Problem-solving is of importance in the teaching and learning of mathematics. Nevertheless, a baseline investigation conducted in 2016 revealed that mathematical problem-solving is virtually missing in South African classrooms. In this regard, a two-cycle design-based research project was conducted to develop a professional development (PD) intervention that can be used to bolster Grade 9 South African teachers’ mathematical problem-solving pedagogy (MPSP). This article discusses the factors that emerged as fundamental to such a PD intervention. Four teachers at public secondary schools in Gauteng, South Africa, who were purposively selected, participated in this qualitative research study of a naturalistic inquiry. Teachers attended PD workshops for six months where PD activities that were relevant to their context were implemented. Between the PD workshops, teachers were encouraged to put into practice the new ideas on MPSP. Qualitative data were gathered through reflective interviews and classroom observations which were audio-recorded with teachers’ consent. Data were analysed through grounded theory techniques using constant comparison. The findings from the study suggested that teachers’ personal meaning, reflective inquiry, and collaborative learning are factors fundamental to their professional growth in MPSP. The major recommendation from the study is that facilitators of PD must acknowledge these factors to promote teachers’ professional growth in MPSP. If PD processes and activities are relevant to teachers’ personal meaning, reflective inquiry, and collaborative learning, teachers find the PD programme fulfilling and meaningful. This study contributes to the PD in MPSP body of knowledge by having worked with teachers in an under-researched context of historical disadvantage.

Historically, South African learners perform poorly in mathematics in national tests of achievement like Annual National Assessments (ANA), in regional tests of achievement such as Southern and Eastern African Consortium for Monitoring Education Quality (SACMEQ), and in international tests of achievement like Trends in Mathematics and Science Study (TIMSS). Of the approximately 270 500 Grade 12 learners who wrote mathematics in the 2018 matric examinations, only 37% passed with 40% and above. These results indicate that South African learners’ performance in mathematics is inadequate and this crisis has existed for over 20 years (Van Jaarsveld & Ameen,

Further findings from the baseline study were that problem-solving is virtually missing in South African classrooms. The findings from the baseline study are consistent with Jagals and Van der Walt’s (

MPSP is of importance since, besides learners’ preconceptions, teaching is widely known to influence learning. Consequently, if teachers are to encourage learners to do problem-solving or to improve learners’ problem-solving skills, then teachers need to deepen their MPSP. MPSP can enrich teachers’ mathematical experiences because it broadens their views of what it means to do mathematics with learners. Teachers’ MPSP can be grown by providing them with PD programmes that focus on the teaching of mathematical problem-solving. This kind of PD is commonly known to improve teachers’ confidence in implementing MPSP and their interaction with learners. Subsequently, this improves learners’ mathematical problem-solving processes and abilities. However, there is no sufficient literature on research on the PD of South African teachers that focuses on MPSP. Over the years, several longitudinal projects that focus on PD of primary and secondary mathematics teachers have been conducted in South Africa (see Biccard,

Context relates to the factors external to teachers and learners that may affect the process of teaching and learning (Lester,

This study focused on designing a PD for teaching mathematical problem-solving. Problem-solving is when a problem-solver is confronted by a mathematical problem for which they have no known solution (Schoenfeld,

In this study, Polya’s (

In order for problem-solving to become an integral part of learners’ experience in school and university, all aspects of the human psyche, cognition, affect, behavior, attention, will and metacognition or witnessing must be involved. Focusing on only one or two aspects is simply inadequate and very unlikely to lead to full-scale integration into learners’ ways of being in the world. (p. 109)

The picture presented by Mason (

Schoenfeld (

Professional development involves the programmes implemented to enhance teachers’ knowledge and practices with the key objective of improving learner achievement (Loucks-Horsley, Hewson, Love, & Stiles,

The above programmes are from around the world; however, in South Africa there are few programmes that explicitly support teachers in teaching mathematical problem-solving. To address this identified gap, in the large project, I developed a PD intervention for teachers’ MPSP. The study reported in this article focuses on the factors fundamental to such a PD intervention. In the context of the study, the PD intervention was about supporting teachers who worked with mathematical word problems found in the CAPS curriculum. Mathematical word problems are verbal or (con)textual descriptions of a problem situation (Verschaffel et al.,

I adopted andragogy, the adult learning theory, as a theoretical framework for this study. In 1968, Malcolm Shephard Knowles proposed a theory that distinguished adult learning (andragogy) from children learning (pedagogy). Andragogy is the art and science of adult learning. It refers to any kind of adult learning (Kearsley,

Consequently, they prefer a more self-directed learning approach rather than being teacher-led. In this study, teachers were contributors to their own PD experiences in MPSP: they were actively involved in the workshops’ activities and the implementation of the MPSP. Thus, teachers had control over their learning. Secondly, past experiences are important in adult learning (Knowles et al.,

Thirdly, adults are ready to learn when there is a purpose and relevance, for example when it is about professional growth connected to their work. Adult learning experiences must be scheduled so that they are concurrent with their readiness to learn (Knowles et al.,

Fourthly, adults are interested in learning practical skills that help them to solve problems. As human beings mature, their orientation towards learning switches from subject-centeredness to problem-centeredness. Teachers were interested in participating in the PD for MPSP because it would help them assist their learners in doing mathematical problem-solving. Finally, adult learners are driven by internal motivation. Adults develop their methods of motivation and are driven to learn for their reasons, for example for promotion or to bolster self-esteem. Knowles (

Knowles (

Sztajn, Borko and Smith (

After identifying the need to design a PD intervention to support Grade 9 teachers in MPSP, the first phase involved a literature review, context analysis, and the baseline study. The second phase involved two cycles of design-implementation-evaluation-redesign of prototypes of the PD intervention. I designed the first prototype from the design principles generated from the literature review, context analysis, and the baseline investigation.

I implemented the first prototype in the first cycle. Next, I designed the second prototype from the design principles generated from the first cycle and I implemented it in the second cycle. From the design principles generated in the second cycle, I redesigned the second prototype to develop the third prototype.

The research design of the study.

In the first cycle, three PD workshops were carried out with teachers on the last Wednesday of the first, third, and fifth months of a six-month period. In the second cycle, the workshops were on the last Friday of the first, third, and fifth months of the six-month intervention period. Each workshop was three hours long and this amounted to nine hours of training for teachers in each cycle. The first workshop had five main activities, which were 30 minutes long, with a 30-minute break between the third and fourth activities. The activities were theoretically informed by the literature, the context, my personal preferences as a researcher, the adult learning theory, and the intended, implemented, and examined curricula that the teachers were dealing with on a daily basis.

Elizabeth is going to her sister’s wedding this morning. She began driving at 7 am on the N1 highway at a constant speed of 80 km per hour. At 7:30 am, Elizabeth’s mother followed her and began driving along the same highway (N1) at a constant speed of 95 km per hour. At what time will Elizabeth’s mother catch up with Elizabeth?

Sipho built a circular patio that perfectly fits inside his father’s square yard measuring 14 meters on the sides. If he needs to plant a lawn in the four corners of the square, how many square meters of the lawn does he need to buy?

The following is a demonstration of how discussions of this problem unfolded in the workshop.

Circular patio fitted in a square yard.

^{2} and area of a circle is ^{2}. By finding these two areas, we can get the amount of lawn by subtracting the area of the circle from that of the square.

The circle fits perfectly inside the square; therefore, the radius of the circle is half the side of the square,

Sipho must buy A_{square} – A_{circle} = 196 – 154 = 42m^{2}.

^{2} is reasonable.

The lesson starts with a problem (preferably word problem) in the textbook, or the teacher poses a problem related to those found in the CAPS curriculum.

Learners read the problem as individuals or in pairs and the teacher determines if all learners understand the problem. If some learners do not understand the problem at hand, the teacher assists learners to understand by using approaches like re-reading the mathematical problems carefully with the aim of understanding before solving them. I encouraged teachers to ask learners questions like: Do you understand what the problem is looking for? Do you know all the words? Can you repeat the problem in your own words?

This stage involves learners devising strategies to solve the given problems. We discussed with teachers how they could help learners to create a plan to solve a given problem. For example, during problem-solving teachers could ask learners questions like: What operation are you going to engage? Do you need to draw a picture? A table? Would you use an equation?

This stage involves learners finding a solution to the problem independently, in pairs or as a group. I discussed with teachers how they could assist learners in carrying out the plan, persisting with a chosen plan, or if a plan does not work to discard it and choose another one. The teacher, during this process, circulates in the classroom, asking learners questions about their work, clarifying misunderstandings, giving suggestions, and helping or giving hints to learners who get stuck. Moreover, the teacher may repeat the understanding process with the stuck learners if necessary. Concurrently, the teacher looks for learners who have interesting solutions with the objective of asking them to explain their solutions in a certain order during the whole class discussions. After getting a solution to the given problems, I persuaded teachers always to help learners review their answers by reflecting on what would have worked and did not work and try to find alternative solution strategies.

At this point, the class conducts a whole class discussion where learners present their ideas and solutions as well as listen to others. Learners are required to look at the similarities and differences among the solutions being presented by their peers and to realise that there are multiple solution strategies to some given problems.

The teacher sums up the solutions presented by learners, highlights the main points, and concludes the lesson or, if there is still time, learners solve another problem.

Translanguaging is ‘the flexible and meaningful actions through which bilinguals select features in their linguistic repertoire to communicate appropriately’ (Velasco & García,

Between the PD workshops, teachers implemented the MPSP for two months. During the implementation of the MPSP, I conducted classroom observations and guided teachers as was required. Following the initial and second implementation, the second and third workshops were conducted, where the objective was for teachers to listen to the audio-recordings of the observed lessons and reflect on how they had implemented MPSP. After the reflective process, teachers watched two more videos showing MPSP and collaboratively solved mathematical word problems similar to those they were currently teaching. Teachers also simulated how to teach mathematical word problem-solving. I discussed with teachers their experiences with learners’ language during the implementation of MPSP and linguistic strategies appropriate to support learners in their context. Finally, I coached teachers on how to reflect on their MPSP and classroom practices incessantly.

The study reported on in this article is a qualitative research approach of a naturalistic inquiry (Salkind,

Thirty-one Grade 9 mathematics teachers (19 female and 12 male) from 20 different South African public secondary schools were conveniently selected to participate in the baseline study. For the PD intervention, two schools (school A and school B) were purposefully selected out of the initial 20 schools based on their accessibility and representativeness of South African public secondary schools that are resource-deficient, multilingual, and have a high learner to teacher ratio. Furthermore, during the baseline study, the Grade 9 teachers of the two selected schools revealed that they misunderstood MPSP, were utilising traditional methods of teaching mathematics, and had shown interest in the project.

Two female teachers (with pseudonyms Mary and Bertha), aged between 31 and 40 years, from school A, participated in the first cycle. One female and one male teacher (Ms N and Mr M), aged between 25 and 30 years, from school B, participated in the second cycle. Both teachers in the first cycle had bachelor’s degrees in Mathematics Education. Mary had 13 years’ experience and Bertha had 19 years’ experience in teaching secondary school mathematics. Mr M had a diploma in Mathematics Education and Ms N had a master’s degree in Mathematics Education. Mr M had six years of experience teaching secondary school mathematics and Ms N had one year of experience of teaching secondary school mathematics.

I employed a mathematics teacher open-ended questionnaire to conduct the baseline study to establish Grade 9 teachers’ views on MPSP (see Chirinda & Barmby,

I engaged grounded theory data analysis techniques using constant comparison to analyse the data, such that the process was both ongoing and retrospective. Grounded theory data analysis techniques involve systematic guidelines for analysing qualitative data to generate codes, themes, and theories grounded in the data (Chamaz,

Permission to conduct research was granted by the overseeing university and Gauteng Department of Education (GDE) with reference numbers 2016ECE002D and D2016/373AA. Regarding informed consent, teachers freely participated in the study without implicit or explicit coercion and were given assurance that they could withdraw from the study at any time without being disadvantaged. I upheld issues of beneficence throughout the research by ensuring that the study was beneficial to mathematics teachers and mathematics education. The teachers were not subjected to any possibility of harm. I viewed teachers as autonomous beings and respected them throughout the study. I did not divulge participants’ information that they disclosed during the study to others or institutions. The audiotapes and transcripts were stored in password-protected files on a password-protected computer. The data or reports did not reveal the participants’ names or their schools.

In this study, I was interested in identifying the factors fundamental to the PD intervention for teachers’ MPSP. The research question was: What factors are fundamental to a professional development intervention for teachers’ mathematical problem-solving pedagogy? The major findings from the study were that teachers’ personal meaning, reflective inquiry, and collaborative learning are factors fundamental to their PD in MPSP. In the next sections, I discuss the findings and present the themes with direct quotations to ‘give life’ to the data analysis and to make teachers’ voices audible.

Personal meaning is what the teacher desires, feels, thinks, considers to be truthful, and includes what meaning they give to certain routines or conflicts (Elbaz,

During the pre-observations, all four teachers indicated that they had previously participated in several PD programmes and viewed PD as not improving their pedagogy or promotion. Teachers had developed negative attitudes towards these PD programmes, which they indicated as short-term, often lasting not more than one day. Teachers believed that these programmes were designed and conducted by PD facilitators without teaching experience in their contexts. Teachers were frustrated that the PD programmes were unrelated to their profession and real classroom challenges. As noted by Mary:

‘I have attended short workshops before, where facilitators spoke about other trivial things in maths. However, I really want to attend workshops based on different strategies for teaching mathematical problem-solving. These workshops should also teach me to design problem-solving activities that can shift the learners’ mindset to love mathematics.’

The above quote implies that teachers’ disposition towards past PD programmes impacted on their professional learning. This was not surprising because personal meaning is known to be a foundation for structuring one’s actions in the world. Kilpatrick, Hoyles and Skovsmose (

This is comparable to what teachers felt about the PD programmes that the school required them to attend. The reflective interviews revealed that the teachers viewed PD facilitators as authorities who came to impart new knowledge to them but seemed not to understand who they were, what context they were working in, what they experienced every day when they were working in overcrowded and multilingual classes, and how they were supposed to learn to enhance their practice. This was put across by Mr M:

‘The school sometimes requires us to attend workshops to improve on mathematics teaching. Unfortunately, these workshops do not cover what happens in our classrooms. The facilitators always speak of models that are helpful overseas and not to me.’

In this regard, I engaged DBR for this study, such that the teachers’ input assisted in developing a PD intervention that was relevant to their lived experiences and context. As the intervention progressed, the reflective interviews disclosed that the teachers had reshaped their personal meaning in professional learning and were interested in the PD intervention because it had relevance and personal significance to their MPSP and classroom practices.

The PD intervention provided teachers with prospects of learning from their MPSP on a daily basis, with and from their colleague. A key factor in PD for teachers’ MPSP was giving teachers the opportunity and voice to pinpoint what practices would best support their strengths in MPSP. This observation suggests that, before implementing the intervention, the PD facilitators should first investigate the characteristics of PD in MPSP that teachers need through informal observations, surveys, focus groups, or discussion groups. This is the only way PD can effectively address teachers’ personal meaning in MPSP. If teachers find the PD programme irrelevant to their personal meaning and non-aligned to their concerns on MPSP, it rapidly disappears from their memory.

At the beginning of each cycle, teachers exhibited what I would term ‘traditional’ approaches to teaching mathematical problem-solving. From the pre-observation lessons, I could not see the teaching of problem-solving in mathematics classrooms (see Chirinda & Barmby,

Initially, teachers did not recognise that doing problem-solving enhances learners’ mathematical understanding of problem-solving processes that can be applied to different problems in the future. In the interviews, teachers also revealed that they taught procedurally because they assumed that learners did not understand since they struggled with the language of teaching and learning (Chirinda & Barmby,

After attending the PD intervention, classroom observations revealed that the teachers were no longer drilling procedures but implementing the problem-solving strategies they learned in the workshops. I discuss, in a vignette (

Reverend Joseph of the Methodist Church of South Africa has passed on, and his wish before he died was that people should wear only white, red, and pink hats at his funeral. 171 women and 93 men attend the funeral. His wife made 13 dozen white hats, five dozen red hats, and three dozen pink hats for the guests. At the end of the funeral, 11 hats were not worn. How many hats were worn by the guests?

Implementation of the mathematical problem-solving pedagogy by teacher Mary.

As the intervention progressed, I observed that teachers began to emphasise doing of mathematical problem-solving rather than focusing on learners finding the correct answers. Learners were now given opportunities to participate in problem-solving and contribute meaningfully during lessons. Teachers began to promote classroom discussions, by providing learners opportunities to discuss solutions or solution strategies in pairs or groups. Learners were required to monitor their work instead of relying on the teachers’ guidance. During problem-solving, I observed that teachers frequently probed learners to be open-minded by asking them to explain their solutions and solution strategies. Learners were given opportunities to present their solutions on the board so that their classmates could learn different solutions.

At the start of each cycle, teachers in the study did not believe that their learners could learn mathematical problem-solving because they did not have basic problem-solving processes from prior grades. Participant teachers believed that they were not supposed to give learners full ownership of problem-solving, but rather demonstrate each procedure step by step and require learners to practise the procedures (Chirinda & Barmby,

Meghan’s old grandmother starts walking at 8am to go to the mall for shopping. Meghan must clean the house; therefore, she follows grandma after a few hours. They both arrive at the mall at the same time.

I observed the learners working in pairs trying to understand the given problem prior to solving it. I could hear most of the learners planning and formulating problems with excitement and looking back at the answers after finding solutions. Nonetheless, a few learners wanted to be shown how to pose the problem; nevertheless, I observed that Ms N identified meaningful ways to motivate learners to persist in problem formulation. This seems to suggest that Ms N had established self-understanding in personal meaning in giving learners full ownership of doing mathematical problem-solving.

Reflective inquiry is the teacher’s act of looking back at the teaching and learning of what has transpired and re-constructing or re-capturing the events’ occurrences, emotions, and experiences (Schon,

‘In my class, I usually give learners problems to work on as individuals. When they get stuck, I quickly give them hints or show them how to get the answer before the bell rings. Now I understand, I will not be giving learners opportunities to engage in problem-solving when I do this. From now on, I will not rush to provide learners answers. Learners need to take their time. They need time to think about the problems and find ways of how to get answers by themselves.’

Bertha also indicated this in one of the reflective interviews:

‘I often show my learners the meaning of a problem before they begin solving it. I now see that I am taking away from learners, the chance to learn to analyse problems on their own. Going forward, I will let learners take the lead in understanding the problems of the day.’

The above excerpts seem to demonstrate that self-inquiry evoked meaningful learning in the teachers. Reflective inquiry as a means of teacher PD is recommended by Muir and Beswick (

Teachers were required to work collaboratively during PD workshops and the implementation of the MPSP. Collaboration means teachers are working and learning together to address challenges they grapple with in their profession (Robutti et al.,

After explaining the characteristics of collaboration, teachers in each cycle rose to the challenge. They accepted the joint responsibility of working on the word problems and other activities during the workshops and implementation of MPSP.

It is well documented in the literature that collaboration brings teachers together to implement new teaching strategies and reflect on their practices (Robutti et al.,

‘It was a great privilege to work with my colleague during this programme. Having real conversations with him before and after the observations of the lessons benefitted me professionally in that he is familiar with the subject, CAPS curriculum, and mathematics problem-solving teaching strategies.’

Teachers observed each other’s successes, challenges, and failures. My observations during the workshops were that as teachers collaboratively solved the word problems and planned how to teach these problems to their learners, they reached a shared understanding of how to implement Polya’s (

The reflective interviews gave insight into teachers’ perspectives about the process of collaboration during the PD intervention. Teachers in both cycles embraced collaboration and valued the discussions they carried out with their colleague on how to teach specific mathematics problems to learners. Teachers reflected that working cooperatively with a colleague strengthened feelings of positive interdependence between them. Positive interdependence, which means the success of one teacher was determined by the success of their colleague, is acknowledged to deepen learning (Drago-Severson,

In the second cycle, Ms N, a novice teacher with one year of teaching experience, reflected that collaborating with her colleague during the PD intervention had vastly improved her MPSP and eliminated feelings of loneliness. As a newcomer both in the profession and in the school, Ms N explained that she had felt isolated. This implies that collaboration eradicates feelings of isolation that beginner teachers usually find themselves experiencing. Ms N valued collaborating with Mr M as she felt she had someone to work with and no longer felt isolated. This finding seems to imply that there is value in pairing new teachers with the experienced members of the teaching staff. The finding agrees with Vygotsky’s (

‘As a beginning teacher, the discussions I conducted with my colleague helped me understand several aspects on the teaching of problem-solving including misconceptions learners usually have in problem-solving and how to handle them.’

The above quote suggests that collaboration supported Ms N’s professional growth. This growth resulted in her teaching being accessible and effective to learners as she was now able to identify and handle learners’ misconceptions in mathematical problem-solving. This finding suggests too that collaboration is a factor fundamental to PD intervention for teachers’ MPSP in the study. Nonetheless, my observations were that teachers’ collaboration in the context seemed limited because of the prescribed curriculum and a tight school timetable. This observation agrees with Cookson (

This study explored factors that are fundamental to a professional development intervention for teachers’ MPSP.

This study’s findings were that teachers’ personal meaning, reflective inquiry, and collaborative learning are the factors fundamental to their professional growth in MPSP. This study’s focus on teachers’ personal meaning in the previous PD programmes they had attended seemed to have effectively addressed their personal meaning in PD in MPSP. In addition, the PD facilitator’s ability to recognise teachers’ personal meaning in MPSP seemed to have facilitated professional growth in their MPSP. The reflective inquiry process, in turn, gave teachers insight into their personal meaning in the activities and processes of the PD intervention. Clarke and Hollingsworth (

This study contributes to the PD in the MPSP body of knowledge by having worked with teachers in an under-researched context. A study in this context is valuable since research conducted in contexts of advantage is ubiquitous in the mathematics education field and few studies originate from countries and contexts of historical disadvantage (Skovsmose,

Because of inadequate monetary resources, the study focused on teachers in one South African district. The limitation of focusing on one district is that I could not explore how a PD intervention for MPSP could be designed and implemented to support Grade 9 teachers in other districts. However, the objective of this study was not necessarily to explore several participants but to design a PD intervention to support teachers’ MPSP. Nonetheless, I recommend that further research can be done with a larger, more diverse sample of South African teachers to determine how personal meaning, reflective inquiry, and collaborative learning promote their growth in MPSP. I feel that there has not been enough critical examination of this aspect in the South African context.

This study revealed that collaboration is a factor fundamental to PD intervention for teachers’ MPSP; however, further research needs to be done to establish how best collaboration can be conducted in contexts where teachers feel that there is not enough time to collaborate because of prescribed school curriculums and tight school timetables.

Patrick Barmby for supervising my PhD. This article emanates from my PhD work.

The author has declared that no competing interests exist.

I declare that I am the sole author of this research article.

This research was made possible by the National Research Foundation (grant number 11439).

Data sharing is not applicable to this article as no new data were created or analysed in this study.

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