The problem posing capacities of pre-service post-primary mathematics teachers: insights from an Irish university

1 Introduction

Mathematical problem-solving is considered to be at the heart of mathematics education (Cockcroft, 1982; Guberman and Leikin, 2013). While there has been much deliberation on the definition of “problems” and “problem-solving” (Lester, 2013; Schoenfeld, 1992), it is important to have clarity. One definition of problem-solving is offered by Lester and Kehle (2003, p. 510);

“Successful problem solving involves coordinating previous experiences, knowledge, familiar representations and patterns of inference, and intuition in an effort to generate new representations and related patterns of inference that resolve some tension or ambiguity (i.e., lack of meaningful representations and supporting inferential moves) that prompted the original problem-solving activity”.

The recognition that problem-solving involves a level of ambiguity is also highlighted by the NCCA (2017) and Schoenfeld (1992). This uncertainty of method to achieve a solution is recognised as a key component of problem-solving (Schoenfeld, 2013) and a detailed definition is provided by Lesh and Zawojewski (2007, p. 760) which aligns with that of Lester and Kehle (2003). Through a synthesis of the literature, Lester (2013) recognises that amongst the many definitions offered on problem-solving, there is a commonality that there must be a goal, a problem solver, and it not being possible to immediately achieve the goal.

Based on a comparison of the commonalities of definitions offered, (Lester and Kehle, 2003; Chamberlin, 2008; Lesh and Zawojewski, 2007; Lester, 2013; NCCA, 2017; Polya, 1945), this study adopts the following ‘Three Key Characteristics’ of mathematical problem-solving: (Owens, 2023, p.16).

• Problem-solving includes a goal,
• It is not immediately clear to the problem-solver how to achieve the goal,
• The problem-solver must organise prior knowledge to generate reasoning towards achieving the goal.

Based on these characteristics a mathematical problem is a task which requires students to engage in problem solving for its completion.

Problem-solving enables teachers to gain insights into students' understanding of mathematics (Silber and Cai, 2017). Schoenfeld (1992) describes how problem-solving builds students' range of techniques and perspectives creating flexible thinkers who can make judgements and arguments based on interpretations of situations. Given that problem-solving holds a key position in mathematics education, problem posing should hold similar esteem (Christou et al., 2005) as ‘at the heart of problem-solving often lies a great problem’ (Leung, 2013, p.2). However, despite this, problem posing does not receive the same amount of attention as problem-solving (Leung, 2013). It has been found that pre-service teachers expressed a desire for increased opportunities to engage in problem posing as they found it more challenging than solving a similar problem (Ellerton, 2013).

Mathematical problem posing can be defined as “the creation of mathematical problems, generated from given situations or from pre-existing questions” (Silber and Cai, 2017, p.163). In order to advance mathematical thinking, imagination is required in generating new questions and reviewing existing questions (Ellerton & Clarkson, 1996). The generation of problems can take place in response to multiple situations, such as aiming to find a solution to a real-life scenario, or, as a self-reflection tool whereby a person evaluates their solution to a similar problem by generation of a new one (Silber and Cai, 2017). Indeed, problems may be posed in the form of extending or challenging students. There are many different situations in which teachers may pose mathematical problems. One such situation is free posing, where a problem is posed for an open mathematical situation. On the other hand, structured posing situations involve a problem being posed encompassing a predetermined situation and operations (Stoyanova and Ellerton, 1996). Another situation is a semi-structured one whereby the problem poser is given a specific situation for which they base their mathematical problem on (Christou et al., 2005). Benefits of problem posing include the improvement of mathematical thinking and problem-solving skills, improvement in attitude towards mathematics, and the increased understanding of mathematical concepts (Christou et al., 2005). Given these benefits alongside the advancement of mathematics through posing problems, it is crucial for problem posing to be valued in mathematics education (NCTM, 2000).

There are many different types of mathematical tasks which can be used by teachers for different purposes. A procedural task, which can also be described as a routine task, is defined by Yeo (2007) as a task which involves the person applying previously learnt skills to a situation which has an immediately clear path to achieving a solution. Although this type of task may be initially a problem for a person who has not yet been taught the procedure, this task becomes an exercise with repetitive practice (Yeo, 2007). Procedural tasks have one specific correct answer and therefore called “closed tasks”, however, there may be multiple methods possible to achieve this goal. In contrast to procedural tasks, open-ended problems have a range of solutions which can all be justified (Silver, 1994). Due to the range of possible solutions, open-ended problems allow students to explore problems in a different manner than simply find a single solution (Foong, 2000; Silver, 1994). Yeo (2007) highlights that mathematical tasks which involve words can be called “word problems” despite them not actually being a task which involves problem-solving but rather share characteristics with procedural tasks. “Word problems” typically used in schools do not involve problem-solving but instead allow students to practise learnt procedures and methods (Moschkovich, 2002). The “word problem” and procedural tasks are more appropriate for the consolidation of methods and facts than the development of new knowledge (Orton and Frobisher, 2004).

Internationally, the high value placed on mathematical problem-solving is reflected through its inclusion in school curricula around the world including Ireland (NCCA, 2017), England (DfE, 2013), Singapore (Cheng, 2001), and Finland (Pehkonen, 2007). Expanding beyond these national examples, problem-solving is systematically evaluated within the PISA framework (Perkins and Shiel, 2014). The assessment delineates six levels of problem-solving proficiency, providing a structured means of describing students' capabilities and competencies in this domain (Perkins and Shiel, 2014). This underscores a widespread recognition—particularly among OECD member states—of the significance of problem-solving. It is recognised that there are multiple benefits of mathematical problem-solving (Guberman and Leikin, 2013) including that it facilitates students making connections between different mathematical concepts that might initially appear to be different (Carson, 2007) and also allow for the problem-solver to bring connections between theory and practise through real-world applications.

Although problem-solving is mentioned as a key skill throughout the mathematics curricula in Ireland, there appears to be disparity between the aims of the curricula and the enactment of problem solving in classrooms. In a study conducted by O'Keeffe (2011) it was found that textbooks are used by over 75 percent of mathematics teachers on a daily basis in classrooms. In an Irish context, there is not a nationally mandated textbook for which schools must use. An analysis of the topics of patterns, series, and calculus tasks, in Irish mathematics textbooks showed that tasks supported the development of proficiency with mathematical procedures but do not offer the same opportunities for supporting mathematical problem-solving (O'Sullivan, 2017). This research involved the analysis of over 7000 mathematical tasks from three popular senior cycle post-primary mathematics textbooks. From this analysis it was clear that there is a need for more novelty in textbook tasks along with a requirement for a greater level of cognitive demand in tasks. In the classification of these tasks, it was found that a high number required algorithmic reasoning which suggests that students are not exposed to tasks which facilitate attainment of skills necessary for solving non-routine problems (O'Sullivan, 2017). Indeed, 83.3% of analysed tasks were classified as low level of cognitive demand. Also, it was found that communicating mathematically is overlooked in the tasks in textbooks.

In Ireland, a new curriculum was launched in 2008 called “Project Maths” which was initially piloted before being rolled out across all post-primary schools in 2010. In this curriculum, there is a strong emphasis on problem-solving specifically the promotion of the skills of “explanation, justification, and communication” (SEC, 2015, p.3). In a report commenting on the impact of Project Maths, it was concluded that whilst there was some evidence of changes in teaching practices, traditional approaches remained prominent. These traditional approaches include the reliance on textbooks with repetitive practice of exercises (Jeffes et al., 2013). Although there was support amongst teachers of the curriculum, there was a profound difficulty in delivering it (Johnson et al., 2019). The position of textbooks in mathematics classrooms in Ireland is not surprising with a general agreement that textbooks have a dominant role across school subjects (Fan et al., 2013). However, the reliance on textbooks, which contain mainly routine exercises which do not elicit problem-solving, along with the PISA and TIMSS results suggest that students may not be receiving exposure to mathematical problem-solving tasks in Irish classrooms. In order to alleviate this situation, teachers of mathematics in Ireland need to be proficient in generating mathematical problem-solving tasks, if their students are to experience mathematical problem-solving, a key skill in the curriculum. This leads to the research question in this study;

What are the mathematical problem posing capacities of pre-service post-primary mathematics teachers in Ireland?

The teacher plays a crucial role in how their students are exposed to mathematical problems as they are responsible for creating the environment, proposing the problem, and asking students questions (Felmer and Perdomo-Diaz, 2016, p.289). A teacher's ability to create new problems or reformulate mathematical tasks has a strong influence on students' problem-solving ability (Chapman, 2015). This is underpinned by the teacher's understanding of the characteristics of a mathematical problem which is paramount in the selection and generation of mathematical problems. In the situation where teachers have sources of mathematical problems at hand, the recognition of a mathematical problem versus a routine task is necessary to employ problem-solving in the classroom. Similarly, it may also be necessary to adapt the available problems to suit their students. The skill of posing mathematical problems is particularly challenging whilst learning to teach mathematics (Crespo, 2003). When selecting or generating problems for students, teachers must consider students' prior mathematical knowledge in order to give students the opportunity to realise the limitations of their knowledge and formulate new mathematical knowledge (Nunokawa and Fukuzawa, 2002). Additionally, it is crucial that the problems posed are meaningful to the students, and provide motivation for them to attempt to solve the problems (Nunokawa and Fukuzawa, 2002).

Acknowledging the many considerations in posing a mathematical problem, teachers struggle to pose problems (Chapman, 2015, 2012; Inoue, 2009). This conclusion is supported by Silver et al. (1996) who found that both practising and prospective teachers have difficulty posing genuine mathematical problems, where they posed ill-formulated, predictable and unsolvable mathematical problems. Strong subject matter knowledge (SMK) as defined by Ball et al. (2008) is important for posing problems. SMK includes common content knowledge (CCK); mathematical knowledge and skill that is also used in settings outside of teaching, specialised content knowledge (SCK); mathematical knowledge and skill that is specific to teaching – for example, understanding of different interpretations of operations, horizon content knowledge (HCK); awareness of how mathematical topics relate to the broader discipline. Each element in the domain of SMK is important for teaching mathematical problem-solving since it involves the ability to link knowledge from different mathematical topics (HCK) (Kilpatrick et al., 2001), and conceptual understanding (CCK and SCK) (Kilpatrick et al., 2001; Lester and Kehle, 2003). Additionally, pedagogical content knowledge (PCK) may also be important for teachers to pose problems. PCK can be described as an amalgam of combining what is known about students, pedagogies, curriculum, and knowledge of mathematics (Ball et al., 2008). PCK encompasses teachers' understanding of the mathematical content and associated mathematical pedagogies which are derived from research and experience of practice (Shulman, 1987). Although strong subject matter knowledge does not necessarily mean effective teaching, a teacher will undoubtedly be limited in their teaching of mathematics if they do not possess strong subject matter knowledge. Ball et al., (2001) outline that PCK is developed through teaching experience rather than studying mathematics at an advanced level. For example, PCK includes anticipating what students may find challenging and how they might understand the material (Ball et al., 2008) which requires classroom experience. Problem posing has been found to enhance both pedagogical content knowledge and content knowledge amongst pre-service teachers (Ran et al., 2025). In relation to posing problems, teachers may need to adapt their tasks in response to students' reactions or in anticipation to students' responses. The aforementioned studies (Chapman, 2015, 2012; Inoue, 2009, Silver et al., 1996) found difficulties in posing problems by both practising and pre-service teachers, suggesting that teaching experience is a prerequisite for increasing PCK but does not guarantee an increase. It may be a lack of subject matter knowledge that is at the centre of the inability to pose a mathematical problem.

To develop this capacity, pre-service mathematics teachers (PSMTs) should experience both solving and posing problems during their initial teacher education (ITE) programme (Zimmerman, 2016). This is supported by Ellerton (2013) who advocates that problem posing skills must be acquired by teachers if problem posing is to be incorporated into school mathematics curricula. One primary concern of ITE programmes is the development of teaching potential of PSMTs to ultimately improve mathematics learning in schools (Lerman et al., 2009) through developing PSMTs' mathematical knowledge (Liljedahl et al., 2009) and a combination of theoretical and practical knowledge and skills (Lerman, 2009). In particular, learning to teach mathematical problem-solving is considered to be challenging (Lubienski, 1999; Zimmerman, 2016), and involves developing an understanding of the complex network of interconnected knowledge (Chapman, 2015). Given that, in Ireland, there is such a high reliance on textbooks which have been shown to contain predominantly procedural mathematical tasks, there is a need to research teachers' skills to pose mathematical tasks which require problem-solving, especially in the context of the prominence of problem-solving in the curricula.

The contribution of this paper is the first empirical investigation into the problem-posing capacities of pre-service mathematics teachers (PSMTs) in an Irish university context. This research holds particular relevance given the central role of problem-solving in Irish mathematics curricula. The international assessments, PISA and TIMSS, show that Irish post-primary students score higher in tasks which require procedural fluency compared to problem-solving. This indicates a potential misalignment between curricular intentions and students' classroom experiences. Since teachers largely determine their students' exposure to diverse range of mathematical tasks, it is essential for teachers to possess the capacity to pose problems through reformulation, generation, and/or extension (Silver, 1995) given the reliance on textbooks which contain repetitive exercises. This research contributes to the field by highlighting the problem posing capacities of PSMTs in Ireland in the view to supporting their future students.

2 Research methods

2.1 Theoretical framework

This paper is concerned with the problem posing capacities of PSMTs, who are undertaking an ITE programme in Dublin City University. The study formed part of the first author's PhD research. The framework for the overall study was provided by Chapman (2015) who conducted an extensive review of the literature on the capacities needed by teachers to effectively teach problem-solving spanning over 100 years. This review led to the identification of three main categories that make up the mathematical problem-solving for teaching (Chapman, 2015); 1) Problem-solving content knowledge (PSCK), 2) Pedagogical problem-solving knowledge (PPSK), and 3) Affective factors and beliefs. These three main categories are comprised of six different categories. 1) PSCK consists of; knowledge of problems, knowledge of problem-solving, and knowledge of problem-posing. 2) PPSK consists of; knowledge of students as problem-solvers, and knowledge of instructional practices, and the final category 3) is affective factors and beliefs. As shown in Figure 1, these categories are interconnected in nature (Chapman, 2015, p. 32). It is important to note that the knowledge held by teachers in these categories must be purposeful in order for it to be beneficial to the teaching in teaching for problem-solving proficiency.

While acknowledging the interconnected nature of these components for enabling effective problem solving, the focus of this study is primarily on the problem-posing dimensions as a sub-component of the problem-solving content knowledge domain. That said, we recognise that problem-posing is only one dimension and that more detailed analysis and understanding of teacher's problem-solving capabilities is only possible with additional research that takes into account the other sub-components and knowledge domains.

Knowledge of problem posing is defined as the generation of new problems and the re-formulation of given problems (Silver, 1994). This capacity involves teachers' ability to generate new problems and to adapt existing problems based on the considerations of students' needs (Chapman, 2015). It is possible for problem posing to take place in three different situations, namely, generating a problem from a given situation, extension after the completion of a problem and presenting a problem in a new context, and posing a problem during the problem solving context due to a change in the original goal (Silver, 1995). Through the lens of a teaching situation, teachers must be cognisant of the mathematical ideas that are involved in the problems and therefore must be proficient in analysing the characteristics of a mathematical problem in anticipation of the influence of the task on students (NCTM, 2000).

The knowledge of problems capacity refers to teachers' understanding of the nature of problems which is a key aspect in selecting appropriate problems for students (Chapman, 2015). It is important for students that teachers can recognise the characteristics of problems to enable them to select suitable tasks (Yeo, 2007), and to be able to anticipate students' questions about the task (NCTM, 2000).

The final capacity in the PSCK category, knowledge of problem-solving, refers to the teacher's proficiency in mathematical problem-solving. It is important for teachers to be proficient in problem-solving and understand the nature of different approaches in order to; understand students' attempts and solutions, understand the implications of different approaches, and evaluate the effectiveness of approaches (Chapman, 2015). While proficiency is required to make connections between mathematical concepts, it is not required for teachers to be expert problem solvers but rather be experienced in a wide range of mathematical problem-solving tasks and to understand the requirements of successful problem-solving (Lester, 2013). Successful problem-solving involves; sufficient resources, metacognitive control, appropriate beliefs, and competency in heuristic strategies (Schoenfeld, 1985).

The category, PPSK, involves the following two capacities; knowledge of students as problem-solvers, and knowledge of instructional practices. The first capacity refers to the teachers' understanding of their students as problem solvers (Chapman, 2015). This understanding of students' thinking improves the quality of the teaching of problem solving (Maher et al., 2014). The second capacity, knowledge of problem-solving instruction, consists of teachers' knowledge of instructional strategies and metacognition to support problem-solving proficiency. Instructional strategies include using heuristics, a general strategy which is independent of any subject topic but helps with understanding the problem (Schoenfeld, 1992). Whilst it is recognised that heuristics are not alone enough to solve problems, they are a helpful resource to employ, albeit dependent on mathematical knowledge (Schoenfeld, 1992). When teaching problem-solving, teachers must analyse the rationale for employing a certain strategy and then use this knowledge to solve the problems (van den Kieboom, 2013).

The final category of Chapman's framework is affective factors and beliefs which has been identified as an important contributor to problem-solving behaviour (Lester and Kroll, 1993). The beliefs of teachers impact beliefs of their students when problem solving, as students tend to rely on extrinsic motivation when they have low motivation when trying to solve problems (Marcou and Philippou, 2005). The cognitive resources available to student when learning are related to their beliefs about what they consider to be useful in learning maths, and if these beliefs are not in favour of understanding, a large amount of information becomes inaccessible (Schoenfeld, 1985). McLeod (1988) highlights that problem-solving situations cause emotions due to the ambiguity of the task which can be either negative or positive.

The framework presented by Chapman (2015) incorporates the seminal work of Polya (1945) and Schoenfeld (1985) and other frameworks which are offered in the area of teaching mathematical problem-solving (Ellerton, 2013; Lester, 2013). Chapman's framework accounts for elements both inside and outside of the classroom (Lester, 2013), heuristics (Polya, 1945; Schoenfeld, 1985; Mason et al, 2011), problem posing (Ellerton, 2013; Polya, 1945) and affective factors (Schoenfeld, 1983) amongst others.

2.2 Setting

This research was conducted in an Irish University setting across two different years with two cohorts of PSMTs; Cohort A (n = 55) and Cohort B (n = 50) and is situated in a larger study investigating the capacities of PSMTs to teach mathematical problem-solving effectively (Owens, 2023). At post-primary level in Ireland, the study of mathematics is compulsory for all students. There are two state examinations; Junior Certificate (age 14–15) and Leaving Certificate (typically age 17–18). In these examinations mathematics is offered at different levels of study, ordinary level and higher level. There are no nationally mandated textbooks which schools must use but there are national curricula which are developed by the National Council for Curriculum and Assessment with the State Examinations Commission responsible for the development, assessment, accreditation and certification of the post-primary examinations.

Each cohort of students, (Cohort A and Cohort B), is made up of a mixture of students in their first year of university-level education and a group of second-year students. Cohort A comprised of 35 first-year and 20 second-year students. Cohort B comprised of 32 first-year and 18 second-year students. Both programmes undertook a concurrent module with a specific focus on mathematical problem-solving – within which data for this study was collected. In both programmes, students study both mathematics and education studies. There are also extensive school placements in both programmes. At the point of this study, the PSMTs would not have been on school placements. All the participants completed their second-level education in the Irish system and thereby completed the Leaving Certificate curriculum. The entry requirements, in terms of previous mathematics grades, are the same for both programmes. Both programmes are concurrent teacher education programmes which means that graduates are fully qualified to teach at post primary level in Ireland without further university education. The differences between these two programmes exist in the second subject that participants study as both are dual-subject ITE programmes. Although one group of PSMTs are in second year of study, this is the first module which includes the study and practice of mathematical problem-solving. Since all participants completed the same secondary school mathematics curriculum and this is their first explicit module focused on problem-solving, it is reasonable to assume that they share broadly comparable prior experience, while recognising that individual experiences may still vary. In the findings, there was no evidence to suggest that participants performed differently based on their programme of study.

2.3 The module

The module was 12 weeks in duration and consisted of three hours of classes per week, which were a combination of lectures and tutorials. The module was designed around Chapman's framework with the view to investigate and develop the capacities of pre-service teachers to teach mathematical problem-solving. The categories of PSCK and affective factors and beliefs shaped the module, while PPSK had lesser emphasis within the module due not being in a classroom setting. It is important to reinforce that these capacities are over-lapping in nature.

The Rubric Writing approach, developed by Mason et al. (2011) held a central role in the module. This heuristic consists of three phases; Entry, Attack, and Review. The Entry Phase proposes three questions; 1) What do I want? 2) What do I know? 3) What can I introduce?. These three questions frame a starting point for the problem solver by identifying the goal of the task, isolating any applicable prior knowledge, and promoting the use of different representations or notation. The Attack phase includes trying different approaches, specialising, generalising, and getting stuck. The final, Review phase, happens on completion of a solution whereby the problem solvers; 1) checks the resolution, 2) reflects on the key ideas and key moments, and 3) extends to a wider context (Mason et al., 2011, p.36).

The participants were gradually and incrementally exposed to the three phrases of the Rubric Writing approach in a variety of different types of mathematical problem-solving tasks. The first two phases align with development of PSMTs' knowledge of problem-solving through the use of representations, eliciting prior knowledge, conjecturing, specialising and generalising which are all elements of problem-solving proficiency. The Review phase prompts the person to pose a problem in the form of an extension which aligns with the knowledge of problem posing capacity.

The capacity of knowledge of problems was interwoven throughout the module. PSMTs received tuition on the nature of mathematical problems and the characteristics of different types of mathematical tasks such as routine tasks (Yeo, 2007), “wordy” problems (Moschkovich, 2002), and mathematical problems (Mason et al., 2011). Characteristics of tasks were scrutinised such as the level of cognitive demand, the syntax of the task, and the spectrum of closed to open tasks. PSMTs were given opportunities to analyse different types of tasks and discuss their characteristics. They also completed a “Task Sorting” activity to further investigate this capacity (Owens, 2023).

The affective factors and beliefs category of Chapman's framework was prevalent in the module, particularly through the Attack phase of the Rubric Writing approach, where being “stuck” is a key element. Mason et al. (2011) state that a positive attitude is essential in overcoming being stuck, whilst negativity inhibits progress. Having appropriate beliefs play an important role in terms of successful problem-solving (Marcou and Philippou, 2005; Philipp, 2007; Lester, 2013; Schoenfeld, 1985). Additionally, teachers' beliefs on the role of problem-solving impact how problem-solving is incorporated in the classroom (Andrews and Xenofontos, 2015; Kloosterman and Stage, 1992; Lester, 2013). The PSMTs received instruction on the impact of the affective domain on problem-solving, growth mindset (Boaler, 2016; Dweck, 2017), completed a survey (Kloosterman and Stage, 1992), and completed activities which articulated their feelings whilst problem-solving (Owens, 2023; Felmer and Perdomo-Diaz, 2016).

The Knowledge of problem posing capacity, the focus of this paper, consisted of different activities throughout the module. Problem posing is evident in the Review phase of the Rubric Writing approach in the form of extension. Extending a problem is defined as a problem-solver generating a new problem based on a completed problem (Mason et al., 2011). A benefit of extending a problem is enhancing the problem-solver's understanding of the solution to the original problem and recognising features of the problem (Brown and Walter, 2005). The skill of extending a problem aligns with the heuristics developed by Polya (1945) and Mason et al. (2011) which were fundamental in the university module. Extending mathematical problems has benefits such as promoting mathematical thinking through engagement with the task (Mason et al., 2011) and making links between different mathematical topics through exploration of multiple adaptation possibilities (Prestage and Perks, 2007). The PSMTs completed tasks on reformulating procedural tasks into mathematical problem-solving tasks along with generating problems based on specific topics. Further detail on the activities for which data was collected is given below.

Coinciding with the elements of Chapman's framework in the design of the module was a focus on mathematical language. This included definitions of mathematical terms, proofs, notation, propositions, and types of errors in communication using mathematical language.

2.4 Data collection

The following instruments were used to address the research question above, in relation to the three different types of problem posing situations as outlined by Silver (1995); generation, reformulation, and extension. Cohort A (n = 55) completed two activities focusing on their ability to pose mathematical problems in the situation of generation (Activity One) and reformulation (Activity Two) (see Appendix A). Activity One was a semi-structured situation (Silber and Cai, 2017) and was completed in Week 11 of the module while Activity Two was a structured problem posing situation (Christou et al., 2005) and was completed in Week 12.

In Activity One, the PSMTs were presented with 12 different teaching scenarios for which they were asked to generate a problem. For each scenario, the year of study, the level of study (higher or ordinary level¹), and the prior knowledge (in terms of content covered, aligned with the relevant Irish second level curriculum) of the hypothetical students were specified. The prior knowledge topics listed covered a broad range to allow the PSMT freedom to develop a mathematical problem without limiting its scope. No specific learning outcomes were specified, again to give greater scope to the PSMTs to compose a problem. The PSMTs were asked to generate a mathematical problem that would be suitable for each set of hypothetical students. The PSMTs had access to the curricula whilst generating problems. The criteria for these tasks were the Three Key Characteristics, which the PSMTs were aware of prior to designing these tasks. An example of a scenario is:

Create a mathematical problem that is suitable for each of the described students below: It is 2nd year ordinary level class who have completed trigonometry, algebra, indices, factorising and geometry.

Table 1 below gives an outline of scenario with the prior mathematical topics, level of study, and year group for which the PSMTs were asked to generate of mathematical problem-solving task for.

Table 1. Requirements of the scenarios in activity 1.

Activity Two, which focused on reformulation, consisted of five questions. The variety of reformulation of tasks allowed for a broader understanding of PSMTs' capacities. Question 1 involved reformulating a routine task to a task which requires problem-solving (Crespo, 2003). Question 2a, 2b, and 3 required reformulations so that specific mathematical concepts were addressed (Greer and McCann, 1991; Leung and Silver, 1997). Question 2a asked the PSMTs to create a problem which involved any aspect of fractions and Question 2b required the division of fractions. Question 3 required the topic of ratio. These topics were specifically chosen as they are fundamental in the Number strand of the national curricula and fractions have an association with mathematics achievement (Copur-Gencturk, 2021). Question 4 required reformulation of an exercise to an open-ended problem as open-ended problems can provide students with important experience in interpreting problems and generating multiple solutions (Foong, 2000; Nicol and Bragg, 2009; Silver, 1994).

To further investigate the PSMTs' capacity to pose problems, the data collection for Cohort B (n = 50) focused on the extension of problems (Silver, 1995). The PSMTs were asked to complete three extension tasks where they attempted a problem (see Appendix B) and then extended said problem. Extending a problem is defined as a problem-solver generating a new problem based on a completed problem (Mason et al., 2011). Problem One involved the topic of area, Problem Two was distance, speed and time, and Problem Three was trigonometry. As mentioned previously, the heuristic of the Rubric Writing approach was central to the module and the Review phase focuses specifically on extension meaning that the PSMTs had increasing exposure to this skill throughout the module. Similarly, this aligns with other heuristics such as Polya (1945), who advocates the “Looking back” phase of problem-solving. There are multiple ways to extend a problem, one such way being the removal of assumptions (Mason et al., 2011). These changes can allow for links to be made between different mathematical topics and offer experience of different approaches (Prestage and Perks, 2007).

The problem posing activities of Cohort A and Cohort B align with the situations for which teachers might pose mathematical problems (Silver, 1994) and allows for the investigation of the knowledge of problem posing capacity within the PSCK category of the framework.

2.5 Data analysis

As outlined by Zhang et al. (2024), there are multiple ways in which problem posing studies have analysed the data due to the diverse nature of mathematical problems. In this study, the analysis for each of the activities undertaken by Cohort A and Cohort B involved a two-step process due to the qualitative nature of the methodology. This was completed by the first author. Every activity (both of the activities undertaken by Cohort A, and the activity undertaken by Cohort B) involved the PSMTs generating mathematical problems. Thus, the first step in the data analysis consisted of determining if the tasks produced by the PSMTs did indeed meet the criteria of a mathematical problem- the Three Key Characteristics above. The Three Key Characteristics adopted in this study align closely with established conceptions of problem-solving in the literature (for example, Lester and Kehle, 2003; Lesh and Zawojewski, 2007; Polya, 1945; Schoenfeld, 1992). These characteristics define mathematical problems as goal-directed tasks for which solution methods are not immediately apparent and which require the application of prior knowledge and reasoning. These elements of mathematical problem-solving are consistent with those identified by researchers of mathematical problem-posing which will be further discussed in due course (Chapman, 2012; Crespo, 2003, Silber and Cai, 2017, Silver, 1994). In Silver (1994), problem solving is described as a process of exploration, conjecture, and testing, explicitly reflecting all three characteristics, particularly the non-routine nature of tasks and the centrality of reasoning. In contrast, Silber and Cai (2017) and Chapman (2012) do not explicitly define problem solving, but assume a conventional understanding grounded in the work of Polya and Schoenfeld, where tasks are goal-directed, non-routine, and require the application of prior knowledge. Thus, the Three Key Characteristics can be seen as making explicit a shared underlying definition that is implicit across these studies by operationalizing a widely accepted conceptualization rather than introducing a novel definition.

The PSMTs were aware of these characteristics and asked to pose a problem based on them. The PSMTs' work (each of the tasks produced) was classified as either Achieved or Not achieved, where Achieved were tasks that met the criteria of the Three Key Characteristics of a mathematical problem. The second step of analysis focused on the characteristics of the posed tasks that were classified as Not Achieved. By utilising an inductive approach (Thomas, 2006), similarities and differences in these tasks were identified. The second author completed this process on a sub-sample of the posed tasks for each cohort with ultimate full agreement between both authors. A minimum of 25% of each activity for both cohorts was also analysed by the second author. In the first step of the analysis, 100% agreement was reached across all activities for both cohorts. In the second step of analysis, there was also full level of agreement but with further discussions around “unsolvable” leading to a separate subcategory of “not solvable due to missing information” which meant repeated analysis of the sub-sample. There was 100% agreement in the analyses and these categories were used in further analysis.

3 Findings

The first step of the analysis of the tasks posed by the PSMTs involved determining whether the tasks constituted mathematical problems or not. This was done by comparing the posed tasks to the Three Key Characteristics of a problem. The results are shown in Table 2 with Achieved referring to tasks classified as mathematical problems and Not Achieved referring to tasks which did not constitute a problem.

Table 2. Percentage of posed tasks classified as achieved or not achieved as mathematical problems.

As seen in Table 2, the majority of mathematical tasks posed by PSMTs in Cohort A and Cohort B were not deemed to be mathematical problems when compared to the Three Key Characteristics and were therefore deemed as Not Achieved. Indeed, in some instances there were no tasks posed which met the criteria of a mathematical problem-solving task.

The second step of the analysis involved using an inductive approach (Thomas, 2006), on the posed tasks which were categorised as Not Achieved in the first step of the analysis. This second step revealed the reasons for which the tasks did not meet the criteria of a mathematical problem and resulted in the identification of the following categories:

Procedural: Included in this category were tasks which for which the students have access to a procedure that allows for the immediate completion of the task or tasks that could be solved using a previously learnt procedure.

Unsolvable: The mathematical task posed was not possible to solve, or there was an unhelpful ambiguity² as to what the question was asking due to the poor phrasing of the task.

On analysis, a prominent subcategory of unsolvable here was the following:

Not solvable due to missing information: Tasks in this category were missing information which was crucial to finding a solution. These are presented as a separate category to allow for understanding of the extent for which missing information caused the tasks to be unsolvable. Tasks which remained in the unsolvable category included tasks which were flawed in reasoning, were ambiguous in the goal of the task, or contained contradictory information.

We note that unsolvable falls outside of the three criteria: for a task to be characterised as a valid mathematical problem it must be possible reach a solution (Silber and Cai, 2017).

The results of the categorisation of tasks in Activity One are presented in Figure 2 below.

Examples of tasks posed by the PSMTs for each of the categories are given below.

The first two examples of tasks posed by PSMTs were classified as Achieved meaning they were judged to meet the criteria of a problem:

“The diameter of the base of a cone is 7 cm, the height is twice the circumference. What is the volume of the cone?”

Example 1 meets the criteria of the Three Key Characteristics for the following reasons. There is a goal; there is a clear goal for the problem solver which is to find the volume of the cone. It is not immediately clear how to achieve the goal; there is a level of uncertainty in this task as the problem solver is not explicitly presented with the data required for use of the formula for the volume of a cone. Some interpretation is required. Prior knowledge is required to generate reasoning towards achieving the goal; the problem solver must use their prior knowledge of the definition of the circumference of a cone and select the correct formula to find the length of the circumference. The problem solver must also halve the diameter to find the radius. There are multiple steps that the problem solver must take to find the solution. The problem is also solvable.

“A driver drove at 90 km per hour. He sneezed for half a second and his eyes were closed. How far did he travel in the half second?”

Example 2 meets the criteria for the Three Key Characteristics of a mathematical problem for the following reasons. There is a goal; there is a clear goal to find the distance travelled in half a second. It is not immediately clear how to achieve the goal; it is not immediately clear how to achieve the goal as changes in units are required and multiple steps. Prior knowledge is required to generate reasoning towards achieving the goal; the problem solver must identify the units and decide how to approach the problem using prior knowledge of the relationship between distance, speed and time.

As seen in Figure 2, with the exception of Question 1, the majority of the generated tasks for each question were classified as Procedural.

“In a right-angled triangle with sides of 4 cm and 5 cm find the hypotenuse.”

In this example, it is immediately clear to the problem solver that the task requires Pythagoras' theorem to reach a solution.

The next task was deemed to be Unsolvable due to the language used and a suggested misunderstanding of the mathematical content. There are two interpretations available: the student has asked for the midpoints (plural) of each of the two lines, or they have asked for the midpoint of a pair of lines. Neither concept is defined, and so there is a failure to articulate correctly a valid mathematical task.

“There are two lines: 2x + y = 9 and x + 4y = 16. Find the slope of the two lines and the midpoint of the two lines.”

In the final example further information is necessary to find a solution. The task does not include the shape of the water tank meaning it is not possible to find how long it would take to empty. Even if the shape of the water tank is considered to be irrelevant, then it would be necessary to know that the tank was full when it started draining. In addition to this, the rate of 0.25 m³ per second is a very fast and most likely not a realistic rate of draining. For these reasons the task is unsolvable and not a mathematical problem. Indeed, it may highlight the PSMT's level of subject knowledge and also an insight into the PSMT's focus on correct mathematical writing.

“A water tank has a radius of 5 m and can hold 20 m³ of water. If water flows out of the tank at a rate of 0.25 m³ a second how long will it take for the tank to empty?”

The analysis of the PSMTs' work on Activity Two proceeded in the same way: rating their tasks as either Achieved or Not Achieved. In the second step, additional categories of Not Achieved tasks were identified. Recall that in Activity Two, the PSMTs were asked to firstly create one problem from one exercise. Next, participants were given three problems to reformulate so that specific mathematical concepts were addressed. Finally, the participants were asked to reformulate one mathematical task to make an open-ended problem. Thus additional categories arose relating to the PSMTs lack of success at meeting these additional criteria.

The analysis of the Not Achieved posed problems in Activity Two resulted in an additional two categories which were specific to the requirements of this activity. The category of “did not address the topic” was only applicable to Question 2a), Question 2b) and Question 3. The category of “not open” was applicable to Question 5 only. An open-ended problem is a mathematical problem which has more than one justifiable solution (Silver, 1994). These two categories were established in addition to the Three Key Characteristics due to the specificity of the requirements of the tasks.

Did not address the topic: The task posed did not address the topic that was specified in the given question

Not open: The task posed did not meet the criteria of being an open-ended problem. This is only applicable to Question 4 and is in addition to the Three Key Characteristics.

Figure 3 shows the categorisation of the tasks posed by the PSMTs in Activity Two.

Both Problem 2a) and Problem 2b) required the PSMTs to reformulate the given questions to mathematical problems involving fractions. Problem 2a) required the knowledge of fractions whilst Problem 2b) more explicitly required knowledge of the division of fractions. In a study conducted by Toluk-Uçar (2009), it was found that despite PSMTs' competence and confidence in their knowledge of fractions, their posed problems contained many inaccuracies and misconceptions. It was clear that the PSMTs had more difficulty in addressing the topic in Problem 2b) with 18.5% success rate compared to 31.7% in Problem 2a). It is evident from the low level of achievement (8%) in Problem 4 that the PSMTs had difficulty in generating an open-ended problem.

3.1 Cohort B

At three different points of the module, the PSMTs were asked to individually attempt a mathematical problem and subsequently to extend the given problem. PSMTs were provided with instruction and direct input on what was meant by extending a mathematical problem. This included a discussion of the purpose of working on extending problems through the lens of Mason's Rubric Writing approach (Mason et al., 2011) and the lens of being a prospective teacher and challenging future students. Time was an influential factor in the number of PSMTs who completed their problem-solving attempt and then generated an extension task. The PSMTs had an hour to complete both. The PSMTs worked on these tasks in classroom sessions of limited duration, and so pressure of time may have been a factor in their success with this activity. Out of a total of 50 PSMTs, in Problem One, 12 PSMTs completed the extension task, 33 PSMTs in Problem Two, and 22 PSMTs in Problem Three. Attrition was seen across all three problem posing tasks experienced by Cohort B which could, firstly, be attributed to time pressures of the task. In particular, attrition is seen in Problem 1 where 12/50 completed the problem posing task. Secondly, this may be explained by the PSMTs' experience in using the Rubric Writing style increasing throughout the module. When completing Problem 2 and Problem 3, the PSMTs would have greater experience and perhaps confidence in using the rubric approach which may have reduced the length of time spent solving the original problem. It is also possible that the topics covered in the different problems influenced the PSMTs' ability to reach the problem posing element of the activity. The non-completion was considered to be random with variation in those who completed the task across the three problems.

Each extension task which was initially categorised as Not Achieved was analysed using an inductive approach (Thomas, 2006) to determine why it did not meet the criteria of a mathematical problem. Three categories emerged, namely: unsolvable, procedural, and measurement change. The characteristics of the categories of unsolvable and procedural align with those described for Cohort A. The category, Measurement Change, referred to instances where the PSMT retained the original problem but simply changed the units of measurement given or the numbers. The percentage of posed tasks in each category are shown in Figure 4 below.

For each of the three problems, the category of Unsolvable had the highest percentage. There was a decrease in the percentage of tasks that were classified as a mathematical problem between Problem 1 and Problem 2, which then persisted in Problem 3. After each problem, the PSMTs received feedback as a group on their posed problems regarding the characteristics of the posed tasks based on the definition of a mathematical problem.

Below are examples of posed tasks which were classified as Unsolvable for each of the three original problems. All three of these examples were missing information required to reach a solution so were therefore classified as Unsolvable.

Problem 1 original (Mason et al., 2011): A goat is tethered by a 6 metre rope to the outside corner of a shed measuring 4 metres by 5 metres in a grassy field. What area of grass can the goat graze?

Problem 1 extension:

A goat is tethered by a 6 m rope to the outside corner of a shed measuring 4 m by 5 m in a grassy field. There is a water trough attached to the shed. Find out the furthest position it can be from the goat, with the goat still able to drink from it.

From the language used in this task, it appears that there is one particular answer expected to be achieved. However, the shape or size of the water trough is not given meaning that one specific answer cannot be reached.

Problem 2 original (NRich, 2019): Roberto drove a total distance of 250 km. The whole journey took him 3 h, including a 20 min stop to get petrol. Before he stopped for petrol, his average speed was 80 km per hour. After his petrol stop, his average speed was 100 km per hour. How long did Roberto drive for after his petrol stop?

Problem 2 extension:

Richard took part in 180 km cycling race. His average speed was 40 km/h throughout the race. During the race he fell and had to stop for some time. Overall, it took him 4.5 h. How long was he off the bike for?

With the given information in this question, it is not possible for the cyclist to have spent time off of his bike meaning that the question is unsolvable.

Problem 3 (NRich, 2019): A well is dug in a courtyard. The distances from the well to three of the corners are 10 metres, 5 metres and 11 metres, as shown in the diagram below. Find the distance from the well to the fourth corner

Problem 3 extension:

There is a well dug in the ground in a courtyard. One side of the courtyard is 10 m. From one corner to the well is 10 m. Find the length of x from the corner to the well.

There are multiple ways in which extra information could be presented to reach a possible solution to this mathematical task, however, there is not enough information currently available. The PSMT does not appear to realise that all the data presented in the original task is needed to answer the question. If the PSMT meant for this task to be open-ended, the language would have written in the form of “find possible solutions”. Similarly, the accuracy of language is seen here in finding “the length of x” where, indeed, “x” is a length.

Chapman's theoretical framework comprises of three main categories, one of which is PSCK. PSCK encompasses three capacities, including the capacity for knowledge of problem posing. Across Cohort A and Cohort B, the problem posing capacities of PSMTs in Ireland was investigated across generation, reformulation and extension.

4 Discussion

The goal of this study was to investigate the capacity of post-primary PSMTs in Ireland, to pose mathematical problems in response to the position of problem-solving in the curricula. As identified by Ran et al. (2025), problem posing is inherently related to problem solving but Leung (2013) has noted that the attention given to problem posing is not proportionate to that given to problem-solving. Likewise, Chapman (2015) identifies problem posing as a component of the suite of knowledge and skills required by teachers for the effective teaching of problem solving. Motivated by this, problem posing was incorporated into the module at the heart of this study. Nevertheless, the analysis of the mathematical tasks posed by Cohort A and Cohort B showed that the PSMTs had difficulty in posing mathematical problems. In a study conducted by Ellerton (2013), it was found that most pre-service teachers desired more opportunities to create problems as they found it more difficult to create a problem than solve a similar problem. The key finding of this study is that PSMTs in Ireland struggle to pose mathematical problems in situations of; generation, reformulation, and extension, demonstrating that this component of the PSCK category in Chapman's framework for effective teaching of mathematical problem-solving needs attention.

In the analysis, the category of Procedural included tasks whereby it was immediately clear on how to approach the task towards achieving a solution or a previously learnt solution could be used to reach such a solution. In Activity One which was completed by Cohort A, the majority of posed tasks, with the exception of Problem 1, were classified as Procedural. This aligns with the findings of Crespo (2003) who conducted a study investigating pre-service primary teachers' problem posing capacities. It was found that on a course in a university setting that the tasks posed by the teachers were straightforward computational tasks (Crespo, 2003). While this may be more expected by non-specialist primary teachers, the PSMTs in this study are specialist mathematics post-primary teachers. The difficulty in posing non-procedural mathematical tasks is not unique to PSMTs but the tendency to reduce the cognitive demand of posed tasks is also demonstrated by in-service teachers (Stein et al., 1996).

The Unsolvable category which consisted of ill-formulated tasks (Schoenfeld, 1987) arose frequently across the problem posing situations expressed by Cohort A and Cohort B. The presence of this category is similar to the findings of Silber and Cai (2017) who investigated the problem posing capabilities of pre-service teachers in free and structured mathematical problem-solving conditions. Activity One, completed by Cohort A would be considered semi-structured situation meaning that a specific situation is given to base the posed problem on (Christou et al., 2005). The high level of Unsolvable posed problems is in contrast to the findings of Silber and Cai (2017) where the majority of posed problems were classified as solvable problems.

In Activity Two, Problem 2a) and Problem 2b) the PSMTs were asked to reformulate given tasks to problems involving fractions. The low level of success (31.7% and 18.5% respectively) aligns with the work of Toluk-Uçar (2009) who found that whilst pre-service teachers were confident in their knowledge and use of fractions, their posed problems showed misconceptions and inaccuracies. The results of this present study suggests that the PSMTs may lack understanding of the topic of fractions and more specifically, the division of fractions. The high level of unsolvable posed tasks exposes knowledge gaps in the PSMTs. It is outlined by Ball et al. (2008) that teachers need to have a deep understanding of the content in the school curriculum allowing them to guide students in problem-solving. It is crucial for teachers to have subject matter knowledge to solve the work that they assign along with using the correct terminology and notation (Ball et al., 2008) which in this case the PSMTs did not express this knowledge. If the PSMTs were to use these tasks with students in a classroom it would be potentially detrimental to students' learning (Ball et al., 2008) through the PSMTs' content knowledge influencing learning with inaccurate tasks (Lester, 2013).

The PSMTs had the most success in reformulating an exercise as a problem. This may be viewed as a positive finding given the reliance of teachers in Ireland on textbooks whereby most tasks in the textbooks are routine exercises. However, it is important to note that whilst this is where the PSMTs demonstrated most reformulation ability, the level of success stood at just 41.9% with a further 27.9% of posed tasks being classified as procedural.

In contrast to this, the PSMTs had the lowest level of success (8%) with Problem 4 which required the reformulation of an exercise as an open-ended problem. Of the problems posed by the PSMTs, 56% were classified as Unsolvable with the remaining tasks being classified as Procedural. These results are in sharp contrast to the findings of a study conducted by Nicol and Bragg (2009) who found that pre-service teachers were competent in posing open-ended problems with a 97% success rate without formal training. The setting of the study conducted by Nicol and Bragg (2009) was similar to this study as it was conducted in a mathematics education course over 13 weeks. This suggests that the PSMTs in this study do not understand the characteristics of an open-ended problem or have difficulty in posing such.

Cohort B focused on the extension aspect of problem posing whereby after attempting a problem, they were asked to pose an extension problem (Silver, 1994). As the Rubric Writing approach (Mason et al., 2011) was central to the module, the PSMTs had increased experience of extending tasks through implementing the Review phase of the rubric between each of the analysed problems. The majority of the extension tasks were classified as Unsolvable (P1 = 50%, P2 = 39.4%, P3 = 59.1%) showing an increase of such tasks between Problem 1 and Problem 3 despite a decrease in Problem 2. It is possible that this decrease in Problem 2 may be due to the topic of distance and time, or there is the possibility that the PSMTs spent a greater amount of time working on finding a solution to the problem than extending said problem compared to Problem 1 or Problem 3. This corresponds to the greater number of posed tasks that were classified as Measurement Change in Problem Two. The mathematical topic associated with Problem Two was distance and time and trigonometry was the mathematical topic associated with Problem Three. In Problem One, the PSMTs had most success with posing a mathematical problem and this may be due to area being the mathematical topic as the PSMTs used a variety of shapes in their posed problems. This possible explanation for the variation across the three extension tasks relates to the differing mathematical demands of the underlying topics with a solid understanding of the mathematical concepts being required (Ellerton, 2013). This may be particularly important when the underlying mathematical domains differ in their structural complexity. While the area problem allowed participants to modify dimensions or shapes within a relatively straightforward geometric structure, the distance–speed–time and trigonometry problems required the coordination of multiple interconnected quantities and relationships. Such tasks place greater emphasis on conceptual understanding and the ability to make connections between mathematical ideas (Kilpatrick et al., 2001), while also reflecting the complexity of modelling situations in which multiple variables and relationships must be coordinated simultaneously (Lesh and Zawojewski, 2007). From the perspective of Ball et al.'s (2008) framework of MKT, successfully extending these latter problems may require a greater degree of SCK. Rather than simply solving a problem, PSMTs must understand the mathematical structure underpinning the task and recognise which relationships must be preserved when modifying it. The large number of unsolvable tasks suggests that participants often altered surface features of problems without maintaining these underlying mathematical relationships. This may indicate limitations in SCK and may also help explain the prevalence of missing information in posed tasks, as weaknesses in understanding mathematical structure can become evident through mathematical writing and the communication of mathematical relationships. Indeed, posing a mathematical problem requires individuals to communicate mathematical concepts, assumptions, and relationships in a precise and coherent manner, suggesting that difficulties in mathematical writing (Powell et al., 2021; Smith et al., 2025) may contribute to the production of incomplete or unsolvable tasks.

An important observation arising from the findings is the limited evidence of improvement in PSMTs' problem posing capacity over the duration of the module, particularly in Cohort B where no clear progression was evident across the three extension tasks despite explicit feedback being provided. Whilst it is possible that the module successfully developed awareness of the characteristics of mathematical problems, translating this understanding into practice proved challenging for PSMTs. This aligns with previous research indicating that problem posing is inherently difficult and requires both strong subject matter knowledge and experience with diverse problem types (Crespo, 2003; Ellerton, 2013). A key note to consider is that the tasks for Cohort B focused explicitly on extension signalling that this form of problem posing might need more to be addressed in more than a single module to support meaningful development.

4.1 Limitations

A limitation of this study is that the time constraints impacted the PSMTs' posing abilities, however, they did still complete the activity in the time frame. Similarly, for Cohort B, the change in mathematical topic that the original problems involved could have influenced the PSMTs' ability to pose problems with perceived more difficult topics. This could potentially explain the fact that there was no improvement in Cohort B's performance. It is important to remember that knowledge of problem posing is one component of the category of PSCK meaning that inferences about teachers' capacity to effectively teach mathematical problem-solving is limited to their capacity to pose mathematical problems through reformulation, generation, and extension in this study.

This study is situated in a single Irish university, although, there is no indication that the cohorts in this study are atypical of pre-service post-primary mathematics teachers more broadly. Entry to the programme requires successful completion of Leaving Certificate mathematics, indicating that participants are mathematically capable relative to the general student population. More broadly, recent research into the supply of mathematics teachers in Ireland has ongoing challenges in recruiting and preparing sufficient numbers of highly qualified mathematics teachers, as well as the prevalence of out-of-field teaching (Prendergast et al., 2026). These findings suggest that cohorts entering initial teacher education are not necessarily characterised by uniformly high levels of specialised mathematical knowledge, but rather reflect a mathematically capable yet diverse population. Recruitment issues are not unique to Ireland with many countries experiencing the shortfall in mathematics teachers. Furthermore, internationally, students also experience school mathematics through predominantly procedural and teacher-centred approaches like that in Ireland (O'Sullivan, 2017) with limited exposure to problem-solving. This has been shown to shape subsequent teaching practices with pre-service teachers entering teacher education programmes with strong beliefs (Jao, 2017). Taken together, these factors suggest that while the findings may be transferable to similar teacher education contexts, consideration of local programme structures and participant characteristics should be accounted for.

5 Conclusion

The difficulties expressed by the PSMTs in posing mathematical problems corroborate findings of international studies of both in-service and pre-service mathematics teachers (Crespo, 2003; Ellerton, 2013; Silber and Cai, 2017). However, this is the first study to investigate this capacity in an Irish setting and is of significant importance since problem-solving is central to the mathematics curricula in Ireland and there is a reliance on textbooks (O'Keeffe, 2011) which are heavily comprised of tasks which do not elicit problem-solving skills (O'Sullivan, 2017). The PSMTs struggled to pose mathematical tasks that met the criteria of a mathematical problem and instead posed procedural or unsolvable tasks. This was evident in both cohorts of PSMTs in the three different types of posing situations: generation, reformulation, and extension (Silver, 1994).

It is worrying to see that there appears to be a deficit in the understanding of the characteristics of a mathematical problem and content knowledge based on the tasks posed by the PSMTs. Given the context of this study, in an Irish setting, where procedural tasks are prevalent in textbooks, it is vital for teachers to understand the nature of a mathematical problem in order to select and design them (Chapman, 2015). Understanding the characteristics of different types of tasks is an important skill for teachers to elicit different mathematical skills and thinking in their students (Yeo, 2007). Although, it is recognised that it is difficult to anticipate the mathematical ideas involved in the problems (NCTM, 2000). One such difficulty includes confusion around the inclusion of words in mathematical tasks and the assumption that this automatically indicates a “word problem”. The “word problems” traditionally used in schools ask students to practise learnt procedures or methods rather than genuine problem-solving (Moschkovich, 2002) which is more appropriate for consolidation of learning (Orton and Frobisher, 2004). The PSMTs' own school experience, including experiences of textbook tasks, may influence their ability to unpick the characteristics of mathematical problems and thereby their ability to pose problems.

From the analysis of the tasks posed by the PSMTs which were not identified as mathematical problems, it was clear that mathematical common content knowledge appeared to affect the tasks posed. This was apparent through the large number of tasks which were deemed to be unsolvable. Subject matter knowledge is an element of pedagogical content knowledge (Ball et al., 2001); however, strong content knowledge does not necessarily imply effective teaching of mathematics (Baumert et al., 2010). It is important to note that PCK is developed through teaching experience and the PSMTs in this study have limited experience of classroom teaching. However, the lack of common content knowledge exhibited in the posed tasks, indicates that the PSMTs would be challenged to perceive the implications of the different approaches that students may employ when solving a problem and even understand the validity of the ideas expressed by the students.

A possible reason behind the PSMTs producing tasks that were unsolvable is their ability to write mathematically. Mathematical writing is defined as writing “about mathematics concepts or procedures” (Powell et al., 2021, p. 418). The process of posing mathematical problems falls under this definition due to problem posing requiring the PSMT to consider mathematical concepts in their writing. It is recognised that much research of pre-service mathematics teachers has focused on mathematical content knowledge with much less on PSMTs' mathematical writing skills. Smith et al., (2025) highlights that PSMTs may enter initial teacher education programs with limited experiences of mathematical writing from their own school experiences. This may be an influential factor in the PSMTs being able to structure a coherent mathematical problem. Whilst the module included an emphasis on mathematical language, it is important to notice that this module was a total of 12 weeks in duration where PSMTs may need longer to overcome habits formed from previous years of experience in mathematics education.

It is worrying to see that there appears to be a deficit in the understanding of the characteristics of a mathematical problem and subject matter knowledge based on the tasks posed by the PSMTs. These are future teachers of mathematics and will have the responsibility of controlling the learning experience of secondary school students in mathematics and building their problem-solving skills. Given that, to the best of our knowledge, that there has been no other research conducted within an Irish context and the prominence of problem-solving in curricula, further research needs to be conducted in developing the capacity of problem posing in mathematics teachers. Such research could focus further exploration of the connection between mathematical content knowledge, mathematical writing, and posing mathematical problems.

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Footnotes

Edited by: Natividad Adamuz-Povedano, University of Cordoba, Spain

Reviewed by: John Sevier, Appalachian State University, United States

Mustafa Zeki Aydoğdu, Trakya University, Türkiye

Footnotes

¹Secondary school education in Ireland spans over 5 or 6 years (1^st year to 6^th year) with an optional fourth year. Mathematics is compulsory for all students to study over the course of their secondary education. The syllabus for mathematics is provided and assessed at three levels; higher, ordinary and foundation level.

²An ‘unhelpful ambiguity’ is uncertainty in the goal of the question due to missing information which differs from an open problem where different interpretations and multiple solutions are possible.