A problem is a task, situation, or activity students get from their teachers or face in life for which they do not have a ready-made formula or strategy to solve (Cai et al., 2015; Hiebert et al., 1996; Kilpatrick, 1987; Schoenfeld, 1992). Avcu and Avcu (2010, p.1282) defined a problem as ‘a situation that one faces with some blockage while solving the problem’. Nurkaeti (2018) referred to PS as a task where the students are engaged in solving either routine or non-routine problems. Routine problems are mostly seen in textbooks and can be solved through basic operations. For instance, a routine problem is a kind of mathematical task that is closed-ended. This kind of problem or task often uses an algorithm or formula to arrive at the desired solution for the problem in question. Non-routine problems require planning, organising, and classifying data, discovering the relations, and determining the rules and generalities. Usually, a non-routine problem has no predetermined algorithm or formula or method to be applied in solving the given task. Precisely, this is an open-ended task that requires conceptual understanding, internalising, and integrating conceptual reasoning into the real-world situation. Rich and open-ended tasks can provide opportunities for rich learning by engaging students in PS and mathematical thinking (Boaler, 2016; Mueller et al., 2014; NCTM, 2014).

Many researchers have advocated that problem posing and PS are central to mathematical thinking, creativity, and discourse in mathematics (Cai et al., 2015; Hiebert et al., 1996; Silver, 1994; Tesfamicael et al., 2020). Problem-solving refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development (Cai et al., 2015).

Van de Walle et al. (2020) presented three approaches to PS which were described by Schroeder and Lester (1989): teaching for PS, teaching about PS, and teaching through PS. ‘Teaching for PS starts with learning the abstract concept and then moving to solving problems as a way to apply the learned skills (explain-practice-apply)’ (Schroeder & Lester, 1989, p. 55). Teaching for PS follows traditional mathematics teaching, where mathematics rules and formulas are applied in word or text problems while teaching about PS is about providing guidance to students to solve problems. George Polya (1945) proposed four steps for PS: understanding the problem, devising a plan, carrying out the plan, and looking back. However, the third approach is of interest to this study. Teaching through PS is an approach where students learn mathematics through inquiry by exploring texts, problems, situations, patterns, and models (Boaler, 2016; Cai et al., 2015; Hiebert et al., 1996).

Teaching through PS typically engages the students to participate in deep thinking and conceptual reasoning as they encounter mathematical problems (Hiebert et al., 1996; NCTM, 2014; Stephan, 2014; McGatha et al., 2018). Teachers are expected to use tasks that lend themselves to multiple representations and strategies (McGatha et al., 2018). This is regarded as teaching through PS, where students ultimately grapple by themselves to find meaningful solutions to the mathematical problems posed by their teacher. Hence, students are engaged with the notion of solving mathematical problems by using critical thinking skills and reasoning to strengthen their solutions (Smith & Stein, 1998; Stephan, 2014).

Facilitating meaningful MD is one of the eight effective mathematics teaching practices promoted by NCTM (2014). Stephan (2014) considers MD as one of the crucial aspects of student-centred instruction that involves using student discourse in whole-class discussion to bring out important mathematical ideas. Mathematical discourse refers to verbal and written communication that is centred around deepening, thinking about, and making sense of mathematics (Sfard, 2012). Through MD, students discuss, brainstorm, and engage in critical thinking and reasoning in pursuit of arriving at the answer to the problem (Ballard, 2017). Celik and Baki (2023) explain MD as a socially accepted association among ways of using language, other symbolic expressions, and ‘artifacts’ of thinking, feeling, believing, valuing, and acting that can be used to identify oneself as a member of a socially meaningful group or ‘social network’, or to signal (that one is playing) a socially meaningful role. This explanation regards MD as a means to unite students by dialoguing, agreeing, and disagreeing to establish consciousness over a critical matter requiring collegial understanding (Jill and Erlina, 2015).

In TCT classrooms, teachers stand at the front of the room and dominate the conversation (NCTM, 2014). According to Drageset (2015), such practice can be described by the Initiation Response Evaluation (IRE) model of discourse, which is a theoretical framework describing a discourse pattern where the teacher initiates the questions, the students respond to them, and the teacher evaluates the responses (Drageset, 2015; NCTM, 2014). In SCT classrooms, teachers should allow the students to explore various strategies and approaches, especially when encountering mathematical problems (Stephan, 2014). The students undertake the autonomy of calculating the mathematical tasks being posed to them by their teacher (Stephan, 2014). Contrarily, if the teacher undertakes full control of what is being said by the students, this can ensure that the lecture includes the intended mathematics goal. On the other hand, with student-led discussion, the teacher carefully guides the students toward discussing the intended mathematical tasks (Cobb, 1994; Stephan, 2014).

In general, leading an effective classroom discussion is a demanding task, and there are several research-based protocols that mathematics teachers can implement in their classrooms (Ballard, 2017; Faria et al., 2024; Hufferd-Ackles et al., 2004; Larsson, 2015; Smith & Stein, 2011). For instance, Hufferd-Ackles et al. (2004) underline four developmental trajectories in the Math-Talk Learning Community that consist of: (1) questioning, (2) explaining mathematical thinking, (3) sources of: mathematical ideas, and (4) responsibility for learning. Chapin et al. (2009) provide talk moves and tools that can help facilitate discourse, including revoicing, repeating, reasoning, and adding on. Additionally, selecting open-ended, high-level thinking, and conceptually focused tasks or questions is vital for facilitating effective discourse in the classroom (McGatha et al., 2018). Smith and Stein (2011) provide a framework that can help orchestrate a productive classroom discussion. It has five elements: anticipating student responses to challenging mathematical tasks, monitoring students’ actual responses to the tasks, selecting particular students to present their mathematical work, sequencing and sharing students’ reactions in a specific order for discussion, and connecting different students’ responses to crucial mathematical ideas (Ballard, 2017; Faria et al., 2024; Larsson, 2015). These demand high competency on the part of the teachers to implement in the classroom. For this study, part of the four-level NCTM (2014) framework, which was developed by Hufford-Ackles et al. (2014), is used to situate the six mathematics teachers’ practices before and after the intervention due to its simplicity in implementing it in the selected South Sudan school context. At Level 0, the teacher dominates the conversation. At Level 1, the teacher encourages the students to engage in discourse with the whole class. At Level 2, the teacher facilitates conversation between students and encourages them to ask one another questions. Finally, at Level 3, students carry the conversation themselves while the teacher guides them from the periphery (NCTM, 2014).

This article emanated from a doctoral study focusing on SCT in the context of South Sudanese schools where the researcher adopted design-based research (DBR) as the methodological approach to guide the study. Design-based research refers to the systematic study of designing, developing, and evaluating educational interventions. According to Campanella and Penuel (2021) and Cobb et al. (2003), DBR helps to foster learning, create usable knowledge, and advance theories of learning and teaching in complex settings (Fowler et al., 2023). The rationale for using DBR as the research methodology is that it bridges the gap between theory and practice. Gravemeijer and Prediger (2019) provided five approaches regarding the concrete realisation of DBR: interventionist, theory generative, prospective, and reflective, iterative, and pragmatic roots, and humble theories.

This study employed theories about SCT and trained the six teachers after observing their teaching practice. The study identified how SCT can be intertwined with DBR and the practical application of the intervention to enhance effective teaching and learning of mathematics in South Sudan elementary schools. It is worth mentioning that intervention requires iteration of the intended programmes or tasks of SCT to satisfy the solutions to the problems identified with TCT. This occurred when the researcher observed the teachers and identified that they were using the TCT strategy and that this was a practical problem at hand. A remedy was designed by the researcher which was the application of the SCT approach. This SCT strategy was introduced to the teachers by the researcher through PD training. After successful training, the teachers were sent back to their respective schools and the researcher followed them and observed how they applied the strategy of SCT in their classrooms. Having completed the observation of these teachers, the researcher again invited them for reflection and the programme kept iterating cyclically. Thus, the reasons for using the intervention feature of DBR are as follows: (1) intervention often facilitates the process of multiple outcomes to be realised in the forms of better achievement, improved student attitudes, and increased teacher satisfaction, and (2) it helps to articulate the process of inputs by designing a conducive learning environment to promote both certain instructional learning materials and teacher development.

Figure 1 shows parts of the design elements and process in this study, indicating the operationalisation of the DBR as illustrated. Firstly, selecting schools and mathematics teachers to participate in the study was completed. After negotiating with educational officials, principals, and teachers in two schools, six teachers, three from each school, were identified. The two schools were conveniently chosen based on proximity, and the six teachers were those engaged in teaching mathematics at the Grade 5 level. So, convenient sampling was used to make access to the schools as simple as possible for the researcher. To design and implement context-based professional learning, the researcher conducted a preliminary study on types of PD and the meaning of effective PD in the South Sudan education system (Soforon et al., 2023). These activities define the pre-DBR phase of the study. Following that, four days of intervention on SCT were planned and conducted as a workshop. The researcher then observed the teachers in their classrooms to assess their actions and collect their reflections. Subsequently, the researcher observed them in their classes again.

Instruments such as interviews, observation, recorders, cameras, and field notes were used for collecting the data. However, they were regarded as tools for social interaction in the conversation (Rubin & Rubin, 2012). The interviews held with the six teachers before and after the intervention formed part of the data. The researcher observed the teachers while teaching, depending on their classroom schedules. Furthermore, audio and video were recorded and the lessons photographed using both recorder and camera during the classroom observations. Lastly, the field notes were compiled for documenting or detailing the information written by the teachers on the chalkboard as the teaching was ongoing in the respective classrooms.

All the collected data in the form of interview, observation, field notes and video (VDG-1 and VDG-2) were analysed using the content analysis method. Content analysis is ‘a research method for the subjective interpretation of the content of text data through the systematic classification process of coding and identifying themes or patterns’ (Hsieh & Shannon, 2005, p. 1278). There are three kinds of content analysis: conventional, directed, and summative (Hsieh & Shannon, 2005). Directed content analysis is guided by existing theory or prior research by identifying key concepts or variables as initial coding categories, while conventional content analysis is used when researchers try to avoid using preconceived categories; in summative content analysis, keywords are selected based on previous research or the researchers’ interests (Cohen, 2018; Hsieh & Shannon, 2005). The researcher employed conventional content analysis to analyse the data using codes, categories, or themes from the textual data available. This conventional content analysis is appropriate because the existing theory on textual data is limited to allowing the categories and names for categories to flow from the data. This includes reading all the data repeatedly to achieve immersion and obtain a sense of the whole, where the data are read word by word to derive codes, by first highlighting the exact words from the text that appear to capture key thoughts or concepts. Table 1 provides the guiding template for analysing the data which summarises the details of the characteristics for SCT versus TCT for the case of PS and MD.

This study followed directed content analysis. The data collected before, during and after the intervention were analysed using the conventional content analysis procedure defined above. The checklist provided by Stephsn (2014) as presented in Table 1 was used as an analytical tool. The extended version of the checklist is provided in Appendix 1. The genuine reason for choosing these instruments was their simplicity. It provided maximum opportunity for the researcher to explore more information. Each characteristic had four levels starting from one end with TCT-dominated practice, Level 0, to full implementation of SCT, Level 3. Level 0 and Level 3 coincided with the two ends ‘directed’ and ‘open’ inquiry according to Stephan (2014). In this work two more levels were introduced to be able to register and discuss the shift in teaching practice by the teachers after the intervention, if any.

The researcher has considerably applied the trustworthiness orchestrated by the four criteria proposed by Lincoln and Guba (1985), which include credibility, transferability, dependability, and confirmability. Credibility refers to adequately representing the constructions of the social world under study. In this case, the researcher designed interview questions that encouraged the participants to participate in answering them and subsequently leading to a conclusion. These questions were related to activities such as recording, photographing, and documenting information that would help improve the credibility of the research findings.

Transferability refers to the extent to which the study hypothesis could be applied to another context. The researcher ensured that the study conducted on PD of mathematics teachers in South Sudan, specifically the case of SCT, could also be carried out in other locations or places. Dependability refers to the coherence of the internal process and how the researcher accounted for changing conditions in the phenomena. It entails the concept of reliability, in which the researcher used the same methods to obtain the same results. The same method was used: the application of DBR with the help of intervention on SCT while keeping a complete record of all phases of the research process. Confirmability explains to what extent the researcher admitted their bias (Bordens and Abbott, 2018). In this study, the researcher ensured that correct data from the participants were collected without influencing the nature of the data to accomplish the intended goals.

The result of the preliminary observation, with the six mathematics teachers within their time schedules, with the goal of understanding the practices of these teachers in connection to PS and MD, was conducted and was provided as follows. In both schools, all six teachers started their instruction by standing in front of the students and introducing the lesson they had prepared for the day. All six teachers were engaged in teaching the mathematical content of algebra and algebraic expressions as presented in the textbook. The teachers’ actions are summarised in Table 2. In all the classes, the students copied and imitated what their mathematics teachers wrote and solved on the blackboard. The students were seated in fixed rows facing their teachers, with their backs to one another in every classroom of the six teachers, as shown in Figure 1.

After designing lessons with tasks and activities that can facilitate SCT, the six teachers were invited to a four-day workshop, the contents of which were presented and discussed with experts in the field. Most of the materials were taken from recent research – informed teaching resources like NCTM (2014), Boaler (2016), YouCubed¹ and others (see Appendix 2). The researcher introduced TCT and SCT during the workshop seminar and demonstrated them with examples. Among other characteristics of SCT, the teachers were exposed to the application of both PS and MD to enhance constructive mathematics teaching and learning. As shown in Appendix 2, different tasks were selected by the researcher for the training.

During the training, the researcher organised the six teachers into groups of two, making three groups of primary teachers. The researcher presented them with different tasks that could contribute to learning algebra, explicitly focusing on generalising patterns that can foster learning through PS and MD. The facilitator encouraged the teachers to discuss, interact, and come up with solutions for the given tasks. The teachers were asked to reflect on their experiences at the end of each day. The researcher-facilitator posed the following question: ‘What features of teaching practice do you experience while teaching in your classroom?’ The responses of the three groups were as follows (summarised by the researcher):

After the four-day workshop, the researcher visited again the six teachers in their respective schools. All the teachers were motivated to implement what they had learned. The results in connection to PS and MD are presented below.

Before the intervention, the mathematics problems that the teachers used in the class were extracted from the Grade 5 textbook and presented to the students as tasks or examples. The nature of the problems that the teachers used did not engage the students in relational understanding (Skemp, 1978). Instead of investigating how solutions were determined, the students relied on the procedures or rules provided by the teachers to arrive at the solutions. The problems these teachers presented before the intervention were not inquiry-based tasks, as they failed to incorporate elements such as explanation, clarification, justification, and argumentation that could have deepened the students’ understanding (Vygotsky, 1978). The problems used were routine, closed-ended tasks from the textbook that required the application of algorithms, procedures, and rules to find solutions (NCTM, 2014; Nurkaeti, 2018; Stephan, 2014).

Stephan (2014, p. 340) stressed that:

Most of the teachers in the two primary schools in South Sudan did that before the training on how to lead SCT. Teachers need to understand the difference between teaching via PS, about PS, and for PS (Schroeder & Lester, 1989; Van de Walle et al., 2020). Most of the routine and closed-ended tasks in the textbook might not provide an opportunity to implement teaching mathematics through PS, which can help students to do mathematics (Fosnot & Dolk, 2001; NCTM, 1989; Stephan, 2014).

After the intervention, most of the teachers demonstrated a more meaningful shift in their teaching practices and beliefs from PS-TCT-L0 and PS-TCT-L1 to PS-SCT-L2. Firstly, the teachers decided to use the open-ended tasks from the intervention (see Figure 3 and Appendix 2). They came to understand that not all tasks can provide opportunities for problem-solving. Closed-ended tasks are less rich than open-ended tasks (Boaler, 2016). Secondly, they allowed the students to struggle with the task instead of solving it for them. Thirdly, the teachers sought different strategies and approaches from the students. These strategies and techniques incorporated how the students were doing group work, posing mathematical tasks without guidance from the teachers, and creating their own meaningful solutions that the five teachers implemented. These approaches can help students to engage in mathematics (Fosnot & Dolk, 2001; NCTM, 1989; Stephan, 2014).

NCTM (2014) highlighted that a feature of PS is to engage the students in solving and discussing tasks that aim to boost mathematical reasoning and allow multiple entry points and varied solution strategies by the students. The six teachers’ engagement in teaching through PS using open-ended tasks is promising. However, this shift observed in their mathematical practice might not be sustained unless they receive assistance through regular follow-up by experts, facilitators, or educators. The tasks teachers used during the observation post intervention were the tasks they learned during the workshop, which were limited to the concept of algebra, specifically generalisation from patterns of figures (Kaput & Blanton, 2008; Boaler, 2016). Such resources are needed if teachers are expected to develop their mathematical teaching practices.

We found that all six teachers were on level zero (MD-TCT-L0) before the intervention. The teachers were observed standing in front of the classroom, dominating the mathematical discourse, acting as questioners, and focusing on correctness. As shown in Table 4, the teachers initiate discussion, then the students respond, and then the teachers provide an assessment, terminating the discourse abruptly. NCTM (2014) highlights that students must also have opportunities to talk with, respond to, and question one another as part of the discourse community in ways that support the mathematics learning of all students in the class. This means that by talking, interacting, and discussing, the students are engaged in the construction of knowledge and skills that foster critical thinking and reasoning (Kaput et al., 2008).

After the intervention, three teachers (T1, T3, and T4) demonstrated a meaningful shift in their teaching practices and beliefs from MD-TCT-L0 to MD-SCT-L2. T2, T5, and T6 presented a somewhat lesser shift in their teaching practices and beliefs from MD-TCT-L0 to MD-TCT-L1. As shown from the discourses presented above, the teachers have followed not only the IRE model of communication (Drageset, 2015), but also others like IRIRE, several initiations (I) and responses (R) before providing feedback (E). T1, T3, and T4 were critically observed both facilitating conversation and encouraging the students to ask questions of one another, and the teachers asked probing questions that facilitated some student-to-student talk. T2, T5, and T6 were seen encouraging the students to share mathematical ideas, and the teachers’ questions began to focus on students’ thinking rather than the answers (Franke et al., 2009). The fact that these teachers have considered to provide more autonomy to their respective students is an encouraging development.

Implementing effective MD in a mathematical classroom is a demanding task. Demirci and Baki (2023) stressed that MD allows students to speak, think, and discuss mathematics, which involves explanation and debate on mathematical ideas. Hufford-Ackles et al.’s (2004) framework includes questioning, explaining mathematical thinking, using sources of mathematical ideas, and taking responsibility for learning. These primary teachers have agreed to provide more autonomy to their students in solving and discussing their strategies, primarily T1, T3, and T4. However, the remaining three teachers faced challenges in applying these developmental trajectories.

Another possible way to orchestrate productive classroom discussions is to implement the five elements of orchestrating mathematical discussions as described by Smith and Stein (2011). However, it is complex and demands interventions and investigations (Ballard, 2017; Faria et al., 2024; Larsson, 2015). Stephan (2014) emphasised that to guide the discussion, teachers need to not only accept both correct and incorrect solutions from the students but also purposely choose particular students’ answers to begin the discussion and create debate in class. Smith and Stein (2011) called this process selecting and sequencing students’ solutions so that students’ mathematical reasoning and justification can build up with one another, providing a high level of student engagement and mathematical thinking.