Preservice teachers (PSTs) training does not equip students with adequate skills and knowledge of geometry to enable them to teach this section of mathematics competently. Inadequate teacher knowledge of transformation geometry, in particular, requires intervention that targets PSTs’ faulty reasoning displayed in errors they make.
The aim of this study was to explore the use of Bachelor of Education (BEd) students’ faulty reasoning in geometric translations, in designing a Van Hiele phasebased instructional programme that could address such faulty reasoning.
The setting for the study was a newly established rural university in South Africa.
Tests on geometric translations were administered to BEd Foundation Phase students, followed up by interviews to explore errors made when responding to the test items. The errors were then mapped to the design of a Van Hiele phasebased instructional programme.
The results revealed that the students had several misconceptions with geometric translations. The misconceptions were delineated into the errors that the students displayed and these were classified under two themes. The first theme was incorrect properties of transformation and under this theme, the errors were coded as confusing translation with rotation, wrong translation method, incorrect interpretation of coordinates and confusing the x and y axis. The second theme was errors involving basic mathematics operations including wrong diagrammatic representation of coordinates and incorrect calculations.
The study showed that if the students’ misconceptions and the resulting errors are mapped to specific instructional approaches, their faulty reasoning in geometric transformations is addressed and effective learning is enhanced.
Low performance in mathematics has been a worldwide challenge for many decades, for instance, in 1989, the then president of the International Commission on Mathematics Instruction (ICMI), Dr Hassler Whitney said ‘for several decades we have been seeing increasing failure in school mathematics education, in spite of intensive efforts in many directions to improve matters’. This scenario is true even today. Mathematics education in South Africa is labelled as being in a state of crisis, with learners performing dismally in several international assessments such as Trends in International Mathematics and Science Studies (TIMSS) (Spaull
Teachers are believed to play a significant role in the quality of mathematics that is offered to students, through the knowledge that they bring to the classroom (Ball, Hill & Bass
This article is derived from a bigger study that sought to examine PSTs’ challenges with transformation geometry in the context of primary school teacher training. The focus here is on exploring problems involving geometric translations, in order to design relevant instructional activities that will address such challenges. Through the researchers’ experience of teaching transformation geometry to students in the Bachelor of Education (BEd) in Foundation Phase (FP) programme, they have seen the errors displayed and misconceptions held by students when learning about this topic. This study therefore seeks to explore these errors and map them to the design of a programme of instruction that is informed by Van Hiele phasebased learning. The error mapping intervention was intended to assist the researchers and other mathematics lecturers and teachers in general in mediating the teaching of geometric translations.
Research indicates that even though teachers are able to identify overall trends and weaknesses in their learners’ work, they lack skills to apply this knowledge in developing a suitable intervention programme (Holmes et al.
Deficiencies in geometric reasoning exist with both students and teachers. Studies involving investigations into South African preservice 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
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
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
Students’ misconceptions and the associated errors in transformation geometry are stubborn and persistent (Bansilal & Naidoo
This study involves PSTs’ faulty reasoning in transformation geometry. Faulty reasoning is detected in the errors students display. Luneta and Makonye (
Examples of instances that might involve nonsystematic 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
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
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.
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
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:
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
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
Most studies applied Van Hiele theory at the schooling level than at the postschool level. Hence, Feza and Webb (
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
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 phasebased 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 phasebased instructional programme:
What errors and associated misconceptions do BEd (FP) students make and have when solving problems involving geometric translations?
What steps can be followed to map student errors and misconceptions in geometric translations to the design of a Van Hiele phasebased instructional programme that can address such faulty reasoning?
This qualitative interpretive study (Cohen, Manion & Morrison 2010) sought to gain an indepth 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 designbased research, a formative approach in which a product or process is designed and developed (Swan
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
Data collection for the study occurred in three stages: tests, interviews and instructional design activities.
The participants wrote two tests, a short multiple choice (MC) test and a longer discussion (D) test where students had to give explanations or show calculations for their answers. Both tests were administered during the first semester of the year, which coincided with the period during which transformation geometry was supposed to be taught as part of the BEd curriculum. The tests were written before teaching any topic and therefore served as diagnostic tools to assess students’ understanding of challenges with geometric translations, so as to answer the first research question (Flick
Soon’s levels and the link between test items and foundation phase teachers’ challenges with transformation geometry.
Levels  Characteristics: The student  Link between test items and errors 


Example of test item  Targeted error/s  
Level 1  Identifies transformation by the changes in the figure, (1) in simple drawings of figures and images and (2) in pictures of everyday applications. Identifies transformation by performing actual motion; names, discriminates the transformation. Names or labels transformations using standard and nonstandard names and labels appropriately. Solves problems by operating on changes of figures or motion rather than using properties of the changes. 
Systematic errors involving understanding of properties of different transformations Lack of visualisation skills The correct answers would reflect both procedural and possible conceptual knowledge 

Level 2  Uses the properties of changes to draw the preimage or image of a given transformation. Discovers properties of changes to figures resulting from specific transformation. Uses appropriate vocabulary for the properties and transformation. Is able to locate axis of reflection, centre of rotation, translation vector and centre of enlargement. Relates transformations using coordinates. Solves problems using known properties of transformations. 
What are the coordinates of the vertex if the figure is rotated 180° clockwise about the origin? 
Nonsystematic errors involving leaving out signs when reading/writing coordinates Systematic errors such as confusing rotation with reflection or translation. The errors and even the answers will also enable the teacher to assign intervention instructions targeted at conceptual knowledge acquisition or more advanced enrichment activities 
Level 3  Performs composition of simple transformations. Describes changes to states (preimage, image) after composite transformations. Represents transformations using coordinates and matrices. Interrelates the properties of changes to a figure resulting from transformations. Given initial and final states, can name a single transformation. Given initial and final states, can decompose and recombine a transformation as a composition of simple transformations. 
B, E and C are midpoints of AD, DF and AF, respectively. Triangle CEF is formed by applying transformation to the triangle ABC. What could this transformation be? 
Systematic errors involving properties of the different transformations Inability to recognise (visually or otherwise) geometric translations Errors displayed here reflect lack of both conceptual and procedural knowledge and would be more systematic. Interventions would require even reteaching of the concepts. Correct answers show advanced knowledge in transformation geometry 
Level 4  Gives geometric proofs using transformational approach. Gives proofs using the coordinates and matrices. Thinks through multistep problems and gives reasons for problems. 
Prove, using diagrams and some explanations, that the rule (x; y)→(x – 2; – y + 3) represents a combination of a reflection followed by a translation. Your diagram and explanation must clearly show the line of reflection as well as the units through which the figure is translated.  Systematic errors involving properties of transformations Inappropriate vocabulary used when describing transformations At this level, the systematic errors are predominantly conceptual and most of them were revealed in the interviews. The intervention involved remedial activities and reconceptualisation of the learners views using the errors 
Adapted from Guven, B., 2012, ‘Using dynamic geometry software to improve eighth grade students’ understanding of transformation geometry’,
Semistructured interviews were carried out with 21 students as key informants (Cohen, Manion & Morrison
Following the analysis of data from the tests and interviews, an instructional design programme, based on Van Hiele phases of learning, was developed. Errors on geometric translations, which were already coded and categorised, were used to develop activities at the different Van Hiele levels. The activities were then linked to the five Van Hiele phases of learning.
Thematic analysis (Braun & Clarke
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: 2018081
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 (
Prominent student errors according to themes.
Examples of coding of data from tests and interviews through thematic analysis.
Question number  Examples of students’ test responses with errors  Related interview data with error/misconception (where applicable)  Initial coding  Emerging themes 

1  ‘This one looks the same as original… must be rotation of 360°’.  Confusing translation with rotation  Incorrect properties of transformations  
2  ‘… Count from the end…. then when your reach the diagram … then your count down…’  Wrong translation method  Incorrect properties of transformations  
8  ‘42 = 2’ … ‘sign of the bigger number’  Wrong diagrammatic representations or interpretations  Errors involving basic operations  
13    Incorrect interpretation of coordinates Incorrect calculations  Incorrect properties of transformations 

14  ‘Vertical line is xaxis … translation over the xaxis’  Confusing xaxis and yaxis during reflection  Incorrect properties of transformations 
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: Nonsystematic 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 phasebased 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.
The Van Hiele programme design was summarised as depicted in
The starting point towards answering this question was the isolation of the errors, as they appear under each theme from the results of tests and interviews, in
Themes generated from the analysis of tests and interviews to answer research question 1.
Themes  Errors and associated misconceptions 


Nonsystematic errors  Systematic errors  
Incorrect properties of transformations    Inability to recognise (visually or otherwise) geometric translations, for example, when a figure has changed orientation, or visualising what a figure should look like after a particular translation. Confusing, swapping or considering only some of the properties (and ignoring others) of translations. Example: Identifying a translation as a reflection. Inability to physically perform a translation Incorrect or inappropriate description of a translation For example, Believing that translations always ‘move from left to right’ A translation always results in an image being ‘in a different quadrant’ than the original figure 
Errors involving language issues  Carelessly reading/writing words without paying attention, such as reading/writing 
Inadequate knowledge of, unfamiliarity with or confusing certain terminology used in geometric translations, such as Inappropriate or incoherent vocabulary used when describing transformations, such as: ‘…measure the size of the shape…’ (challenges with communicating using the correct language) 
Misreading or misunderstanding instructions  Reading instructions carelessly without paying attention or focusing on what is required. 
Interpreting instructions incorrectly resulting in performing the wrong action or transformation 
Errors involving basic operations    Incorrect calculations resulting in plotting of incorrect points or incorrect transformations 
Incorrect plotting of points  Leaving out a negative sign when writing coordinates. Plotting incorrect points, and then realising the error on their own. 
Swapping xcoordinate with ycoordinate, or leaving out a negative sign from the x or y coordinate, leading to plotting of incorrect points and drawing of incorrect figures. 
Missing information  Leaving out a negative sign when writing coordinates and then realising their careless mistake on their own 
 
Overview of the Van Hiele phasebased programme design.
Van Hiele phase  Focus of design programme activities: Examples at Van Hiele level 2 

Teacher determines if students could discover properties of changes to a point resulting from translation and if they recognise specific properties of shapes.  
Teacher guides students towards discovering the relationship between physical translation of points on the coordinate system of axes and corresponding algebraic changes in coordinates.  
Students express their understanding of both algebraic calculations and physical shifting of shapes when working with translations.  
Students use properties of translation to provide own solutions and argue convincingly about whether given figures are translations of each other or not.  
Students answer questions based on the key concepts and knowledge acquired on translation at the current Van Hiele level 2, before the phasebased instruction process is repeated for the next Van Hiele level 3. 
Mapping errors and misconceptions to Van Hiele phasebased activities.
Errors and misconceptions  Van Hiele phasebased activities 

Reading or listening carelessly to instructions without paying attention or focusing on what is required. Interpreting instructions incorrectly resulting in performing the wrong action or transformation. Inappropriate or incoherent vocabulary used when describing transformations 
Students play a game called ‘I am. You are’. Students work in pairs and each pair is given a card on which coordinate pairs are written. The cards are related to each other according to some ‘rule’. The teacher/lecturer chooses the first pair of students to read the information on their card whilst all other student pairs are listening carefully and working out if the information being read relates to or refers to their card. Then the next pair whose card relates to the information read by the previous pair will follow by reading their own card. This continues until the information in all pairs’ cards has been read and shared. (‘This game encourages students to listen carefully to instructions, as well as challenges them to do algebraic calculations quickly in their heads, because they cannot delay much with their answers since the other pairs are waiting to see who had the next card. At the same time, the game forces them to use algebraic rules and calculations because the system of axes is not drawn and there is no time for them to plot the points and physically count the number of units shifted. The purpose of using pairs is to allow students to help each other so that they could do the calculations faster more competently’). 
Incorrect calculations resulting in plotting of incorrect points or incorrect transformations 
‘I am point M with coordinates (2; 4). Who is 2 units to my right and 6 units below me?’ 
Inability to recognise (visually or otherwise) geometric translations, such as not visualising what a figure should look like after a particular translation. Believing that translations always ‘move from left to right’ Inability to physically perform a translation Inappropriate or incoherent vocabulary used when describing transformations, such as: ‘…measure the size of the shape…’ A translation always results in an image being ‘in a different quadrant’ than the original figure 
Could the figure on the left be a translation of the figure on the right? Why do you say so? 
The main purpose of this study was to explore BEd in FP PSTs’ faulty reasoning in transformation geometry, with a focus on geometric translations. The understanding of these faulty reasoning, as displayed through errors made by students, was to be used to design Van Hiele phasebased instructional activities that were mapped to the errors and misconceptions identified. The instructional design was meant to assist teachers and PST educators with the development of instructional material used to facilitate an intervention that would seek to address and help improve students’ faulty reasoning in transformation geometry. The instructional design was critical considering the lack of such interventions at the South African primary school level of mathematics education, as well as the scarcity of relevant textbooks for targeted intervention. The mapping of activities to errors formed the major part of the study and the results were a programme that can be replicated with other aspects of transformation geometry such as reflection and rotation.
The results of the study revealed that BEd in FP students have several misconceptions associated with learning and solving tasks involving geometric translations. These errors were classified under two main categories: nonsystematic errors and systematic errors, confirming previous studies (Makonye & Luneta
The use of Van Hiele phasebased instructional design in this study provides a novel way of using students’ errors and misconceptions to enhance learning outcomes. Student errors were mapped successfully to activities designed using both Van Hiele levels and Van Hiele phasebased activities. Stepby step guidance is provided for teachers and other potential beneficiaries in mathematics education on how to map student errors with activities.
We propose for other studies that could work with teachers and PST educators to actually implement an intervention based on Van Hiele phasebased instructional design. Such research could use the activities developed in this study. However, the strength of such implementation would lie in their ability to be innovative as well as to envision how their students might participate in the proposed activities (Gravemeijer & Van Eerde
In this research, a series of activities, based on the Van Hiele phases of learning, were developed, and used during the implementation of an intervention programme to address students’ errors in transformation geometry. These activities were organised into a comprehensive document that provides guidelines and activities on teaching transformation geometry to student teachers in the FP. This research thus contributes to knowledge creation. This is important because it gives teachers an idea of how to design and use instructional programmes such as one based on Van Hiele theory to improve learners’ reasoning and understanding of geometry, a component of mathematics that presents with challenges throughout the schooling years.
Teacher educators should rethink the strategies they use to address challenges they experience with their students’ competence and reasoning in geometry. This study has shown that if student errors and misconceptions are the starting point of lesson planning and design, there are better chances of focusing instruction where it will make an impact, given that one would already know what concepts to target, what concepts to spend more on or be more innovative with. We further point that one should not start from scratch because there are available theories, such as Van Hiele phasebased instruction, that have been well researched and give clear guidelines on how to adapt and use them for one’s unique situation.
The authors would like to acknowledge the participation of the students from the university where the research was conducted.
The authors have declared that no competing interests exist.
N.P.S. collected and analysed data. Both authors contributed equally towards writing this article.
This research received no specific grant from any funding agency in the public, commercial or notforprofit sectors.
The data that support the findings of this study are available from the corresponding author, N.P.S., upon reasonable request.
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.