This article reports on the design and findings of the first iteration of a classroom-based design research project which endeavours to design a professional development intervention for teachers’ mathematical problem-solving pedagogy. The major outcome of this study is the generation of design principles that can be used by other researchers developing a professional development (PD) intervention for mathematical problem-solving pedagogy. This study contributes to the mathematical problem-solving pedagogy and PD body of knowledge by working with teachers in an under-researched environment (an informal settlement in Gauteng, South Africa). In this iteration, two experienced Grade 9 mathematics teachers and their learners at a public secondary school in Gauteng, South Africa, participated in a 6-month intervention. Findings from the data are discussed in light of their implications for the next cycle and other PD studies.

Mathematics is an essential requirement for entry into South African universities and other tertiary institutions and is a ‘critical competency for the development of sorely-needed high-level skills’ (Centre for Development and Enterprise,

Many curricula (DBE,

Design an effective PD intervention for mathematical problem-solving pedagogy.

Explore the learning of participant teachers and learners from participating in the PD intervention.

Examine and evaluate the potential impact of the PD intervention.

Generate design principles that can be used to develop a PD intervention on mathematical problem-solving pedagogy for Grade 9 teachers in a particular local context.

This article reports on the first iteration of the PD intervention on mathematical problem-solving pedagogy. We formulated and sought to answer the following research questions:

What is the impact of the PD intervention on learners’ learning processes?

What is the impact of the PD intervention on teachers’ teaching of problem-solving?

What factors facilitated learners’ learning and teachers’ development of pedagogy?

What are the possible design principles required to generate a PD intervention on mathematical problem-solving pedagogy for Grade 9 teachers in a particular local context?

Traditional methods of teaching mathematics, prevalent in the South African context (Adler & Ronda,

Polya (

Understanding the problem.

Devising a plan or deciding on a strategy for attacking the problem.

Carrying out the plan; that is learners follow through with the strategy selected, carefully taking each step along the away.

Looking back at the problem, the answer and what one has done to get there.

During the intervention, we worked with teachers on how to use Polya’s steps in their teaching. We discussed how teachers could work with learners on understanding given problems. Teachers were encouraged 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? We discussed with teachers how they could help learners to create a plan to solve a given problem. Teachers could ask learners questions like: What operation are you going to engage? A table? Do you need to draw a picture? Would you use an equation? Teachers were encouraged to assist learners during carrying out the plan that is, doing calculations. We discussed ways teachers could help learners to persist with a chosen plan and if a plan does not work to discard it and choose another. After getting a solution to the given problem teachers were persuaded to always facilitate learners to review their answers by reflecting and looking back at what worked and what did not.

Our objective for the main study is to design an effective PD intervention that can be used to support mathematics teachers in the teaching of mathematical problem solving. Day (

Effective PD should be based on constructivism (Villegas-Reimers,

Effective PD must be perceived as a process that takes place within a particular context (Villegas-Reimers,

Social constructivism informed this study. The social constructivist perspective to learning mainly originates from the work of Vygotsky (

Social constructivism emphasises that knowledge is mutually built and constructed (Vygotsky,

The conceptual framework we drew on is that of problem-centred learning. The problem-centred teaching and learning approach is a learner-centred educational method that uses problem solving as the starting point for learning and as a ‘vehicle for learning’ (Murray et al.,

Design-based research (Kelly,

A number of researchers have attempted to give a definition of DBR and there is a discussion underway of what constitutes DBR (Van den Akker et al.,

a systematic but flexible methodology aimed to improve educational practices through iterative analysis, design, development, and implementation, based on collaboration among researchers and practitioners in real-world settings, and leading to contextually-sensitive design principles and theories. (p. 6)

This definition implies that in DBR, researchers work as a team with practitioners to provide solutions to practical challenges that face a particular educational context. There is general agreement that DBR should generate valuable educational interventions and useful theory (Van den Akker et al.,

The larger classroom-based design project focuses on designing a PD intervention on mathematical problem-solving pedagogy that can be further modified and used with schools in challenging contexts in South Africa. The larger project has three iterative cycles. In March 2016 we conducted a baseline investigation with 31 teachers at 20 schools in the district of interest. The baseline investigation examined how Grade 9 mathematics teachers in this district were using problem-solving in their teaching of mathematics. We ‘purposefully’ selected three schools A, B and C, out of the initial 20 schools. These schools were chosen because they could be conveniently accessed by the researchers and the Grade 9 mathematics teachers reported that they were using traditional methods of teaching. This article reports on the first iteration of the larger project, which is the phase of the intervention in which we worked with two teachers in school A. Teachers in school B and school C were investigated in cycle B and cycle C respectively. The goal was to work with at least two Grade 9 mathematics teachers from each school; however, this depended on the number of teachers teaching Grade 9 mathematics at a particular school.

The PD intervention is designed to take place within a period of 6 months. The goals we set for the PD intervention were to improve learners’ performance in mathematics and support teachers’ mathematical problem-solving pedagogy. We also aimed to explain and agree with participant teachers what mathematical problem-solving pedagogy is and what it is not. PD took place during the process of classroom instruction in order to link with classroom teaching (Barber & Mourshed,

In the first workshop, we initially presented the workshop’s contents to the participant teachers who then watched two short videos on mathematical problem-solving pedagogy in action. The baseline investigation we conducted before implementing this intervention unearthed that participant teachers believed that teaching mathematical problem solving was about explaining to learners each and every concept, step-by-step, breaking down the topic, working out examples on the chalkboard and giving learners practice exercises to work on. Therefore, these videos were to show the teaching of mathematical problem solving in action. We discussed the videos, focusing on what genuine mathematical problem-solving pedagogy entails and how to apply Polya’s four steps of problem solving as a teaching process. Teachers expounded on ways of introducing or posing the problems in such a way that learners understand the given problems. Teachers collaboratively solved at least two ‘rich’ and open-ended mathematical tasks relating to the work they were teaching, and with our guidance discussed how to teach problem solving as a process.

After attending the first PD workshop, teachers were encouraged to go and implement the new ideas in their lessons for a month. During this implementation stage, we observed, supported and guided the participant teachers as was necessary and audiotaped the lessons. It is imperative that teachers are supported during the implementation stage to address the specific challenges of changing classroom practice (Gulamhussein,

We encouraged teachers to reflect-in-action and reflect-on-action (Schön,

After the first implementation, we conducted the second workshop where the aim was for teachers to further collaboratively reflect on their teaching experiences and to review the audio tapes of the observed lessons. We selected crucial and relevant audio recordings that foregrounded participant teachers’ use of problem solving in their teaching. Participant teachers analysed how they had taught mathematical problem-solving and they watched two further videos showing mathematical problem-solving pedagogy in action. Once again teachers collaboratively solved mathematical tasks relating to what they were teaching and planned on how to teach similar tasks to their learners. After workshop 2 the teachers implemented new ideas for a month while being observed, supported, audiotaped and interviewed by the researchers. We also encouraged the teachers to continuously reflect on their experiences and classroom practices. The third workshop and the implementation process were similar to the second stage.

Teacher data were collected through classroom observations and semi-structured reflective interviews. In order to get first-hand experiences when teachers implemented the problem-solving pedagogy, we observed them delivering lessons and recorded information on the spot as it occurred on the observation comment card. The observation comment card (see

Semi-structured reflective interviews with the participant teachers were conducted with each teacher once a month and were conducted in their classrooms since it was during their free periods. In total, we had five interviews with each teacher. Opie (

In terms of assessing the possible impact of the PD intervention on participant learners, we gave them mathematics attainment tests pre- and post-intervention and a self-reporting mathematical problem-solving skills inventory (MPSSI) at the beginning and at the end of the intervention. We used teachers’ pre- and post-intervention mathematics attainment tests that covered the topics that the learners were doing (geometry and data handling). The questions on the pre-intervention test were different but similar in all respects to those of the post-intervention test. Both tests had 15 questions and the marks were converted to a percentage. The attainment tests were useful in evaluating learners’ ability to use mathematical problem-solving skills because they required learners to supply the answers thereby avoiding guesses.

The MPSSI (see

Our raw data included recorded notes on the observation comment card, audio tapes from the classroom observations and semi-structured interviews, test marks and responses from the MPSSI. Audio tapes from the classroom observations and semi-structured interviews were transcribed verbatim into written notes in order to be able to identify common patterns and experiences. We employed inductive data analysis to analyse the teacher qualitative data from the observations and the semi-structured interviews (Hatch,

This was an ethical study; therefore, the researchers took precaution to protect the autonomy and anonymity of participant teachers. Letters of permission were sent to and subsequently returned from the Gauteng Department of Education, Johannesburg North District of Education, selected school principals, participant mathematics teachers, learners and their parents or guardians. Participants were given detailed information about the proposed study and were clearly informed of the confidential nature of the research. We ensured that participation was voluntary; confidentiality was prioritised, and participants could freely withdraw from the study at any time without incurring any negative consequences, although none did. All responses were anonymised before analysis; neither the participant schools nor the teachers’ names were identified in any report of the results of the study.

The MPSSI results exhibited substantial gains in percentages for each question (see

Results from the mathematical problem-solving skills inventory.

Test | Q1 | Q2 | Q3 | Q4 | Q5 | Q6 | Q7 | Q8 | Q9 | Q10 | Q11 | Q12 | Q13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Pre intervention | 34 | 10 | 2 | 10 | 18 | 32 | 18 | 14 | 26 | 14 | 28 | 22 | 22 |

Post intervention | 86 | 54 | 42 | 48 | 76 | 72 | 54 | 70 | 80 | 72 | 66 | 86 | 84 |

Q, question.

There was an increase in the learner attainment test scores and this indicated that there were gains in learner attainment.

Results from the pre- and post-intervention attainment tests.

Test | Average percentage on test | Standard deviation |
---|---|---|

Pre-intervention test | 19.2 | 16.2 |

Post-intervention test | 37.8 | 17.5 |

The findings from the teacher classroom observations are presented under the three themes: understanding the problem, collaborative learning and encouraging metacognition.

The findings from the teacher semi-structured interviews are presented under the four themes that emerged from the interview data. The four themes are: changes in perceptions about mathematical problem-solving pedagogy, appreciation of collaborative learning, increased awareness of learners’ needs and PD activities that had a positive impact on teachers’ professional development.

Initially, I was confused and showed learners how to work a problem using drilling methods but now I know how to teach mathematical problem solving as a process. (Mrs Y, female, teacher)

Working with my colleague during the workshops and teaching was superb because we could support each other and we shared ideas and obstacles. Collaboration made my learning effortless. I liked it. I became a better teacher by learning from my colleague. (Mrs X, female, teacher)

As the intervention progressed, participant teachers began to implement collaboration in their own teaching. Learners were required to work in pairs or in groups and this kept them on-task. Teachers were no longer the only source of knowledge and this helped with classroom management as learners had a responsibility of completing tasks in pairs or groups, as was highlighted by Mrs Y in one of the interviews:

As I arranged learners to work in pairs, they were no longer moving in and out of the classroom. Pairing them helped me with discipline and class control. (Mrs Y, female, teacher)

This professional development intervention opened my eyes. I used to assume that learners understood the given questions but now I check if they really understand. If they don’t we read the problem together, paraphrase the questions and I sometimes require learners to verbally tell me what the question would be asking for. (Mrs Y, female, teacher)

If my learners do not understand a given problem, I now re-read the problem for them or use their mother tongue to explain. This is to make sure that they understand the given problem. (Mrs Y, female, teacher)

To be able to see mathematical problem-solving teaching in action was intriguing. I saw where I was getting it wrong. It helped me to realise areas that needed improvement in my own teaching. (Mrs Y, female, teacher)

In addition, the teachers appreciated that we, the researchers who delivered the PD intervention were respectful to them and acknowledged their experiences, as expressed by Mrs X:

You treated me like an adult during the intervention. I have been to other workshops where I was treated like a child who knows nothing. You showed me respect. You were aware that I have fully trained to be a teacher and that I teach large classes. (Mrs X, female, teacher)

Finally, the teachers valued solving problems and simulating how to teach problem-solving during the workshops, as explained by Mrs Y:

To practically learn how to teach mathematical problem-solving during the workshops was helpful for me because it built my confidence before implementing the new ideas in the classroom. (Mrs Y, female, teacher)

The pre- and post-intervention tests and pre- and post-intervention MPSSI demonstrated that the PD intervention had a positive impact on learners’ performance. This finding resonates with Barber and Mourshed’s (

Despite the apparent success of encouraging problem-solving approaches, we found that English was a major obstacle to learners’ grasping of the given problems. The finding on language as an obstacle to learning in South African mathematics classrooms is in agreement with what a number of researchers have disclosed in the past (Adler,

The pre-intervention observations revealed that the lessons were teacher-led and teachers implemented traditional methods of teaching. This finding aligns with what other researchers have exposed (Adler & Ronda,

In situations where learners did not understand the given problems because of the language, teachers code-switched. This study fills a gap in the literature on professional development and problem-based learning as it unearthed that language and code-switching are important aspects to be considered when implementing a PD intervention in a multilingual context. We recommend that a PD programme should include a segment that supports teachers on how to appropriately conduct code-switching and to support learners with their language.

Initially, teachers were reluctant to participate in the intervention and implement the mathematical problem-solving pedagogy. They indicated that they were worried that they would fail to cover the CAPS syllabus within the prescribed time. This finding aligns with what Slattery (

We observed the participant teachers for a month before implementing the intervention and this created trust between us and the teachers, which resulted in teachers working comfortably with us. Teachers appreciated that we respected them and acknowledged their experiences. It was important that we were responsive to respecting the participant teachers because respect is a key aspect in an African culture. When implementing the PD we took into consideration the lack of resources in our context. These are important aspects to add to the PD literature: that respect is an important aspect of the African culture and should be considered when supporting African teachers in a PD programme. We recommend creating a positive relationship with teachers before implementing a PD intervention and building on teachers’ experiences when training them.

Collaborative learning was beneficial to both participant teachers and learners. This outcome concurs with Cordingley, Bell, Thomason and Firth’s (

The semi-structured reflective interviews with participant teachers were imperative to the research process. As teachers looked back on classroom events during the interviews and made critical judgments about them, they modified their teaching behaviour and this resulted in them constructing knowledge about themselves, their teaching practices and their learners (Schunk,

Following up on teachers in the classrooms to check if they were correctly implementing the mathematical problem-solving pedagogy was advantageous as we were able to support teachers as necessary. This is different from the traditional ‘one-shot’ workshops and we recommend that PD practitioners should support the teachers in the classroom during the implementation stage.

Design principles are one of the major outputs of DBR and McKenney and Reeves (

A baseline investigation must be conducted to establish teachers’ perceptions and practices on the teaching of mathematical problem solving before implementing the PD intervention.

Facilitators of PD must create a positive relationship with participant teachers before implementing the intervention.

PD should be built from teachers’ experiences and current knowledge of mathematical problem solving.

Respecting participants is important in an African context when implementing the PD intervention.

Facilitators of PD must observe teachers practically implementing the mathematical problem-solving pedagogy and support them as necessary.

PD should be organised around collaborative problem solving.

PD should support teachers on how to implement mathematical problem-solving pedagogy in a multilingual context.

One of the most significant outcomes from this cycle is that language stands so much in the way of learners’ learning of mathematics. Polya’s (

The findings of the study may prompt other researchers to develop PD interventions in local contexts. The study, having been done at one school, means that the transferability of the findings to larger contexts can be challenging. However, DBR does not make generalisability claims as ‘the effectiveness of a design is no guarantee of its effectiveness in other settings’ (Collins, Joseph & Bielaczyc,

We acknowledge the funding received from the National Research Fund (NRF) and the support we got from the school principal, Mathematics head of department and the two participant teachers from school A.

The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.

B.C. is the lead author and was also involved in the coding and analysis of the data. P.B. was involved in the coding and analysis of the data and contributed to the writing of the article.

Stage | What teacher should do | What teacher really did (evidence/indicators) | Comments |
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Before | Give instructional strategies and activities that incorporate learners’ prior knowledge. Pose the problem to learners orally or in a written form. Read or have a learner read the problem. Make sure that learners understand the problem before they begin to work on it, for example ask learners to identify what is asked for in the problem, discuss any unfamiliar terms in the problem, or ask learners to paraphrase the problem in their own words. Brainstorm possible solution strategies. Clarify the task at hand. |
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During | Move among learners. Listen prudently to learners’ thoughts and discussions. Embrace the dynamics of group work. Question individual learners or groups about the strategy they are using and their findings. Probe learners with suitable questions to assist them clarify the direction they are taking in solving the problem. Provide hints to learners who require them. Persuade learners to seek and value alternative modes of problem-solving. Encourage learners to make extensions or generalisations. |
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After | Encourage learners to reflect on their solutions and on the processes they used. Invite learners to justify their solutions. Engage the class in productive discourse by letting learners communicate their ideas in words and diagrams, and allowing them to share their ideas and strategies. Probe the learners how the problem is similar to and different from problems they have previously solved. Have learners discuss the critical aspects of the problem. Summarise the chief points of the discussion and establish that all learners comprehend them. |

Dear learner

Please fill in the table below, where 1 = strongly disagree, 2 = disagree, 3 = neither agree nor disagree, 4 = agree and 5 = strongly agree

No. | Mathematical problem-solving skills inventory | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

1 | I always read the problem carefully to understand it | |||||

2 | I underline important words when doing a problem | |||||

3 | I draw pictures to understand a problem | |||||

4 | I imagine the problem I am doing in my head | |||||

5 | I can separate different parts of the problem | |||||

6 | I carefully plan how I will do a problem | |||||

7 | I remember other problems that I have solved before that look like the problem I am doing | |||||

8 | I can easily explain what I am doing when solving a problem | |||||

9 | I keep checking if the way I am solving the problem is correct | |||||

10 | If I get stuck, I go back to the problem to check if I understood it correctly | |||||

11 | I try and find different ways to solve a problem | |||||

12 | I look back at the way I solved a problem to see if it makes sense | |||||

13 | I check to see if my answer is correct by going back to the given problem |