Many pre-service mathematics teachers in South Africa are apprehensive about the content of Euclidean geometry, because they did not study Euclidean geometry in high school but will be expected to teach the content when they start their teaching career. This article reports on a study that explored the role of semiotic representations in pre-service teachers’ reasoning about the similarity relationship between triangles. Data were generated from the written responses of 65 pre-service mathematics teachers as well as three semi-structured interviews. Duval’s notions of conversions and treatments were used as a framework to understand the pre-service teachers’ struggles with negotiating movements between the visual and symbolic registers of representation. The findings revealed that many pre-service teachers struggled with identifying the similarity relationship between triangles appearing in various configurations of geometric objects. While some participants were easily able to draw upon the two registers to express the relationships, one student who initially made many errors was only able to discern the necessary relationships with the help of a concrete representation that could be physically manipulated. The study therefore provided an example of how a student’s errors could be used as a learning resource to lead to meaningful learning.

Mathematics outcomes in South Africa are very low and many researchers and stakeholders have expressed concerns about the poor performance in mathematics at school level, especially in geometry (Luneta,

In South Africa, curriculum developers did not seem to be convinced about the important role of geometry in the mathematics curriculum. In 2006, as part of the many changes brought in by Curriculum 2005, for instance, the Euclidean geometry strand was made optional for those learners who opted to study mathematics in the FET band (Department of Education,

When it was brought back into the core mathematics curriculum, teachers did not feel as confident about the strand since it had not been taught for such a long time. Some researchers note that teachers avoided the teaching of geometry in school because of poor mastery of Euclidean geometry (Atebe & Shaefer,

In this study we delve into the area of semiotics which is the study of signs and sign symbols, how these signs are used to signify actions or objects, and the interpretation of these signs (Moore-Russo & Viglietti,

The purpose of the study was to explore how semiotic representations influence students in their reasoning about the similarity relationship between triangles. To achieve this goal, the following research question was addressed:

Geometry is an essential part of mathematics and provides unique opportunities for mathematical modelling by drawing upon real-life examples (Usiskin,

Many learners find the study of Euclidean geometry challenging (Ngirishi & Bansilal,

Many students are daunted by the learning of the formal logic and deductive reasoning that are necessary elements of Euclidean geometry. One of the reasons why students find geometry difficult is the emphasis on the deductive aspect without a corresponding focus on the underlying spatial abilities (Del Grande,

Another characteristic feature of the study of geometry is the necessary intertwining of the visual and symbolic or analytic representations where the one representation supports and underpins the others. As pointed out by Del Grande (

Bansilal and Naidoo’s (

Sinclair et al. (

Rivera (

Gal and Linchevski (

Shapes requiring different mental transformations.

Identifying and understanding the errors that students make during the process of constructing their knowledge has occupied the attention of many researchers. However, such an enterprise is also valuable for teachers since knowledge of these errors can be used as a learning resource in their classrooms. Chauraya and Brodie (

According to Duval (

An example of a treatment could be carrying out a calculation while remaining in the same notation system. Transforming the representation

Duval (^{2}

This interpretative study was located within a six-week intervention that was designed to help pre-service mathematics teachers improve their understanding of FET Euclidean geometry. The purpose of the study was to explore how students used semiotic representations in reasoning about the similarity relationship between triangles. The participants of the study were 65 students who enrolled for the intervention. The data for the study were generated by the written responses of the participants to one task based on similar triangles, as well as semi-structured interviews that were conducted with 13 participants who volunteered to be interviewed about their understanding of the concept. The interviews were video-recorded and then transcribed verbatim by the first author.

In order to ensure reliability, the transcripts were checked by the second author against the original recordings. For the purpose of this study, we draw upon interviews with three participants: Sabelo, Celo and Vince, to highlight particular ways in which the semiotic representations were used to reason about the similarity of triangles. Sabelo and Celo had not studied geometry in the FET phase of their schooling while Vince had written the third mathematics paper in Grade 12, which was optional, and which included the study of Euclidean geometry. The purpose of the interviews was to probe their reasoning about the concepts in the written tasks. However, the interviews were also used to help improve the interviewees’ understanding, hence they were interspersed with explanations of key concepts where necessary to clarify the thinking and strategies used in responding to the questions.

Ethical clearance for this research was obtained from the Research Ethics Committee of the Education Faculty at the relevant university (ethical clearance number HSS/0425/018A). After obtaining approval, each participant gave their written informed consent to participate in the research, allowing the use of their responses to the written task and interview extract for research purposes and assured anonymity in the use of these data.

We discuss the participants’ responses to the task which was based on identifying and naming the similar triangles that emerged from various geometric figures. The identification of the equal angles within the triangles to confirm the similarity of pairs of triangles, required knowledge of the properties of these figures which are typically studied in FET mathematics.

The item analysis for the task based on the pre-service teachers’ written responses is presented in

Frequency of Correct, Wrong and No responses for items in the task.

Items | 2.1 | 2.2 | 2.3 | 2.4 | 2.5 | 2.6 | 2.7 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

% | % | % | % | % | % | % | ||||||||

Correct response | 30 | 46.2 | 39 | 60.0 | 32 | 49.2 | 20 | 30.8 | 19 | 29.2 | 17 | 26.2 | 14 | 21.5 |

Wrong response | 28 | 43.1 | 20 | 30.8 | 23 | 35.4 | 37 | 56.9 | 35 | 53.9 | 38 | 58.5 | 42 | 64.6 |

No response | 7 | 10.8 | 6 | 09.2 | 10 | 15.4 | 8 | 12.3 | 11 | 16.9 | 10 | 15.4 | 9 | 13.9 |

From the results in

In

Which of the six figures do you find very easy to solve?

Question 2.2.

Why?

Because the theorem involved is very easy; angles of the same segment are equal.

Then which angle is equal to what?

[

Details of the task.

A representation of Sabelo’s response.

As seen above, Sabelo had no problems identifying the angles that were equal in the two triangles using the ‘angles in the same segment are equal’ result and thereafter representing the symbolic relationship between the similar triangles correctly. The interview continued, where Sabelo was probed about other questions:

What are we required to do in Question 2.1?

We are required to show that Δ

How do we find the unknown triangle?

The diagram is confusing.

Let’s start by identifying the equal angles.

We start by identifying the equal angles; like ∠

Why is ∠

Because ∠

Do you mean that ∠

Yes, Then Δ

[

That question is confusing because of that 90°. I wanted to say that ∠

In the above extract, Sabelo’s misconception that ‘∠

What of Question 2.7, how do we get the unknown triangle?

[

The above interview extract shows that Sabelo has once again made an incorrect deduction when working with the visual representation and identified angles incorrectly as being equal (). Furthermore, his understanding of ‘common angle’ is similar to the misconception identified in the study by Gal and Linchevski (

Vince was one of the students who correctly represented the similarity relationships for all the questions in the task and agreed to be interviewed. During the interview he was asked to explain how he arrived at the correct answer for some of the items.

Looking at the given task [

There are six different shapes included in different triangles and we are required to name triangles similar to the given triangle in correct order.

How do we find the unknown triangles? Let’s take 2.3.

In 2.3; Δ_{1} since line _{1} also corresponding angles. Therefore, Δ

Now in questions 2.4, 2.5, 2.6 and 2.7, how do we get the unknown triangles similar to given triangles?

[

From the interview extracts above, Vince is comfortable with working with the visual representation and effortlessly moves to the symbolic representation. For Question 2.2 he was able to connect the visual and symbolic representations without any hesitation, although for Question 2.4 he first spent some time working within the visual register so that he could identify the equal angles using the visual representation before expressing the relationship symbolically.

Celo’s responses in

If we look at your answer for Question 2.3, it is Δ

This was easy because you have two parallel lines and

A representation of Celo’s response.

His response is focused on how he identified the two triangles that are similar, but he has not mentioned the order of naming the triangles. He was then probed about this.

But why would you say Δ

I just wrote it.

Hence, it was clear that Celo did not assign any significance to the symbolic representation Δ

The researcher spent much time explaining the significance of the order when using the ||| notation. The symbol ||| is a specialised notation, for example Δ

In continuing, the researcher then asked him to try and work out the correct representation for the similarity of the triangles in Question 2.1. In considering Δ

So, let’s try to get the angles matched for Δ

Okay er _{2} [

Right so Δ

But the order was not correct.

Celo then went on to correctly represent the similarity relationship for Question 2.2. He was then probed about Question 2.4 and Question 2.5 which he found difficult to work with. The researcher explained how the order could be found by taking ∠_{1} to be 90º − _{2} =

My method is different from yours – I use logic. I have Δ

Are you trying to move it in your head, like inverting it?

[

Can you draw it for me? Can you tell me your order?

[

It was evident that Celo was trying to visualise how the triangle could be moved around so the two triangles were oriented so that the similarity of the shapes could be easily discerned. However, it was difficult to mentally transform the image in his mind to match the symbolic representation that was needed. At this stage, the time was up so it was arranged that Celo would meet up again and demonstrate his method. At the next meeting, Celo came prepared with three different coloured triangles that were arranged as shown in

Celo’s cardboard models of the triangles.

In

Thereafter, Celo then reflected the light grey triangle across the line

Reorientation of the light grey and white triangles.

By physically manipulating the light grey triangle, he was therefore able to easily match the pairs of equal angles corresponding to one another in the two triangles. He then proclaimed, ‘Now you can see that Δ

Rearrangement of the three triangles.

This demonstration provided insight into Celo’s reasoning because it showed that Celo was dependent on the cardboard cut-outs as a physical representation that could be manipulated. He needed to ‘see’ the orientation of the triangles so that he could draw out the symbolic representation. This means that he could not work within the symbolic register only – that is, to carry out a treatment, using the equality relationship between the angles of the triangles to express it using the similarity notation. However, when he had the physical triangle models that could be manipulated then he was able to use the similarity notation to express the relationship between the triangles. His problem is that he needed the visualisation processes to be merged with the symbolic representation. Working within the symbolic register requires one to be convinced that if angles

Celo’s experiences show how important it is for students to be able to make connections between different representational registers. Duval highlighted that it is sometimes necessary to move to another register, that is, to carry out a conversion, so certain properties of an object can be discerned. Although an object in one register is the same as the object in another register, each register conveys certain properties that may not be so easily discernable in the other register. This function of conversions was illustrated by Celo’s use of the concrete manipulatives which enabled him to manipulate the triangle models so that he could ‘see’ that one was an enlargement of the other. He was not able to discern the equal angles based on the visual representation only and needed to perform the transformations on the physical representation so that the matching angles could be identified. The interview with Celo revealed that he found the mental transformation required in the visual representation too difficult. By drawing upon his cardboard cut-outs (physical representation) he was able to carry out a physical manipulation on the triangle models. Furthermore, he was not able to engage properly with the symbolic notation of ||| because he used it as a sign indicating that two shapes are similar without giving any consideration to the order of the naming. The symbolic register did not give Celo access to the objects and he needed to use the manipulatives comprising a physical or concrete representation so that he could work out the properties of the objects.

It was clear that Celo needs more opportunities for working across the three registers of representation for the concept of similarity. However, it was to Celo’s credit that he recognised that the physical representation would help him access the properties of the objects unlike Sabelo, for example, who was stuck when faced with Question 2.4 and Question 2.5 and did not have the means to move beyond this barrier. Sabelo’s problem was that he did not understand the properties of the geometric figures well enough and needs more opportunities to improve his skills in this area. Clearly the understanding of geometry requires fluency in moving between the visual representation using geometric figures and the symbolic representations which make use of symbolic notations for congruency, similarity, etc. Sabelo was able to work with the symbolic register and carried out the treatments within the symbolic register but expressed the similarity relationship incorrectly because of his incorrect deductions.

In this article, we studied the responses of 65 pre-service mathematics teachers to a Euclidean geometry task based on similar triangles and focused on the role played by semiotic representations in identifying and naming similar triangles which arose in various configurations of geometric objects. It was found that most students struggled with the symbolic specialised similarity notation (|||). The symbol ||| is a specialised notation that denotes which two triangles are in a similarity relationship, for example Δ

Sometimes students can carry out treatments that are based on incorrect deductions which leads to incorrect results as in the case of Sabelo. His use of treatments in the symbolic register led to incorrect results because he incorrectly identified angles as being equal. His knowledge of the relationships and properties of the underlying geometry concepts was weak, so his incorrect deductions led him to incorrect formulations of the similarity relationships between triangles.

Duval (

For one student, the symbolic and the visual representation did not provide him enough access to the objects to allow him to discern the relationship and it was only after carrying out the rigid transformations using concrete representations of the triangles that he was he convinced about the similarity relationship and the order of the naming. Celo needed the comfort of the physical representation of the objects that can be manipulated or rigidly transformed in order to facilitate the visual representation showing that the one triangle is an enlargement or dilation of the other as shown in

A salient point relates to the fact that Vince had elected to study Euclidean geometry in school although it was not compulsory while Sabelo and Celo did not have that opportunity. The geometry workshop was designed to help students such as Celo and Sabelo. The results of this study showed that the students need much more help in navigating these concepts forming part of the Euclidean geometry curriculum. Celo’s initiative in making the cardboard cut-outs helped him to concretise some of the relationships embodied in the similarity of triangles concept. Perhaps such teaching aids may be useful for other students such as Sabelo who did not seem to see the connections between equiangular triangles and enlargements or reductions of the triangles, which are key to the concept of similarity. Much of the earlier work in school in the earlier grades that focus on rigid transformations and enlargements or reductions of figures are meant to form the basis for this later work on similarity. Hence, the use of these concrete manipulations is necessary for students to develop a more robust understanding of the concept. This suggestion resonates with the advice given by Zazkis et al. (

In conclusion, we hope that this contributes some new knowledge in the field in terms of how learners’ misconceptions or errors could be turned into a resource to promote meaningful learning (Chauraya & Brodie,

We acknowledge Mdutshekelwa Ndlovu of the University of Johannesburg for proofreading the manuscript before submission, as well as Moses Mogambery who designed the activities and conducted the workshops.

The authors declare that they have no financial or personal relationships that might have inappropriately influenced the writing this article.

I.U. was responsible for the initial analysis and conducting of interviews as well as drawing up the first draft. S.B. refined the analysis as well as the article.

This research was funded as part of the postdoctoral research support provided by University of KwaZulu-Natal.

Data sharing is not applicable to this article.

The views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors.