers have a limited conceptual understanding of geometric concepts required to teach effectively [14,15,16,17]. Hence, it is important to explore ways of improving geometric conceptual understanding to promote meaningful mathematics learning.

2.2. Similarity and Proof

While similarity is a general concept applicable to pairs of rectilinear figures that have: (1) equal corresponding angles and (2) corresponding sides in proportion, many textbooks focus mainly on the similarity of triangles [8]. When context-based similarity tasks appearing in textbooks in Brazil and the United States books were examined, it was found six different books, three from each country, focused more on the similarity of triangles [8] than other figures. In addition, the context-based similarity task made up less than 30% of the similarity tasks, which were also of low cognitive demand. A study in Israel [7] found that many of the students differentiated between definitions and theorems. Some students felt that there is only one known and accepted definition of similar triangles, while some believed that the essence of the concepts of similarity lies primarily in the lengths of the sides of a triangle, so they could not accept a definition based on the equality of two pairs of equal angles, seeing it as incomplete [7].

The concept of similarity in the South African FET mathematics curriculum starts with a broad non-minimal definition of similarity and progresses towards establishing two equivalent definitions that two triangles are similar if and only if they have equal angles or if and only if their corresponding sides are proportional. This progression is directed by the formal proof that each of the conditions is sufficient for two triangles to be similar. Dealing with proofs requires special reasoning skills as proof is a logical argument statement of complete thinking procedures in a step-by-step manner explaining why these steps are achievable until a new valid conclusion is reached [12]. Mudaly and de Villiers [18] justified the inclusion of formal geometric proofs in mathematic curricula as a medium for teaching and learning deductive reasoning. Proof can lead to further discoveries demonstrating the need for better definitions, or they can yield useful algorithms. However, understanding learners’ cognitive thinking is important to prevent them from resorting to memorisation when teaching proofs [12]. Research into geometric proof shows that students generally find the rigour of proofs difficult, especially when deep abstract thinking is required [12] or many steps are required [10]. Sometimes students rely on the appearance of the figures alone to generate a justification for a claim [10].

2.3. Errors and Misconceptions as Part of the Learning Process

In terms of the constructivist perspectives of learning, errors and misconceptions are naturally formed as part of the process of learning, and teachers should try to learn more about these so that they can better support their learners [19,20,21,22]. As learners construct new knowledge, they do this based on what they know already and understand or do not understand yet. Hence the process of productive knowledge construction includes the process of developing misconceptions since learners’ conceptions are embedded within their own individual cognitive systems, which are not single units of knowledge [22]. When students make persistent errors, this may be signalling an underlying misconception, although some errors may be a result of a careless slip. So errors can be seen as “opportunities for deepening one’s understanding and as important components of learning process” [21] (pp. 44–45). In this study, we look at misconceptions that arise when students have not developed the appropriate reasoning skills required to solve problems based on similarity. According to the van Hiele model [23], as students receive instruction in geometry, the ways in which they reason about a concept change, starting from a visualisation perspective and shifting towards an increasingly abstract approach. Not all students reach higher levels, as detailed in the next section.

2.4. The Van Hiele Model: Levels of Geometric Thought

The van Hiele model of Levels of Geometric Thought is used to explain students’ struggles with geometric reasoning and can guide the planning of instruction that can develop learners’ thinking in geometry [24]. Van Hiele identified five hierarchical and discrete levels of geometric thought that learners must progress through to reach the stage for accessing the critical reasoning for formal proof-writing [23]. For this study, we hypothesise a trajectory in the development of reasoning skills as learners’ progress through the van Hiele levels related to the concept of similarity in Euclidean geometry:

Level 1 (Visualisation): This is the elementary level where figures are recognised as a whole, disregarding their constituent properties. At this level, a person could name and reproduce shapes like a rectangle but cannot identify essential properties that make a specific shape a rectangle [25]. In terms of the concept of similarity, a person will recognise similar figures based on whether a figure looks like an enlargement or a reduction of the other.

Level 2 (Analysis): At this level, the person can identify properties of shapes. However, when mentioning properties to illustrate the shapes, the person mentions all characteristics of the figure they know [23]. At this stage, a person can state the properties of similar shapes and can investigate properties of similarity by translations, reflections and rotations.

Level 3 (Informal deduction): At this level, a person can provide informal arguments to substantiate their reasoning. Relations between figures begin to make sense for a person. For instance, a person can accept that a square is a rectangle because it meets all the properties of one [23]. Hence they are able to understand the notion of class inclusion. A person can use the ratio to see if the sides of a figure are in, or not in, proportion and can use informal deduction to calculate angles to check if the corresponding angles of two figures are equal or not. Students at this level are able to identify equal angles using informal deductions and can grasp the significance of the similarity symbol [9]. For example, in this context, the symbol for similarity (///) is a specialised symbol that goes beyond the identification of similar triangles by indicating the order of equal angles and corresponding proportional sides [9]. They can use the similarity relationship between two triangles to make deductions about the proportional relationships between the relevant sides, which depend on the order of the naming. For example, if ΔABC is similar to ΔPQR, then AB/PQ = BC/QR = AC/PR, but if ΔABC is similar to ΔPRQ, then the relationships are AB/PR = BC/QR = AC/PQ. Hence in signifying the similarity relationship, the order of naming directs the derivation of the relationships between the sides in this context.

Level 4 (Formal deduction): At this level, a person acknowledges the importance of definitions in devising formal deductive proofs and can present proofs on demand without memorisation. Furthermore, “the interaction of necessary and sufficient conditions is understood; distinctions between a statement and its converse can be made” [23] (p. 3). A person understands the minimum conditions for two figures to be similar and can construct formal proofs about why two figures are similar.

Level 5 (Rigour): This is the most advanced level of geometric thought where a person can go beyond Euclidean geometry and study non-Euclidean systems, such as spherical geometry.

According to the Van Hiele model, developments along these levels are based on teaching instruction and experimental opportunities rather than age, and not all students are able to progress across these levels. Progression from elementary to a more advanced level is best facilitated through strategically planned instruction using relevant language and terms [26].

3. Research Methodology

3.1. Research Approach and Sampling

This study was qualitative in nature with the aim of exploring pre-service teachers’ understanding of similarity in geometry. The study was conducted with 34 PST’s from one South African university. The participants were of mixed abilities and gender who volunteered to take part in the study. Seven of the PST’s further participated in semi-structured interviews.

3.2. Collection and Analysis of Data

Firstly, the written responses to the online worksheet were used to investigate the degree of success of PST’s in solving tasks based on similarity and to identify misconceptions. Secondly, the semi-structured interviews, also conducted online, were used to probe the PST’s’ further about their written responses to the items. The written responses were coded from PST1 to PST34 in ascending order of their performance and to preserve their anonymity. Each item was classified according to the VH level that it was most suitable for and appears in

Table 1. Items with VH level and overall results.

Number	Item		Targeted VH Level	% Correct
	Question One
1.1	Explain the meaning of similarity in mathematics as the best you can.		L2 Know similarity property	85
1.2	State the conditions required to show that two polygons are similar.		L2 Know similarity property	88
1.3	Given any two arbitrary squares of different sizes, are they similar or not? Explain		L3 can use informal deduction to see that properties of similarity hold	82
1.4	Given any two arbitrary rectangles of different sizes, are they similar or not? Explain		L4 Can understand if the minimum conditions for similarity are not met—corresponding sides are not always in proportion	0
1.5	The sides in the diagram have lengths as indicated. Are these two triangles similar? Explain why you say so.		L3 can use calculations to establish that corresponding sides are in proportion	62
1.6	State with reason whether or not the following triangles are similar.		L2 can see that triangles are equiangular	76
	Question Two For each diagram below, write down a triangle similar to the given triangle. Naming must be in the correct order:
2.1	PQ is a tangent to the circle at Q. ∆BQP///∆__		L3 Can use informal deduction to identify the equal corresponding angles and, thus, order of naming	62
2.2	∆PRS /// ∆______		L3 Can use informal deduction to identify the equal corresponding angles and, thus, order of naming	71
2.3	∆ABE /// ∆______		L3 Can use informal deduction to identify the equal corresponding angles and, thus, order of naming	74
2.4 (2.5)	$\mathrm{Q}\widehat{\mathrm{P}}\mathrm{R}=90°$$\mathrm{and} \mathrm{PS}\perp \mathrm{QR}.$ ∆PQR /// ∆______ /// ∆______		L3 (L3) Can use informal deduction to identify the equal corresponding angles and, thus, the correct order of naming,	56 (53)
	Question Three. Two circles touch each other at point A. The smaller circles pass through O, the centre of the larger circle. Point E is on the circumference of the smaller circle. A, D, B and C are points on the circumference of the larger circle. OE//CA.
3.1	Prove, with reasons that AE = BE		L4 Can devise a formal proof using previous theorems	59
3.2	Show that $∆AED/// ∆CEB$		L3 Can use informal deduction to show that the corresponding angles according to the naming order are equal	91
3.3	$\mathrm{Hence}, \mathrm{or} \mathrm{otherwise}, \mathrm{show} \mathrm{that} A{E}^2=DE.CE$		L3 can use informal deduction using the results in 3.2 and 3.3 to reach the required result	97

. We used the descriptors in Section 2.4 as the framework to decide on the categories, and we have included a brief explanation of this process in Table 1. To ensure reliability, the items were first coded by the first author and then checked by the second author. Where there were disagreements, these discrepancies were interrogated until an agreement was reached.

The percentage of PST’s who completed each item correctly was computed and appeared in Table 1. In identifying the difficulties encountered by the PST’s, common patterns were identified across the scripts. During the interviews, PST’s were asked to explain how they solved the problem so as to extract information about their calculations, reasoning and proofs relating to problems based on similarity.

4. Results and Discussion

4.1. Overall Performance across the Items

The worksheet comprised 13 items, and Table 1 below shows the items, the van Hiele level that it was predicted to be suitable for, as well as the performance of the students in each item.

As seen in Table 1, the participants were generally able to cope with most of the problems. The three items which they found easiest were 3.3, 3.2, and 1.2, respectively, while the items which proved most challenging to them were Items 1.4 (0% correct), 2.5 (53% correct) and 2.4 (56% correct). We now discuss these items in greater detail.

4.1.1. Least Challenging Items

Item 1.2

Item 1.2 was experienced as the third easiest, with 88% of the group providing the correct answer. The question for Item 1.2 was: State the conditions required to show that two polygons are similar. Most were able to state the two conditions that corresponding pairs of angles and sides should be equal and in proportion, respectively. This showed that they were comfortable with the non-minimal nature of the definition of similarity that they first encountered, similar to the study in [7]. Some who did not get it correct limited their descriptions of the concept of similarity to triangles only, as shown in Figure 1.

In the interview, PST14 acknowledged his error and admitted that “Similarity can be applied to other figures like squares and rectangles, not only in triangles”.

Item 3.2

Item 3.2 was the second easiest item and was the second item of Task 3 which was a geometry rider, as shown in Figure 2.

Students found Item 3.2 relatively easy; this may be because proving that a pair of triangles is similar is a well-known procedure that they have mastered since this type of question is frequently encountered in secondary school. Additionally, the order of the naming was provided in the instruction. Ubah and Bansilal [9] also noted that the students in their study found it relatively easy to show similarity in the same situation.

Item 3.3

Item 3.3 was experienced as the easiest and was answered correctly by all except one participant. The item required students to make a deduction from the similarity relation between $∆AED \mathrm{a}\mathrm{n}\mathrm{d} ∆CEB$, which was proved in Item 3.2. They also needed to use the property that AE = BE, which was proved in Item 3.1, and hence available to them as a fact that they could apply in Item 3.2. The high success rate may be because PSTs have routinised the application of proportional relations within similar triangles after studying it at school and encountering these concepts at university. Since the correct order of naming the triangles was provided in the previous item (3.2), deducing the proportional relationships between the sides became a routine exercise, so the only challenge would have been to use those relationships to derive what was required. Hence even students who did not work out Item 3.2 completely were able to apply the proportional relationship between the sides of the two triangles to arrive at the required result. Students may have found it easier to arrive at the result since they had a starting point ($∆AED/// ∆CEB)$and the endpoint ($A{E}^2=DE·CE).$

It is interesting to note that in contrast to the above two items, Item 3.1 was much harder since students needed to formulate a proof from scratch to show that the two lines were equal. Only 20 students were able to produce this proof, showing that many PSTs find it challenging to present sound geometric proofs based on geometric facts. Some used theorems haphazardly where their conditions were not applicable, as shown in the response by PST 7 in Figure 3.

The student wrote: OE$\perp$AB with the reason “line from centre $\perp$ to chord bisect thchord”, therefor AE = EB with the reason ” OE$\perp$AB and bisects AB”. That is PST7 claimed OE$\perp$AB because the line from the centre perpendicular to a chord bisects the chord. What is missing is the reason why OE is perpendicular to AB. Instead, it seems to be presented as a known fact. This response illustrates the Department of Education’s [27] assertion that the structured requirements of providing appropriate reasons for statements make this subject section seem complicated. Furthermore, the result is consistent with findings [13] which revealed that school learners struggle to answer proof questions, especially when more than two steps are required. Despite spending years at secondary school and a semester at university, it can be seen in this study that many of the PST’s have not yet developed an understanding of the nature of geometric proofs.

4.1.2. Most Challenging Items

Item 1.4

The question for Item 1.4 was: Given any two arbitrary rectangles of different sizes, are they similar? Explain.

This question required PST’s to analyse the properties of rectangles and then decide if the conditions were satisfied or not for the figures to be similar. Most students assumed that they were similar, as shown in the response by PST7 in Figure 4.

The response of PST7 showed they did not carefully think about the possibility that the ratios of corresponding sides of rectangles may not be in the same proportion. During the interview, the student was probed about his assumptions:

Researcher: Will it be always the case that two arbitrary rectangles are similar?

PST7: Yes, Sir, because their corresponding angles are equal, and their corresponding sides are proportional.

Researcher: Will it be always the case that rectangles with corresponding sides are in proportion?

PST7: Yes, Sir.

Researcher: Consider the typical example of two rectangles; the first is 16 by 4 units, and the second is 8 by 3 units. Are their corresponding sides proportional?

PST7: The ratio of corresponding sides will not be same then these rectangles are not similar. It is clear now how these conditions of similarity can apply.

As seen in the interview above, PST7 firmly believed that the corresponding sides would always be in proportion for any two rectangles. However, when PST7 was presented with a counterexample of two rectangles whose corresponding sides were not in proportion, he agreed that any two rectangles are not necessarily similar. Another student, PST29 (Figure 5), did realise that the sides may not always be in proportion but could not express his argument precisely:

PST29 indicated “not similar”, meaning that any two rectangles need not be similar, but it conveyed another message that all rectangles are not similar. Although he included that rectangles are only similar if there is a consistent ratio between all the sides, the first part of the response was not completely correct. It would have been completely correct if he had said “not necessarily” or “not always”. The gaps in the response by PST29 also show that the reasoning that is required at the formal deduction level is quite complex and not easily attained.

In contrast, Item 1.3, which asked whether all squares are similar, was answered correctly by 82% of the PSTs. Many PSTs incorrectly assumed that equal corresponding angles are sufficient to prove that two figures are similar and that corresponding sides will always be proportional. In an interview with PST32, he asserted that to prove that any two figures are similar, “only one condition needs to be satisfied, which is corresponding angles must be equal”. If PSTs applied this incorrect reasoning to Item 1.3, they would get the correct answer because all squares are similar since both similarity conditions automatically hold. However, this incorrect reasoning cannot lead to the correct answer for Item 1.4. Exploring the similarity of two rectangles requires checking if the two minimum conditions of similarity are satisfied since the equal angles condition on its own is insufficient. This difference between the results for Items 1.3 and 1.4 indicates that showing that a pair of figures may not necessarily be similar has a much higher cognitive demand than confirming that a pair of figures are similar. Many studies have shown that a task requiring students to show a statement is false is of higher cognitive demand than one requiring them to identify (in the same setting) it is true [28,29,30]. In another study [31] about functions, it was noted that students found it easier to identify that a function satisfied certain conditions rather than showing that a function does not satisfy a condition. The authors [31] recommended that students be given more opportunities to construct arguments for why an object does not satisfy a condition. Similarly, PSTs needed more opportunities to argue about situations where polygons do not satisfy both similarity conditions.

Items 2.4 and 2.5

These items were the second most challenging items achieving 55% and 52% correct answers, respectively, and appear in Figure 6.

The problem was not with identifying the triangle but in naming the triangles in the correct order, as shown in the response by PST11 in Figure 7. Note that usually, it may not be necessary to specify the correct order when naming the triangles that are similar if one is just interested in identifying the figures that are similar. However, in the current context, the similarity relationship between two triangles is used to make deductions about the proportional relationships between the sides, which depend on the order of the naming. For example, if ΔABC is similar to ΔPQR, then AB/PQ = BC/QR = AC/PR, but if ΔABC is similar to ΔPRQ, then the relationships are AB/PR = BC/QR = AC/PQ. Hence in signifying the similarity relationship, the order of naming is very important.

The challenge was related to identifying the corresponding pairs of equal angles, which would help to name similar triangles correctly. Students often need to engage in dimensional deconstruction in order to attend to the properties of figures [32]; for example, in Items 2.4 and 2.5, where the three triangles overlapped, students needed to deconstruct or separate the three triangles in their minds and re-orientate them so that they could see the triangles separately. This task of dimensional deconstruction of figures is made more difficult when triangles share parts (lines or angles), that is, when they have a common side or common angles. A study [33] found that students commonly identified intersecting lines as a shared side and a combination of two angles, with one falling in the one triangle and the other in the second triangle, as being shared or common. Likewise, in another study [9], a student who initially was unable to identify the correct naming order of the pairs of similar triangles was successful after creating physical triangles that could be manipulated to identify the pairs of equal angles. The manipulations allowed him to visualise the similarity relationship between the triangles by seeing them as reductions of the outside triangle.

4.1.3. General Insights about the Overall Performance

The items experienced as the least challenging required either a procedural or basic understanding of similarity in mathematics, including knowing the definition of similarity and identifying the similarity relationship between triangles. The results suggest that PSTs considered these items as familiar situations they had encountered during their school studies, specifically related to the similarity of triangles. It seems that they have routinised the steps involved in setting up a similar relationship between pairs of triangles. Another study [34] found that many students preferred questions that require procedural rather than conceptual understanding and those based on familiar procedures. These findings seem to support [35], who claimed that students might perform well on procedural questions; however, their conceptual knowledge is poor when deeply probed. It was noted, too, in [36], that students resorted to memorisation to pass through the examination without engaging in problems that involve insight and understanding. It is likely that PSTs apply their knowledge as their teachers taught them [37]. This may limit their learners’ opportunities to engage in questions that require higher levels of cognitive analysis.

4.2. Misconceptions Associated with Particular VH Levels

In analysing the responses, we observed that many misconceptions arose because certain students had not yet developed the appropriate reasoning skills aligned to particular VH levels. We look at five such misconceptions which seemed to arise because students had not developed the appropriate reasoning skills linked to a VH level.

4.2.1. Misconception That Similarity Applies to Triangles Only

Many students believed that similarity could only be applied to triangles, which may be because they were using reasoning linked to the visualisation level where shapes are seen as a whole and not in terms of their properties. This misconception was identified when PST: (i) gave a special case of similar triangles instead of the general definition of similarity in mathematics in Item 1.1 (ii) provided conditions of similarity which is only applicable to triangles in Item 1.2; and (iii) incorrectly assumed that all figures with equal corresponding angles would have corresponding proportional sides as is the case for triangles, in Item 1.4. The above reasoning suggested a narrow understanding of the similarity concept that was limited to triangles. Examples of this reasoning can be seen in the work of PST 3 in Figure 8 and earlier in the work of PST14 that appeared in Figure 1.

This finding is supported by the study by [7], which explored student conceptions of congruent and similar triangles and found a poor understanding of geometric definitions and theorems. Learners did not accept similar triangle theorems as formal definitions of similarity because of the concern for one uniform concept definition. However, this lack of understanding may be attributed to the fact that some textbooks discuss similarity only in relation to triangles. For example, a study [8] that examined context-based similarity tasks in Brazil and the United States books found that six different books, three from each country, focus more on the similarity of triangles while context-based similarity tasks accounted for less than 30% of the similarity tasks. Many of those contextual tasks were of low cognitive demand.

4.2.2. Misconception That All Rectangles Are Similar to One Another

The misconception that rectangles are always similar, emerges when students rely on reasoning linked to the informal deduction level. As discussed in Section “Item 1.4”, many PSTs incorrectly assumed in Item 1.4 that all rectangles are similar because they have corresponding angles equal to 90$°,$ which satisfies one similarity condition. Many believed that one similarity condition is enough to conclude any two figures are similar without checking whether the second condition of corresponding proportional sides was satisfied. This narrow understanding of similarity resulted in a misconception that all rectangles are similar in Item 1.4. PSTs did not recognise that for rectangles, it is not always true that if corresponding angles are equal, then corresponding sides are proportional, implying that one condition may not be sufficient. This misconception about the similarity of rectangles may be because these students had not developed the reasoning that is linked to the formal deduction level. They did not have a general understanding of the similarity definition in mathematics and the minimum conditions of similarity for different figures, which requires reasoning associated with the formal deduction level. Additionally, this indicates that stating the concept definition does not guarantee the conceptual understanding of the concept amongst PSTs.

4.2.3. Misconception about Naming Figures in a Similarity Relationship

Many incorrect responses for Items 2.1 to 2.5 were due to the incorrect order of naming similar triangles. This may indicate that the students are still working at the analysis level because they have not been able to see how the symbolic mode is linked to the iconic mode. They are able to identify the equal angles but have not grasped the significance of the similarity symbol [9]. A person who is able to understand that mathematics is a symbolic language where symbols denote specific meanings shared by the mathematics community has taken an important step in informal deduction. For example, in this context, the symbol for similarity (///) is a specialised symbol that goes beyond the identification of similar triangles and indicates the order of equal angles and corresponding proportional sides [9]. For example, $∆$ABC///$∆$DEF indicates that $\widehat{A}=\widehat{D},\widehat{B}=\widehat{E} and \widehat{C}=\widehat{F}$ and sides AB/DE = BC/EF = AC/DF. This matching of angles and sides is crucial when identifying the proportional relationships arising from similar figures. The correct naming of similar triangles requires an intertwining of both visual and analytic skills.

4.2.4. Misconception about Class Inclusion of Quadrilaterals

Understanding the notion of class inclusion of shapes is vital in geometry because it enables one to reason about the relationship between geometric shapes and their properties. Class inclusion reasoning is usually developed as students advance to the informal deduction level, as described by van Hiele. If they have not developed informal deduction reasoning skills, they are unlikely to understand the notion of class inclusion. In this study, a few items required an understanding of quadrilaterals (Items 1.3 and 1.4). The students PST 6 and PST 9 showed misconceptions about the class inclusion of quadrilaterals, as shown in Figure 9 and Figure 10. Participant PST 6, when asked about the similarity of squares in Item 1.3, gave a response about rhombus and parallelograms (Figure 9), while PST 9 (Figure 10), when asked about the similarity of squares, spoke about a rhombus and rectangle. Furthermore, when asked about the similarity of rectangles, participant PST9 referred to the properties of a rhombus and parallelogram.

It seems that the student PST 6 tried to use a rhombus and parallelogram as illustrative examples of the class of squares or rectangles; however, the error is that a rhombus and parallelogram are not included in the class of squares or rectangles. These participants were trying to consider all quadrilaterals, which belong to a family of squares and rectangles, but could not comprehend the relationships between the classes of figures. They could not identify which sets of quadrilaterals were included within the other. It seems that they have not developed the appropriate reasoning consistent with the Informal deduction level about the concept of class inclusion.

Although the notion of class inclusion and other concepts related to geometry has been reported as a difficulty experienced by many learners [13], many teachers also struggle. Bowie [2] asserts that a significant factor contributing to learners’ poor achievement in geometry is teachers’ lack of understanding of geometry. This view was supported by [38], who found that the quadrilaterals whose attributes were known most by the teachers were the square and rectangle. However, they had problems with their diagonal properties. Similarly, some researchers [13,39,40] found that pre-service primary mathematics teachers’ understanding of hierarchical classifications of quadrilaterals is poor. They were not able to establish relationships among quadrilaterals. Others [38,41,42] found that pre-service teachers were able to present formal definitions of quadrilaterals but only recognised prototypical examples of these quadrilaterals, thus making it difficult for them to understand their relationships. One study [42] found that the item on hierarchical relationships of quadrilaterals was experienced as much more difficult than other VHL3 items, suggesting that reasoning skills about class inclusion are harder to develop than other reasoning skills required at VHL3.

4.2.5. Misconceptions Related to Visual Appearance and Orientation of Diagrams in Proofs

It was found that the PSTs generally struggled with presenting sound geometric proofs. They used reasoning related to the visualisation level, where their arguments are directed by the appearance of the figures rather than an analysis of properties. An example of this is the argument used by student PST 2 in Item 1.5.

From Figure 11, student PST 2 investigated the ratios of sides which he perceived to be the corresponding ones of the two triangles and found that these were not proportional. His selection was not based on an analysis of which pairs could be the corresponding ones. He was not able to select the relevant corresponding sides, which would have been in the same proportion.

Student PST 1 also alluded to making judgements based on what the figures looked like. Figure 12 shows her response to Item 1.7, which asked whether two given triangles were similar.

Student PST 1 was not clear about which angles were being compared and claimed they were not equal without providing any sound mathematical justification. The interview confirmed that PST 1 focused on the appearance of the figures when he concluded that these triangles do not have equal sides and angles:

Researcher: How did you know that these triangles do not have equal sides and angles?

PST 1: I was looking at them.

The description by PST 1 makes it clear that the judgements were based on the appearance and not on an analysis of the diagram.

Another student, PST29, seemed to choose pairs of angles based on the orientation of the figures in Item 1.6.

Student PST 29, in Figure 13, opted to compare the angles of the triangles based on the orientation of the triangles. She chose to compare angle C from each triangle, where each had a common horizontal line and angles A and E, which were the two angles formed by a vertical line in each case. This suggests that she chose to compare the two lower and upper angles based on the appearance of the figure, again suggesting reasoning aligned to the visualisation level.

These examples show that some PSTs despite their extensive geometry experience at school and university, were not able to accept that figures are not necessarily drawn to scale. They based their reasoning on the appearance only without doing an analysis using the information about the figures. This shows that the PSTs found it challenging to connect the visual and analytic representations.

5. Conclusions

In this study, we set out to explore the understanding of similarity in Euclidean geometry by PSTs in one South African university. We studied their performance on a set of items based on the concept of similarity and identified a series of misconceptions during the analysis. Although most of the PSTs were able to answer all the items except one, it was found that they did better in those items which were familiar to them or those which were more procedural in nature than those which required conceptual understanding.

The study identified some misconceptions exhibited by some PSTs as a result of limited reasoning skills that are necessary at certain VH levels. Some PSTs believed that the similarity concept applied only to triangles, showing that they saw triangles as prototypes of the similarity concept. These reasoning skills are associated with the visualisation level, where shapes are seen as a whole. The misconception that all rectangles are similar to one another seemed to emerge when students were not yet ready for the formal deduction level, which requires reasoning about necessary and sufficient conditions for a property to hold. For those students who struggled to name the similar triangles in the correct order, they may be still using reasoning linked to the analysis level, where they could identify the pairs of similar triangles but they were not able to link the symbolic mode (naming) with the analytic modes (identifying the pairs that were in a similarity relationship). Misconceptions related to class inclusion suggest that the PSTs have not fully attained the informal deduction level. Those who considered the visual appearances of figures as evidence of relationships suggest that they are still reasoning at the visualisation level since their arguments are based on the appearance of figures rather than on an analysis of properties. Hence PSTs, in similar situations, will need more opportunities to make connections between the visual and the analytic representation of similarity.

The van Hiele theory emphasises the role of instruction in enabling individuals to progress through the levels of geometric thought. Without targeted instruction that utilises appropriate language and symbolism, an individual may not be able to deepen their geometric understanding of concepts. Hence it is important for geometry educators to try to identify the levels of reasoning that the learner can engage with and what they still need to develop so that well-planned activities can be offered to the learners to deepen their understanding.

6. Limitations

One of the limitations of the study was the online nature of the responses. If it were conducted face to face, then the worksheet would have been written under stricter conditions.