The purpose of this article is to report on an investigation of the perceptions and performance of mathematics teachers in a teacher professional learning (TPL) programme based on realistic mathematics education (RME) principles, which included a topic on transformations, undertaken by the researcher. Fortyseven Senior Phase (Grade 7–9) teachers took part in the mixedmethods study in which they answered a questionnaire with both closed and openended items. Fifty teachers took an achievement test at the end of the programme. The TPL programme used the RME approach in the design and delivery of mathematical tasks intended to enhance teachers’ mathematical knowledge for teaching. The sessions were conducted in a manner that modelled one way in which RME principles can be adopted as a teacher professional development strategy. The significance of the study is that continuing TPL is acknowledged to contribute to improvement in teaching and learning to address the concern about unsatisfactory learner achievement in mathematics. The responses suggested that the majority of teachers experienced the sessions positively in relation to all but one of the six RME principles. The teachers reported that they took an active part both as individuals and in small groups and expressed their willingness to adopt the type of activities and materials for their classrooms, which is an essential first step in Guskey's first level of evaluation of a teacher TPL programme. The teachers’ average performance in an achievement test at the end of the topic was 72% which was indicative of modest learning gains at Guskey's second level of TPL effectiveness.
The poor performance of South African Grade 9 learners in the annual national assessments for mathematics in 2012 (Department of Basic Education,
Sowder (
As part of a probable solution this study sought to articulate the design and implementation of a realistic mathematics education (RME)informed TPL programme and to investigate, at Guskey's (
This study mainly sought to carry out Level 1 and Level 2 evaluations of the initial contact or workshop session. Guskey's (
The curriculum materials used for mathematics inservice teacher education for the senior phase in this study were specifically designed to allow for a realistic mathematics education (RME) approach, which originated from the Freudenthal Institute in the Netherlands (Van den HeuvelPanhuizen,
Clarke and Hollingworth (
For example, the uptake of RME principles from a professional learning programme into classroom practice might become possible if the programme helps them modify their selfefficacy beliefs about their ability to enact the principles in their teaching through ‘vicarious experiences and contextualised practice’ (Posnanski,
However, note was made of the following criticisms of an RME approach directed to primary mathematics education as summarised by Van den HeuvelPanhuizen (
In the present study the workshop sessions for teachers were designed and conducted in accordance with Van den HeuvelPanhuizen's (
This principle refers to the interpretation of mathematics as a human activity (Freudenthal,
The reality principle emphasises that RME is aimed at having students be capable of applying mathematics (Van den HeuvelPanhuizen,
The level principle underlines that learning mathematics means that students pass through various levels of understanding (Van den HeuvelPanhuizen,
One of the enduring strengths of the level principle, thus, is that it guides growth in mathematical understanding, from the concrete or enactive, to the iconic and, ultimately, to the symbolic representational forms espoused by Bruner (
This principle means that mathematical domains such as number, measurement and data handling are not considered as isolated curriculum chapters but as heavily integrated (Van den HeuvelPanhuizen,
This principle signifies that the learning of mathematics is not only a personal activity but also a social activity (Van den HeuvelPanhuizen,
This principle means that students are provided with a ‘guided’ opportunity to ‘reinvent’ mathematics by ‘striking a delicate balance between the force of teaching and the freedom of learning’ (Freudenthal,
The didactic approach to be used for the sessions was explained to the teachers in advance and teachers were encouraged to identify its strengths and challenges. With that understanding, the following research questions guided the study:
What were the Senior Phase mathematics teachers’ reactions to the usefulness of the RMEbased TPL programme?
How did the participants perform in an achievement test at the end of the topic on transformations?
A concurrent mixedmethods approach was adopted using a semistructured questionnaire with both closed and openended items and an achievement test with assorted items on transformations. The research design took the form of a survey and an achievement test to elicit the quickest responses with the least strain on teachers and yet be informative enough for subsequent sessions later in the year.
A convenience sample of 47 (out of 53) Senior Phase mathematics teachers that participated in the workshop sessions responded to the feedback questionnaire.
Demographic data.
Variable  No. of teachers 



Female  27 
Male  19 
Blank  1 


< 30  7 
30–39  11 
40–49  14 
50–59  7 
≥ 60  2 
Blank  6 


< 5 years  8 
5–9 years  9 
10–14 years  4 
15–19 years  6 
20–24 years  4 
25–29 years  5 
≥ 30  5 
Blank  6 


7  1 
8  7 
9  19 
10  13 
11  3 
12  2 
Blank  2 


PDE  3 
HDE  3 
NPDE  1 
Dip. Ed.  3 
ACE  4 
Degree  12 
PGCE/Hons  5 
Blank  13 
PDE, Primary Diploma in Education; HDE, Higher Diploma in Education; NPDE, National Professional Diploma in Education; Dip. Ed., Diploma of Education; ACE, Advanced Certificate in Education; PCGE, Postgraduate Certificate in Education; Hons, Honours.
Fifty out of 53 teachers took the achievement test on transformations at the end of the TPL sessions.
The main instruments for the study were:
A questionnaire consisting of eight Likerttype items intended to elicit answers from both closed and openended questions and an openended item (see
An achievement test at the end of the twoweek TPL sessions during which the topic was presented.
Feedback questionnaire.
Item total statistics.
% agreement  

Item no.  Short description  SD  D  U  A  SA  Corrected itemtotal correlation  Cronbach's alpha if item deleted 
1  Reality principle  –  –  –  66  34  0.712  0.701 
2  Activity principle  –  –  –  74  26  0.670  0.711 
3  Level principle  –  –  2  79  19  0.475  0.777 
4  Intertwinement principle  4  15  2  40  39  0.537  0.728 
5  Interaction principle  –  2  2  60  36  0.347  0.751 
6  Guidance principle  –  2  2  81  15  0.263  0.761 
7  Relevance to CAPS  –  2  2  66  30  0.500  0.726 
8  Teaching materials  –  2  2  68  28  0.568  0.714 
SD, Strongly disagree; D, Disagree; U, Undecided; A, Agree; SA = Strongly agree.
For all items Cronbach's alpha = 0.758; Standard deviation = 3.161.
The results show that most of the items or principles received approval as having been in evidence in the presentations of geometrical transformations. The Cronbach's alpha reliabilities for all item deletions all fell within an acceptable range of between 0.7 and 0.8, including the overall value. Two items, in respect of the level and the guidance principles, had Cronbach's alpha values higher than the overall value, suggesting that their deletion would improve the reliability of the scale. However, the differences were not statistically different so the items were maintained for completeness of the reporting.
Distribution of marks per subtopic.
Transformation concept  Symmetries of quadrilaterals  Symmetries of irregulars  Transformation rules  Rotations of 90 degrees  Construction of plane shapes  Congruency and its applications 

Possible mark  11  9  9  7  7  7 
Mean mark achieved  9.4  4.5  7.4  6.3  4.8  5.0 
% achievement  86%  50%  82%  89%  70%  72% 
Not achieved  1.6  4.5  0.2  0.7  0.2  2.0 
Boxandwhisker plot of the overall performance.
The fivenumber summary of the distribution was: a minimum mark of 40%, a lower quartile mark of 63%, a median mark of 74.5% (compared to a mean mark of 72.1%, meaning scores were skewed to the left by 0.412), an upper quartile mark of 82% and a maximum mark of 96%. The distribution had a standard deviation of 13.34 and a standard error of 1.887. The mean was still lower than would be expected of teachers teaching these concepts but the variance was understandable given that some teachers were not necessarily qualified to teach mathematics.
Some responses gave evidence of awareness of the importance of linking mathematics to the everyday life experiences of learners (reality principle, as exemplified in sample task item 1 in
Teacher 13:
Especially for the FET Phase where learners do not have a grasp of linking it into everyday life.
Teacher 15:
Very good examples, the examples is very practical and relates to reallife situations.
Teacher 42:
They were challenging, especially with transformation you ended up taking a tracing paper so that you can make sure if your transformation is correct.
Whereas some participants felt the tasks were easy, others felt that they were challenging or at least would be challenging enough for their learners. Tasks that are challenging should scaffold learners to move from one level of understanding to the next (level principle, as exemplified in task item 2 in
Teacher 1:
It start with selfdiscovery and it end with problems of a ‘higher order’ level. It allows critical thinking. There were some stage that I felt so ‘stupid’ but eventually I got it right. Specifically the reflection in
Teacher 23:
Plus or minus 20% challenging. But I am currently teaching Grade 9 and 10 maths (for 32 years!!). The ones I found challenging were module 30: rotations and enlargement; similarity activities.
Teacher 45:
It took a lot of different cognitive skills. e.g. When you have to do the rotation of 180° clockwise and anticlockwise.
The level of participation by participants as individuals (activity principle) as well as in small groups (interaction principle) was perceived as in the following sample responses:
Teacher 15:
I took active part as an individual in my group, I had some particular time where I was explaining to my colleagues. This took part for the whole group where one member would be explaining to us where we were not understanding.
Teacher 9:
I found the answers first then consulted.
Teacher 37:
As an individual I have to draw making the diagrams, answers then as a group we make comparisons of our answers.
The importance of the interaction principle at work was illustrated by the following responses, amongst others, which show how collaborative work helped the participants overcome their challenges:
Teacher 8:
At times I differed with group members about some answers. We explained to each other and learned from each other.
Teacher 10:
I as a teacher struggle with some of the concepts and the team mates and facilitator made it clear to me. Thanks!
Teacher 13:
Although we spoke different home languages and Afrikaans is my mother tongue my peers helped me by explaining the meaning of difficult words and concepts and formulas.
Although most of the guidance principle was built into the materials and the overall approach, some of it was evident in the following sentiments:
Teacher 23:
I have picked up/been exposed to a lot of new ideas how to present my lessons, especially on Grade 9 and 10 level. Be more practical in the class! More constructions. My own worksheets in class must be clear and well planned.
Teacher 30:
If something wasn't so clear to us we consulted each other in groups or with the facilitator.
Teacher 39:
The material can also be used in the classroom for own lessons as it contains adequate scaffolding.
The overwhelming majority of the teachers who attested to the relevance of the topic content to the Senior Phase described such relevance in varied ways, as the following examples show:
Teacher 19:
It includes topics outlined in the work schedule and it had many activities which will assist learners in problem solving.
Teacher 23:
It was relevant because it is on the syllabus of Grade 8 and 9. It also emphasises the starting point for this topic.
Teacher 34:
It is relevant in the Senior Phase but looking at the FET (Grades 10, 11 and 12) Maths the transformation is no longer done in CAPS.
The majority of participants reported that the materials they used were adequate for the tasks and some declared their readiness to adopt some of the activities and materials used:
Teacher 17:
Everything we needed to use was available to us. The tracing paper were available to do the construction, instrument box to draw circle and triangle.
Teacher 4:
The problem we encounter in our school have large numbers and we don't have some of the equipment to demonstrate these types of transformations (e.g. mirrors) line of symmetry same as with patterns. I learnt a lot from this class because everything was demonstrated to us.
Teacher 8:
It was adequate and appropriate because the chapter of transformation were covered and we had all the material needed to complete the activities. The material needed are things you have in your class.
Baturo, Cooper, Doyle and Grant (
Many participants felt that the time allocated for the topics covered was inadequate. An almost equal number apiece were (1) happy with the delivery mode as it was, (2) needed more whole class facilitator explanations and discussions or (3) would have preferred their home language (predominantly Afrikaans) to be used for instruction and materials. Some even felt they were competent enough to be engaged as facilitators. The examples below show some typical recommendations:
Teacher 39:
We need more time … was squashed!!! I was not able to complete the activities of any day except for the last day (Friday) (That was when I concentrated mostly on myself.) This was not good in my opinion as we were at the course to SHARE and ENRICH each other.
Teacher 8:
I am happy with the workshop as it is. Such workshops are good for us. You also need to do them even for Grade 11 and 12 as well in future. Topic we can be covered again in Grade 9 is finance and conversion table (metric + imperial).
Teacher 40:
Well only worked small group; but maybe there could have even more of a plenary where we could hear from the class as a whole.
Teacher 11:
Notes in Afrikaans will be much appreciated. You can go through us with the answers. More explaining. Teachers forget things
Teacher 3:
Would enjoy being part of your presenters team – I am a 100% educator.
Whilst the quantitative results showed an approval of the RMEbased teacher professional learning opportunity the teachers received, the qualitative aspects of the data gave some details about the specific instances in which the teachers perceived the contact session to have fulfilled their expectations at Guskey's Level 1 of professional learning effectiveness. All the main principles of RME were in evidence in the teachers’ feedback with the exception of the intertwinement principle, which was largely built into the materials. The participants overwhelmingly felt that they were actively engaged both as individuals (activity principle) and as small groups (interaction principle). They also felt that the activities were based on reallife experiences that could be of interest to their learners (reality principle) and challenging enough to their learners and to some of the participants (level principle). They felt that the materials used helped them to understand the geometry of transformations much better and they thought that they received enough guidance from the curriculum materials used (guidance principle). However, they felt that there could have been more wholeclass discussions to tie up the loose ends. This was not surprising for teachers who came from a background of traditional teachercentred approaches that still dominate the overcrowded, underresourced mathematics classrooms in disadvantaged communities.
In line with their varied levels of experience and expertise, the teachers varied widely in their levels of mathematical selfefficacy, some feeling overwhelmed because of teaching the subject for the first time or inadequately supplied with curriculum materials whilst a few others felt on top of the game and wished they could be involved as resource persons. That was an exciting prospect, which should open up the possibility of transformative models for use in sustained, collaborative, professional development (Johnson & Marx,
Lessons drawn from the workshop activities were principally that tasks should revolve around learners’ experiential world in order to be of interest and relevance to them (reality principle). There was considerable positive attitude expressed towards activitybased learning (activity principle) and collaborative learning (interaction principle). TPL programmes should thus relentlessly model the teaching strategies intended for teachers to adopt in their classrooms. As Windschitl (
The purpose of this study was to investigate inservice teachers’ perceptions of the relevance of a teacher professional learning programme modelled on RME principles. A major positive outcome of the study was that many teachers felt better prepared to adopt some activityoriented tasks in their classrooms in the teaching of not only transformations but geometry and mathematics in general. Unless teachers can commit themselves to instituting change in the ways they teach the subject in their classrooms little change can come about. Given the endemic shortage of qualified mathematics teachers, it is not surprising that many teachers of mathematics are not specialist mathematics teachers especially in the Senior Phase and thus require constant support. Of interest too was the fact that teachers are sometimes alert to syllabus changes that bring about discontinuity in mathematical content that is taught in different phases of the schooling system. The recent removal of transformation geometry from the Curriculum and Assessment Policy Statements for the Further Education and Training Phase (Grades 10–12) has caused some teachers to question the future relevance of the topic to the Senior Phase. ‘If it is not going to be examined later at the National Senior Certificate, why bother?’ seems to be a rational question in a system increasingly driven by high stakes examinations. These are questions that curriculum planners have to ponder to convince teachers about the wisdom of including topics that the latter have historically not been comfortable with in the first instance.
Finally, the fact that teachers in this study engaged with mathematical content meaningfully helped to address some of their classroom mathematical knowledge needs. Although a few still scored marks below 50% the average mark of 72% was presumably modest enough to inspire confidence in their feelings of selfefficacy in transformation geometry (Guskey's Levels 1 and 2). Further followup research is needed to explore the impact of the TPL programme on organisational teacher support (Guskey's Level 3), implementation support and monitoring (Guskey's Level 4) and ultimately on student learning outcomes (Guskey's Level 5).
This study was limited to teachers’ perceptions of the effectiveness of a contact session of a teacher professional learning programme based on RME principles and scores obtained in an achievement test. It therefore does not cover the full spectrum of the entire lifespan of the programme which included class visits and another contact session later in the same year.
Permission was granted by the Western Cape Education Department to conduct this study and ethical clearance was obtained from Stellenbosch University's Research Ethics Committee. The participants in this study signed letters of consent and were advised of the objectives for the research, which primarily sought to improve future inservice training. The participants were assured of confidentially and anonymity. Their names were neither required in their questionnaire responses nor were they going to be used in the analysis of the achievement test results.
I acknowledge that my involvement in this study was part of my research duties at the Stellenbosch University Centre for Pedagogy (SUNCEP). I am also grateful to my colleagues, Ramesh Jeram (for allocating me the responsibility of the geometry modules covered in this study) and Cosmas Tambara (for helping me in the administration of the questionnaire).
I declare that I have no financial or personal relationship(s) that might have inappropriately influenced me in writing this article.
Sometimes there are good and practical reasons to make an object symmetrical.
What type of symmetries do you find in each of the following four playing cards: 8 of clubs, 10 of diamonds, Queen of spades, and King of hearts.
Why is it convenient when a playing card is symmetrical?
What sort of symmetries occur in this nut and spanners?
Investigate each of the figures below for line (reflectional) or rotational symmetry by showing the following, where applicable (for line symmetry check your answer by folding or using the provided mirror, for rotational symmetry use the tracing paper provided):
line axis (or axes) of symmetry,
the centre point of rotation
the angle of rotational symmetry,
the order of symmetry.