Deficiencies in geometric reasoning exist with both students and teachers. Studies involving investigations into South African pre-service mathematics teachers’ levels of reasoning in geometry revealed that many teachers struggle to solve geometric problems that require logical abstract reasoning in the proof of statements made (Van Putten 2008). The findings of another study conducted by Adolphus (2011) in Nigeria concluded that most mathematics teachers have a poor foundation in geometry and are therefore not capable of teaching geometry well. Moss et al. (2015) supported the assertion that teachers, especially at the primary school level, have inadequate knowledge of geometry, resulting in geometry not receiving adequate teaching time.

Transformation geometry deals with the way geometrical shapes or objects are changed into their various images under a transformation. Transformation geometry covers translations (all points of the shape are moved through the same distance in the same direction on a plane), reflections (a line is used like a mirror to reflect a figure) and rotations (a shape is turned about a point through a given angle) (Aktaş & Ünlü 2017; Bansilal & Naidoo 2012; Evbuomwan 2013). Transformation geometry has been identified as one aspect of geometry where students experience challenges (Ada & Kurtulus 2010; Bansilal & Naidoo 2012; Evbuomwan 2013; Luneta 2015b). For example, students lack understanding of the geometric meaning of translations and rotations, even though they seemed capable of working with algebraic calculations involving these types of transformations (Ada & Kurtulus 2010). On the contrary, in Naidoo & Bansilal’s (2010) study, students did not work competently with algebraic problems because of confusing rules of transformation geometry such as the rules of reflection with those of rotation. A study by Luneta (2015b), which focused on Grade 12 students’ knowledge of geometry, outlines some of the problems students encounter when solving tasks or involving transformation geometry. Amongst the errors students exhibited were confusing the line y = x with the y- and x-axis.

Foundation Phase student teachers, in particular, tend to have limited basic knowledge of transformation geometry, which leads to the confusion of the properties of, and applicable rules between the different types of transformations (Luneta 2014). This confirms earlier studies by Gomes (2011), who discovered that even though elementary teachers seemed aware of some of the properties such as the preservation of length, orientation and parallelism, they could not correctly perform a horizontal translation of a horizontal flag. A study by Kurt-Birel, Deniz and Önel (2020) further alluded to primary school teachers’ weak knowledge of properties of transformation geometry (translation, reflection and rotation). They concluded that most prospective primary school teachers’ knowledge of translation was fragile and they exhibited difficulty in explaining and understanding concept involved, such as defining translation whilst relating it with a plane, motion, transformation or vector. Bansilal and Naidoo (2012) also highlighted the lack of visualisation skills with teachers and the subsequent inability of teachers to use visualisation as a reasoning process and tool. Visualisation skills are a significant part of learning transformation geometry because physical changes between an object and the image involve visual differences that need to be discerned (Mudaly & Rampersad 2013).

Students’ misconceptions and the associated errors in transformation geometry are stubborn and persistent (Bansilal & Naidoo 2012). Further studies and invention initiatives are, therefore, necessary. This research attempted to provide further insights into the notion of effective mathematics teaching approaches by using an intervention strategy. The intervention was a design of an instructional programme that isolated student teachers’ errors and mapped them onto a step-by-step procedure aligned to specific activities.

This study involves PSTs’ faulty reasoning in transformation geometry. Faulty reasoning is detected in the errors students display. Luneta and Makonye (2010) described an error as a mistake, slip, blunder or inaccuracy and a deviation from accuracy. Errors are often displayed in students’ written work or in oral communication. It is from the displayed errors that teachers can associate the underlying misconceptions responsible for them. Hence, misconceptions are not easily detected by simple observation. Hansen (2017) assets that misconceptions might relate to misapplication of certain rules, an over-generalisation or under-generalisation, as well as an alternative understanding of a certain concept. In other reviews, the link between errors and misconceptions is that errors are slip-ups and non-systematic, whilst misconceptions are regarded as the more serious systematic errors (Makonye 2011). Fundamentally, errors are as a result of misconceptions and are formed in a learner’s head and displayed on a piece of paper or in a conversation.

Examples of instances that might involve non-systematic errors include the misreading of information by students or unintentionally leaving out an important piece of information. In such cases, students are likely to correct the error themselves, because there is no existence of underlying faulty conceptual understanding associated with the error. On the other hand, systematic errors involve the lack of understanding of underlying concepts (Makonye 2011). In such cases, the teacher often needs to guide the student towards identifying and correcting the error.

Analysis of errors was critical in this study because as the preceding literature indicates that students’ challenges with transformation geometry were because of some underlying misconceptions that resulted in errors. Hence, such misconceptions need to be explored, if intervention efforts are to be successful. The importance of analysing students’ errors lies in the benefit that they derive from being made aware that, for example, the conception they have been holding onto has been arrived at by flawed methods, which then impedes their performance in the subject (Schepper & McCoy 2013). However, the intention is not to make students feel guilty for making errors. Instead, they should consider errors as bouncing boards for conceptual knowledge acquisition and form an important part of the learning process (Hansen 2017). Matteucci, Corazzal and Santagata (2014) concurred that student errors cannot be avoided. Instead, teachers should use the existence of errors to redesign lessons by making errors catalysts for learning (Bray & Santagata 2014).

The misconception and the errors that result in students’ incorrect or faulty reasoning have to be isolated and unpacked, so that attempts at intervention target the correct concept or procedures needed to address them. One way of unpacking the errors is by designing learning experiences that are likely to provoke the errors concerned, so that the hidden misconceptions are revealed and corrected (Zehetmeier et al. 2015). Therefore, students must be engaged in conceptual experiences through assessment tasks and other learning activities to allow the misconceptions to emerge. Such engagements provide teachers with opportunities to revise or adjust their teaching approach (Shalem, Sapire & Sorto 2014). The intervention programme that resulted from this study served the purpose of facilitating prolonged engagement with PSTs and points to various strategies that addressed misconceptions and resulting errors that were because of the lack of conceptual understanding of transformation geometry.

The use of error analysis to design a comprehensive instructional programme that uses students’ errors and faulty reasoning in transformation geometry as a starting point is one of the strengths of this study. This, especially, in the absence of error analysis acts as a tool to diagnose and address learner challenges in transformation geometry in the context of South African PST education programmes (Luneta 2008). It is further envisaged that this study would benefit lecturers in teacher education and teachers of mathematics in general. PSTs and teachers can use the study and the instructions programme developed to enhance the learning and teaching of transformation geometry. Alternatively, teachers can adapt the programme and use it towards developing their own instructional programmes based on analysis of student errors. Riccomini (2005) declared that whilst teachers might succeed in describing and understanding errors made by their students, most of them still lack the skills needed to align their lessons with the errors identified.

A Dutch couple, Pierre van Hiele and Dina van Hiele, in their work as mathematics teachers, conducted extensive research with the aim of understanding students’ reasoning in geometry. The results of their research culminated in the formulation of the now seminal Van Hiele theory, which is premised on the idea that students go through five levels of reasoning when working with geometric concepts: Recognition or visualisation, Analysis or Descriptive, Ordering or Informal deduction, Formal deduction and Rigour (Pusey 2003; Van Hiele 1986).

The strength of Van Hiele levels lies in the role of the teacher during the teaching and learning situation, where the student is guided using language that is appropriate for a specific level, towards achieving the next, higher level of reasoning.

In an attempt to help students proceed from one level to the next, the Van Hiele theory proposes five phases of learning that teachers can use to plan instruction (Abdullah & Zakaria 2013; Alex & Mammen 2016; Crowley 1987; Pusey 2003):

The Van Hiele theory proved to be a valuable framework in various studies that used it to explore and address students’ problems with geometry (Panaoura & Gagatsis 2009, Bansilal & Naidoo 2012, Luneta 2015b, Alex & Mammen 2016). A common outcome that is realised using Van Hiele theory is that, for instruction to be effective, the communication between the teacher and the students has to be at the level of development of the students. Teachers need to use a language that matches the reasoning capabilities of the students, and then gradually guide them towards using the correct mathematical vocabulary by the end of the lesson. Failure to do this could result in the teachers’ intended communication being interpreted incorrectly (Kuzniak & Rauscher 2011), which could lead to a lack of conceptual understanding and the development of misconceptions (Mateya 2008). Continuous and focused engagement with students, using a carefully planned instructional programme has a potential to enable the teacher to pick up possible miscommunication and address it.

Most studies applied Van Hiele theory at the schooling level than at the post-school level. Hence, Feza and Webb (2005) suggested more application of the theory in pre-service and in-service mathematics teacher training. This study therefore has the potential to contribute to knowledge in this field. Some teachers registered their difficulty in implementing effectively a geometry curriculum which they did not design themselves (Yilmaz et al. 2011). If such teachers could start by identifying their own students’ faulty reasoning in geometric concepts and contribute towards the design of the instructional programmes, such as the one in the current study, there is a possibility of an increase in competence and confidence amongst teachers. The added benefit of such contextualised contributions could be the reduction of dependency on textbooks, which is either not easily available (Ilaslan 2013), or provides simplistic and routine-like approaches to the teaching of mathematical concepts (Luneta 2015a).

Despite its significant contribution to the teaching and learning of geometry, the Van Hiele theory has been questioned for some of its potential weaknesses (Sinclair & Bruce 2015). For example, concerns have been raised that there is a possibility of students being in transition between levels or advancing to higher levels with some concepts and not with other (Mayberry 1983). This study took cognisance of these important observations by ensuring that the interventions at various levels of the Van Hiele theory were comprehensive and included various assessment tasks and illustrations and the teaching approach was research-based (Pegg 1985).

Based on its success in analysing student reasoning with geometry, as well as its potential in addressing challenges that students might have with understanding transformation geometry, the Van Hiele theory is used in this study to achieve the objective as articulated below.

The objective of this study was to explore BEd in FP students’ faulty reasoning in geometric translations in order to design a Van Hiele phase-based instructional programme.

Therefore, the study sought to answer the following main research question:

How can BEd FP students’ faulty reasoning in geometric translations be used to design a Van Hiele phase-based instructional programme:

This qualitative interpretive study (Cohen, Manion & Morrison 2010) sought to gain an in-depth understanding of students’ reasoning when working with geometric translations. This understanding would then inform the design of an instructional programme that addressed faulty reasoning amongst students. Therefore, the research approach adopted in this study was design-based research, a formative approach in which a product or process is designed and developed (Swan 2014), with the aim of educational improvement of learning, using a local instruction theory (Gravemeijer & Eerde 2009) of Van Hiele phase-based instruction.

The participants for the study were BEd in FP students in their second year of study at a newly established rural university in South Africa. Just under half of the students did Mathematical Literacy (ML) in their matric and the rest, except for one, did Mathematics. This distinction according to the two subjects could have had an effect on student performance because quite a number of students who did ML voiced their concern over the fact that they never did transformation geometry during their Further Education and Training years.

Even though all second year students registered for the mathematics module in the BEd programme formed the target sample and gave consent to participate in the study, 82 students participated in all the phases of data collection. Sampling was purposeful (Creswell 2012). All the students were sampled because, firstly, that was the year in which the topic of transformation geometry was taught and secondly, an intervention programme that was planned for later on had to include these students.

Data collection for the study occurred in three stages: tests, interviews and instructional design activities.

Thematic analysis (Braun & Clarke 2006) was used to analyse the tests and the interviews. Students’ responses, as recorded in test scripts, were analysed according to Braun and Clarke’s (2006) six phases of thematic analysis. That is, the researchers first familiarised themselves with the data, then they generated initial codes, organised the codes into possible themes, reviewed, defined and named the themes to produce the final report. Analysis of interviews started with students’ verbal recordings from the video transcripts. A similar process of thematic analysis was used in analysing the tests. The analysis was inductive because the themes were data driven (Clarke & Braun 2014). The themes created from the analysis of tests and interviews were used to answer the first research question. Analysis of data from the Van Hiele phase-based instructional design entailed content analysis of the activities that appeared in each of the phases, at a more micro level (Wilkinson 2000) to see if they were mapped to the correct theme (which represented some type of error). The analysis resulted in a record of clear, step-by-step teaching approaches derived from the analysed errors. The error analysis further gave guidance on how learning can be facilitated using various assessment tasks on transformation geometry.

Permission to conduct research was sought and obtained from the University of Johannesburg where the participants were enrolled. Written consent was obtained from all participants in the study, before the commencement of the research activities. The participants were informed of the purpose of the study, the procedures to be followed during the period of participation, such as video recording if they gave consent, as well as the freedom to withdraw from participating in the study any time if they chose to. The participants were further informed of confidentiality regarding sharing information that they might come across during the research, such as their names and their identities. Ethical clearance number: 2018-081

The first research question for the study was: What errors and associated misconceptions do BEd (FP) students make and have when solving problems involving geometric translations?

After completing Braun & Clarke’s (2006) six phases of thematic analysis, Table 2 was created, depicting the overall themes generated from the analysis of data from tests and interviews in response to the first research question. In response to the question students displayed different forms of errors. Figure 1 shows how thematic analysis was used to formulate themes, which fell under two main categories: non-systematic errors and systematic errors. The figure further illustrates two themes of the most prominent errors, where n represents the number of students that displayed the errors out of 82. The first theme was incorrect properties of transformation and under this theme the errors were coded as: confusing translation with rotation (n = 18); wrong translation method (n = 29), incorrect interpretation of coordinates (n = 21) and confusing the x and y axis (n = 8). The second theme was errors involving basic mathematics operations and under this theme the errors were coded as: wrong diagrammatic representations or interpretations of coordinates (n = 11) and incorrect calculations (n = 21).

Further analysis of the tests and interviews provided the answer to the above question. The first and second columns indicate the number of the question from the original test and an example of a students’ faulty response to the question, respectively. Additional researchers’ comments or explanations, where applicable, appear in the second column, in bold. The third column indicates, where applicable, key phrases and sentences used by students during interviews to substantiate their (faulty) responses as given in the second column.

The last two columns in the table answer the research question. That is, the errors and associated misconceptions are listed under the two main categories: Non-systematic errors and systematic errors.

The second research question for the study was: What steps can be followed to map student errors and misconceptions in geometric translations to the design of a Van Hiele phase-based instructional programme that can address faulty reasoning?

The programme overview is summarised to indicate how the activities that form part of the programme design were linked to the different Van Hiele phases as well as Van Hiele levels.