Abstract
In this paper, we examine how problem posing is presented in the Singapore mathematics curriculum across four levels: planned, intended, enacted, and attained curriculum. Our review shows that problem posing receives limited attention in existing policy documents as well as from researchers in Singapore as reflected by the limited number of research publications on problem posing. Although mathematics instruction in the classroom does not explicitly highlight the practice of problem posing, some teachers are already using problem posing under other paradigms, such as problem-solving, without making problem posing explicit in their lesson enactment. Problem posing is still not a widespread practice in Singapore. We further argue that problem posing could facilitate the development of other goals, such as mathematical investigation, modeling, and computational thinking, and advocate for a more explicit emphasis on problem posing in the curriculum and the systematic study of its implementation to ensure its sustainability and widespread adoption as a pedagogical practice.
1. Introduction
In a recent review of international studies on problem posing and problem-solving, Silver (2023) noted that citations for problem posing, based on Google Scholar citation index data, amount to only about 10% of those for problem-solving. Silver further noted that studies published after 2000 show nearly equal citation numbers for each topic. At ICME14, the Topical Study Group on Problem Posing and Problem-solving featured a comparable number of presentations for both topics. Similarly, the book Problem Posing and Problem-solving in Mathematics Education: International Research and Practice Trends (Toh et al., 2023), which compiles papers developed from the selected presentations in the Topical Study Group, includes nearly an equal number of chapters dedicated to problem posing and problem-solving. These observations may suggest a growing global interest in problem posing among mathematics education researchers, although it begins much later than problem-solving.
Silver (1994) defined problem posing as “both the generation of new problems and the re-formulation of given problems” (p. 19). Mishra and Iyer (2013) further described problem posing as the process in which learners create new problems based on a given situation and noted that it can take place at various points of problem-solving: during the “Planning” phase or the “Looking back” phase described in Polya’s (1945) problem-solving model or through drawing on prior experiences to create a problem. Based on the definitions of problem posing by Silver and Mishra and Iyer, in this paper, we broadly recognize any formulation of a new problem or creating a new problem based on a given situation, as problem posing. Based on this understanding of problem posing, we explore how problem posing is presented in the Singapore mathematics curriculum.
Specifically, in this paper, we explore how problem posing is presented in the K-12 mathematics curriculum of Singapore based on the four levels of curriculum proposed by Cai and Hwang (2021): (1) the planned curriculum, (2) the intended curriculum, (3) the enacted curriculum, and (4) the attained curriculum. Specifically, this paper aims to answer the following research question: How is problem posing presented at each of the four levels of the mathematics curriculum?
2. Singapore mathematics curriculum framework
Problem-solving has been a cornerstone of the Singapore mathematics curriculum since the introduction of the pentagon framework in the early 1990s. Represented symbolically as a pentagon, the framework outlines five essential attributes that students must develop for effective problem-solving: Concepts, Skills, Processes, Attitude, and Metacognition. Each attribute is annotated by specific descriptors that elucidate its characteristics. Although the mathematics curriculum has undergone regular revisions, the pentagon framework has remained largely intact, with only the descriptors of each attribute being refined over each revision. The latest version of the framework, displayed in Figure 1, underscores the enduring focus on mathematical problem-solving in the curriculum.
The most recent Singapore mathematics curriculum framework (Ministry of Education, 2023, 2024).
The Singapore mathematics syllabus document (e.g., Ministry of Education, 2006) describes problem-solving as the “acquisition and application of mathematics concepts and skills in a wide range of situations, including non-routine, open-ended, and real-world problems” (p. 3). Readers might gather from this description of problem-solving that, within the Singapore mathematics curriculum, the term “problem” is understood as any mathematics task. With this as the definition of a problem, the syllabus uses terms such as “routine problems,” “non-routine problems,” “open-ended problems,” and “real-world problems” to categorize various types of problems that students will encounter in their career as a student.
3. Methodology
Given the close relationship across research, educational policy, and practice (Cairney & Oliver, 2020; Tseng, 2012), we examine how problem posing is presented in research, policy documents, school textbooks, and authentic lesson enactment. Although it is not possible to observe actual classroom lessons directly for the present study, local publications on problem posing can provide an avenue to study authentic classroom practice in Singapore. A summary of the material for each level of the curriculum that we studied is shown in Table 1.
Material examined for each level of the curriculum.
We reviewed problem posing in the enacted and the attained curriculum through a survey of existing publications on problem-posing studies in the Singapore context (Tables 2, 3, and 4) using the ERIC database and Google Scholar to identify the existing literature. Words and phrases that could be related to problem posing (including “problem posing,” “pose problem,” “create question,” “create problem,” “pose questions,” and other variants of such terms, together with the word “Singapore”) were used as keywords to search for existing publications on problem posing. In addition, we searched the library resource database of the Singapore National Institute of Education to identify key studies on problem posing, including postgraduate studies which consist of thesis and dissertations submitted by students for their graduation. The publications on problem posing were included in our survey if they met the following three criteria:
The context was set in Singapore; The studies reported in the publications were conducted in Singapore classroom settings (although the researchers could be located either overseas or in Singapore); and The publications were journal articles, book chapters, conference papers, or theses or dissertations.
Types of publications versus grade levels.
Mathematical topics of problem posing versus grade levels.
Types of problem-posing studies observed in the publications.
aCitations that are marked with an asterisk are empirical studies, whereas the rest are conceptual/discussion papers.
Based on Criterion 2 above, two publications of our search in which the researchers were based in Singapore (at the time their study was carried out) but their study was neither based on Singapore classrooms nor on Singapore students (Jiang & Chua, 2018; Kapur, 2015) were not included in our survey. As the objective of our survey was to gain insight into the enactment of problem posing in the Singapore classroom and students’ problem-posing attainment, we included book chapters and conference papers in addition to peer-reviewed journal articles. Based on the above selection criteria, we found a total of 21 publications on problem posing in Singapore. We classified all these publications according to the types of publications, the mathematical topics of the studies, and the grade levels of the studies, and we present a breakdown of the studies in Tables 2, 3, and 4 below. Table 2 shows the types of publications versus grade levels described in the publications. Table 3 shows the mathematical topics used in problem posing versus the grade levels of the studies. Table 4 provides a brief description of each of the types of studies on problem posing and includes the references for the publications.
Our search shows that, compared with studies on problem-solving in Singapore, there are relatively fewer studies on problem posing. The number of existing problem-solving studies conducted up to 2019 in Singapore, as reviewed by Toh et al. (2019), was 64 (Fig. 7.1, p. 145), and this figure increases to 71 if publications up to 2024 are included. In comparison, the number of published studies on problem posing up to 2024 was only 21 (Table 2).
The sheer length of history could be the main reason for the relatively lower number of studies on problem posing compared to problem-solving. Historically, literature on problem-solving appears much earlier than problem posing. Problem-solving can be traced to as early as the first edition of Polya's seminal book How to Solve It, which was published in 1945. Meanwhile, problem posing, which appears several decades after problem-solving, could be traced back to the seminal work of Brown and Walter (1985) and Silver (1994).
4. How problem posing is presented in the Singapore mathematics curriculum
In this section, we attempt to answer our research question about how problem posing is presented in the Singapore mathematics curriculum at the four levels of curriculum.
4.1 Planned curriculum
At the level of the planned curriculum, the term “problem posing” is not found in the policy documents (Ministry of Education, 2000, 2004, 2007a, 2007b, 2012, 2023, 2024), which include teaching guides and syllabuses published by the Singapore Ministry of Education. Similarly, related terms such as “pose problem,” “create problem,” “generate problem,” “generalize problem,” and “formulate problem” or their variants—terminologies which align with problem posing—are also absent except for one reference in the Ministry of Education (2000).
Except for the single instance mentioned above, little is mentioned about problem posing or opportunities to engage learners to create problems in the Singapore mathematics curriculum documents. Not long after the publication of Silver's seminal (1994) work, problem posing was described as one of the key skills in the Thinking Programme in Mathematics (Ministry of Education, 1997; as cited by Yeap & Kaur, 1997). The mathematics syllabus at that time advocated that students should be engaged in activities to “extend and generate problems” (p. 17, Ministry of Education, 2000, as cited in Ho et al. (2000)). However, this single reference to providing opportunities for students to create, extend, or generate problems has not been continued in subsequent revisions of the Singapore mathematics curriculum or in local textbooks. Notably, the latest curriculum documents (Ministry of Education, 2023, 2024) make no mention of problem posing or activities to engage students to formulate their own problems.
This lack of emphasis on problem posing in the planned curriculum of Singapore stands in stark contrast to practices in countries like the United States and China. In the United States, the curriculum standards explicitly encourage students to “have some experience recognizing and formulating their own problems” (e.g., National Council of Teachers of Mathematics, 1989, p. 189). Similarly, in China, the curriculum highlights the goal of transitioning students from passive to active learning by engaging them in both problem posing and problem-solving (e.g., Chinese Ministry of Education, 2011; cited by Cai & Hwang, 2021).
4.2 Intended curriculum
At the level of the intended curriculum, we examined instructional materials consisting of (1) school textbooks and (2) mathematics teachers’ resource books, both of which reflect the writers’ interpretations of the planned curriculum. We reviewed the sole series of primary mathematics textbooks (Chan, 2025a, 2025b; Curriculum Planning and Development Division [CPDD], 2021a, 2021b, 2022a, 2022b, 2023a, 2023b, 2024a, 2024b) and two of the three series of secondary mathematics textbooks (Chow et al., 2020a, 2020b, 2020c, 2021a, 2021b, 2022a, 2022b; Yan et al., 2020a, 2020b; Yeap et al., 2020a, 2020b, 2021a, 2021b, 2022a, 2022b) to explore how problem posing was presented. These textbooks have been officially endorsed by the Singapore Ministry of Education. We reviewed the mathematics teachers’ resource books (Lee, 2008; Lee & Lee, 2009) produced by the Singapore National Institute of Education, the sole teacher education institute in Singapore. These resource books are used in initial teacher training programs in the Institute. These books are accessible to all preservice teachers and practicing schoolteachers.
4.2.1 School mathematics textbooks
We searched for instances of problem-posing activities by reviewing all the visible activities and using keywords such as “pose problem,” “create problem,” “generate problem,” “generalize problem,” and “formulate problem” and variants of these terms that could possibly describe problem-posing activities. Notably, language related to problem posing was absent from the textbooks.
We further examined the content in greater depth of the primary mathematics textbooks currently in use for Grades 1 to 5 (Grade 6 textbook was unavailable at the time of this study). The activities that were closest to problem posing were only found in Grade 1 textbooks (CPDD, 2021a, 2021b): There were four activities where students were prompted to “make as many stories as you can” for the four basic operations, addition, subtraction, multiplication, and division.
We also found that the textbooks for Grades 2 to 5 focused solely on solving problems, especially problems in real-world contexts, with no exemplar activities explicitly engaging students in activities of the problem posing-type or activities similar to those the Grade 1 textbook that engage students to make up their own stories about a mathematical concept. Interestingly, the four problem-posing activities in the Grade 1 textbook were not categorized under regular features within this textbook series. The prompt to “make as many … stories as you can” was not followed up in the higher grades.
For the secondary level, we examined two series of the mathematics textbooks currently used for Grades 7 to 9 (Grade 10 textbooks were not available at the time of this study). We identified only two instances of activities resembling problem posing at the Grade 7 level (Yeap et al., 2020a). These two activities appeared in the “Challenge Yourself” feature at the end of the chapters. In one task (p. 176), the students were asked to “write a story” describing a given Cartesian graph. In another task (p. 202), the students were instructed to create their own number sequence and investigate how the sequence behaves. Because these tasks are presented in the “Challenge Yourself” section, it is likely that these tasks would be perceived as optional and hence overlooked by most students.
The rare occasional inclusion of problem-posing activities possibly suggests that problem posing was not prioritized or even recognized as a core component of mathematics instruction. We believe that the sporadic and infrequent presence of such activities is unlikely to cultivate students’ disposition toward problem posing. Furthermore, these tasks appear to target higher achieving or motivated individuals rather than the broader spectrum of the student population, which is likely to render problem posing an exclusive rather than inclusive skill.
4.2.2 Mathematics teachers’ resource books
We examined the mathematics activities presented in two volumes of mathematics teachers’ resource books: (1) A Resource Book for Primary Mathematics Teachers (Lee & Lee, 2009) and (2) A Resource Book for Secondary Mathematics Teachers (Lee, 2008). The two books have been used as instructional materials in the initial teacher education program at the Singapore National Institute of Education. Each book comprises chapters discussing theoretical aspects of mathematics education followed by chapters focusing on pedagogical strategies for teaching specific strands of mathematics in the Singapore mathematics syllabuses. These latter chapters further propose various teaching and learning activities based on sound pedagogical principles.
Our review of the resource books shows that there was hardly any mention of the term “problem posing” or descriptions of problem posing-type activities. The primary-level resource book (Lee & Lee, 2009) contains no activities that engage students in problem-posing-related activities. In the secondary-level resource book (Lee, 2008), we found only one activity related to problem posing located in the chapter on Teaching Algebra (Yeap, 2008). In this activity, students are asked to “write ten different equations which have a solution as x = 3” (Yeap, 2008, p. 46). However, the definition of problem posing or the pedagogical principles or rationale underlying problem posing is not mentioned in the resource book. Beyond this isolated example, no other activities resembling problem posing or explicit references to problem posing were identified. In comparison, problem-solving is consistently addressed throughout most of the chapters of both resource books.
4.3 Enacted curriculum
4.3.1 Discussion of enacting problem-posing in the classroom
The publications reporting problem-posing studies conducted in Singapore have provided exemplary activities that infuse problem posing into classroom mathematics instruction. For Category 1 in Table 4, four out of the five papers showcase specific problem-posing tasks, which are deemed suitable for classroom use (Yeap, 2000, 2009; Yeap & Kaur, 1997; Zhao & Lee, 1999). In addition to presenting these tasks for practicing teachers, the authors discuss key principles underlying their design. The three papers by Yeap and Kaur (1997) and Yeap (2000, 2009) offer a progressive exploration of problem-posing tasks for classroom instruction. In contrast, Zhao and Lee (1999) is the only paper discussing problem posing at the tertiary level.
Yeap and Kaur (1997) argued that problem posing is both a feature of mathematical thinking and creativity and that it should be both the goal and means of mathematical instruction by citing the works of Silver (1994) and Kilpatrick (1987). In supporting their argument, Yeap and Kaur presented a collection of five problem-posing tasks adapted from activities proposed by prominent researchers (e.g., Brown & Walter, 1985; Ellerton, 1986; Green & McCaan, 1991; Silver & Cai, 1993) and identified the core thinking skills using the framework of the Dimensions of Thinking and Learning (Marzano, 1988, 1992). Yeap (2000) further built on this work by categorizing problem-posing skills in three ways: (1) whether the problem posing occurs before, during, or after problem-solving; (2) the way the numerical information is presented; and (3) how open the problem-posing tasks are. This discussion was expanded in his third paper (Yeap, 2009) where he proposed different problem-posing processes to engage primary school students: (a) posing primitive, (b) posing related problems, (c) constructing meaning, (d) engaging in metacognition, and (e) connecting to one's experience. He further demonstrated how problem posing could be used to achieve various goals such as concept development, drill-and-practice, problem-solving, assessing understanding, and provision of differentiated instruction.
Parallel to the discussion of primary and secondary school mathematics, Zhao and Lee (1999) emphasized the importance of problem posing in undergraduate mathematics education. In addition to finding a quality mathematics problem, Zhao and Lee suggested that students should be educated to explore and create their own quality problems so as to promote active learning in undergraduate mathematics courses. They detailed their efforts through exemplars to develop students’ problem-posing techniques in undergraduate algebra courses through four stages: (1) preparation, (2) posing problems, (3) presentation, and (4) group discussion. Notably, this paper by Zhao and Lee is the only available study addressing problem posing at the undergraduate level. A common thread across the three works in this category is that they reflect the authors’ personal enactment of problem posing supported by their review of existing literature on the subject and their collective classroom experience as mathematics instructors.
In a related paper on supporting the enactment of problem posing, Yeap and Kaur (1999) proposed a taxonomy for classifying problem-posing tasks based on cognitive processes. They further introduced a framework for coding students’ responses to problem-posing tasks involving word problems. This framework was informed by their review of existing approaches for analyzing students’ problem-posing responses (e.g., Ellerton, 1986; English, 1998; Leung, 1993; Silver & Cai, 1993).
4.3.2 Problem posing to assess students’ understanding of mathematical concepts and processes
In addition to presenting teaching ideas on problem posing, three reports of empirical studies in Category 2 of Table 4 (Cheng, 2013; Kwek & Lye, 2008; Yeo, 2012) use problem posing to achieve the objective of assessing students’ understanding of mathematical concepts. The main aim of enacting problem posing described in the three papers under this category was not so much to develop problem-posing dispositions of students but to provide a means for teachers to assess their students’ understanding of various mathematical concepts (Table 5 presents a summary of the papers).
A summary of the empirical studies conducted on problem posing.
4.3.3 Problem posing to achieve other pedagogical goals
In addition to the above two categories of problem posing, Category 6 in Table 4 consists of two reports of empirical studies that present the use of problem posing to achieve other educational objectives. Both Huang et al. (2016) and Lee (2013) asserted that problem posing could be a useful tool for supporting other pedagogical practices. In particular, problem posing, which has been generally perceived to be key in laying the foundation for mathematization dispositions, is crucial for mathematical modeling (Bonotto, 2010; Niss, 2010 as cited by Lee (2013)).
4.4 Attained curriculum
Based on the publications that we reviewed on students’ problem-posing attainment, it was reported that students were generally able to perform problem posing when required. The complexity and difficulty of the problems they posed increased with the students’ grade levels, and the studies include examining the problems posed by the students and their regulation of cognition during problem posing.
Quek (2000, 2002) argued with a philosophical slant for the importance of the context associated with problem posing in addition to focusing on the nature of the mathematical and cognitive elements involved. To assess students’ problem-posing attainment, the context must be made clear to students. Based on his study of preservice mathematics teachers in Singapore, Quek stressed the importance for researchers to focus on how students interpret problem-posing activities and on their beliefs about mathematics. In other words, how individuals respond to a problem-posing task is dependent not only on their ability but also on their understanding of and beliefs surrounding the situation of the problem-posing task presented to them. He illustrated with his experience that when a professional mathematician was presented with a problem-posing task, the latter gave a trivial mathematics problem, which was clearly not reflective of the mathematician's ability. This was because the context of solving the task was not clearly understood by the mathematician.
4.4.1 Problem posing at the primary level
In the study by Cai (2003) (Category 3 of Table 4), students from Grades 4, 5, and 6 were generally able to pose problems, with the level of difficulty and complexity increasing with grade level. Most of the students in his study were able to pose problems beyond the confines of a given problem stem on number patterns. The students were able to pose more extension problems with increasing grade levels; however, this increase was only statistically significant from Grade 4 to Grade 5 but not from Grade 5 to Grade 6.
4.4.2 Problem posing and other variables
Three (Ho et al., 2000; Yeap, 2002; Yeap & Kaur, 2000) out of the five papers in Category 3 of Table 4 reported empirical studies to explore the correlation of problem-posing ability with other variables: students’ grade level, mathematics achievement scores, and problem-solving ability. Students at higher grade levels were able to pose more complex problems than those at lower grade levels. However, there was no significant correlation between students’ problem-posing ability and their problem-solving ability, between students’ achievement scores and their problem-posing ability, or between their achievement scores and the complexity of the problems posed. These studies point to a similar conclusion that problem posing likely comprises a set of skills that is different from other areas such as students’ mathematics achievement and problem-solving ability. Moreover, problem posing is a skill not exclusively reserved for higher achieving students; students’ ability to pose problems is independent of their mathematics achievement.
4.4.3 Problems posed by students
Three (Chua, 2011; Chua & Toh, 2022; Chua & Wong, 2012) out of the five papers in Category 4 of Table 4 examined problems posed by students at the secondary level. It can be gathered from the above empirical studies that, generally, problem posing might not be a set of skills that Singapore students are familiar with at the secondary level. Chua and Wong (2012) attributed this lack of problem-posing skills to the lack of emphasis of problem-posing skills taught in the school mathematics curriculum.
4.4.4 Students’ regulation of their cognition in problem posing
Parallel to Schoenfeld’s (1985a, 1985b) examination of regulation of cognition during problem-solving, Chua (2019) identified four phases of regulation of cognition in problem posing: property noticing, problem construction, checking solutions, and looking back, which are parallel to the four phases in Polya's (1945) problem-solving framework. Chua (2023) further expanded on examining the four phases of regulation of cognition in problem posing on a specific case study and discussed these descriptors in problem-posing instructions. A summary of the empirical studies on problem posing described in Table 4 is provided in Table 5 below.
5. An in-depth look at problem-posing activities in the enacted and attained mathematics curriculum
In examining the nature of the publications on problem posing in Singapore (Table 2), 9 (43%) out of the 21 papers were published in conference proceedings, with three of these conference papers ultimately developed into the PhD theses of the respective authors. We could not gather any evidence that the other conference papers were later developed into more comprehensive research published in journal articles. Only 5 (23.8%) out of the 21 were reviewed journal articles. We also note that several of the conference papers were published within the same conference proceedings. We conjecture that these authors were responding to a particular theme in a conference.
The above observation could be an indication that most of the studies carried out by the researchers were exploratory in nature, lacking subsequent in-depth follow-up publications. It is highly probable that such studies on problem posing did not arise out of demand from the education community in Singapore. The limited number of studies on problem posing could be attributed to the near complete absence of any mention of problem posing in the official mathematics curriculum.
Studies on problem posing reported in the Singapore education literature continued to be sporadic and lacking in focus, quite a contrast to problem-solving research in Singapore. In the classification of problem-solving publications conducted by Toh et al. (2019), which we discussed earlier in this paper, research on problem-solving from 2012 onward emphasized the holistic approach of teaching problem-solving. This includes the introduction of mathematical modeling and an increased focus on mathematics teacher education. Various fully developed journal articles were published on the later development of problem posing.
However, we would like to caution readers that one should not come to the conclusion that problem posing is not enacted in the Singapore mathematics classroom. Most mathematics educators would agree with Polya that problem posing can be seen as an extension of problem-solving, which is the core of the Singapore mathematics curriculum. We argue that, in fact, in the Singapore context, problem posing might be embedded in mathematics classroom instruction without the use of a visible label of “problem posing” attached to the activities. This is especially possible if we view problem posing as a natural progression of problem-solving. Thus, we next turn to the enactment of problem-solving in Singapore.
5.1 Embedding problem posing within a problem-solving paradigm
Among the numerous studies to enact problem-solving in the Singapore mathematics classrooms, Toh et al. (2008) and Leong et al. (2013) proposed the enactment of problem-solving through their conceptualized problem-solving module based on a new science practical paradigm. The theoretical framework underpinning their problem-solving module was a modified version of Polya's problem-solving model (Figure 2).
Polya's model modified in Toh et al. (2008).
In the mathematical problem-solving module proposed by Toh et al. (2008) and Leong et al. (2013), in addition to positing that the four stages in Polya’ model are cyclical rather than sequential, Polya's original fourth stage “Look Back” was replaced by “Check and Extend,” which more accurately reflects the original spirit of Polya (1945). This modified Check and Extend stage stresses that, in addition to merely checking the reasonableness or correctness of the answers to the original problem, formulating a new problem based on understanding the features and solution of the original problem is equally important. In other words, in this modified approach of enacting problem-solving which includes Polya's full (modified) model, problem-solving does not end with a closed goal and closed answer (Evans, 1987) but with an open goal for students to formulate their new problem based on their experience of solving the original problem. This emphasis on Polya's modified fourth stage (Leong et al., 2013; Toh et al., 2008) was introduced to tackle the issue that most students generally fall short of checking their solution and extending the problem after solving a given problem.
5.1.1 Unpacking problem formulation at the Check and Extend stage
In this new paradigm of problem-solving, problem posing begins with Polya's fourth (modified) stage, Check and Extend. Toh et al. (2011b) elaborated three approaches to the Check and Extend stage in creating a new problem. Because this stage follows the solving of a given problem, formulating a new question here is based on modifying the original given problem. In other words, it is not a free problem-posing situation. Students in their problem-solving classes were taught three types of new problem formulation: (1) adaptation, (2) extension, and (3) generalization. Adaptation involves modifying some features of the existing problem (e.g., changing numbers or conditions or proving the converse). Extension involves creating problems that have a higher level of difficulty or a wider scope than the existing questions. Generalization involves creating a new problem that includes the original given problem as a special case.
5.1.2 Assessing students’ problem-solving and posing
In addition to conceptualizing and enacting the problem-solving module which concludes with the Check and Extend stage, the authors further conceptualized a new approach for assessing students’ problem-solving processes. Under this new assessment approach, the authors asserted that all four of Polya's stages (including the Check and Extend stage shown in Figure 2) must be assessed and given credit in addition to credits given solely for correctness of answers and solutions. In other words, both the processes and products of problem-solving are to be assessed (Toh et al., 2011a, 2011b). Note that such an assessment scheme, which assesses both the processes and products of problem-solving, is generally not a common practice in Singapore mathematics classrooms. Here, “creating new problems,” the process of generating a new problem by learners based on the given situation (Mishra & Iyer, 2013), forms an indispensable part of the assessment of students’ performance in the proposed problem-solving course. A criterion-based assessment rubric was used to assess the open-ended processes rather than a typical marking scheme.
An interview with the students who participated in the problem-solving course was reported in Toh et al. (2011a). The students were generally positive about the problem-solving module and the assessment method, although the low-progress students were concerned that this new assessment approach (which includes assessing the process) could adversely affect their final performance grades. On the other hand, the higher achieving students were intrigued by the affordance of the module to enable them to “work like mathematicians” in formulating a new problem based on a given problem. The students’ feedback suggested that formulating a new problem had empowered them to learn more mathematics than the usual method of solving the problems.
5.2 Embedding problem posing in the curriculum for low-progress learners
Addressing the learning difficulties of low-progress learners has been a priority for teachers, educators, and policy makers in Singapore (e.g., Kaur & Ghani, 2012). In addition to deploying the usual pedagogical approaches to address their learning needs, teachers of low-progress learners have been attempting unconventional innovative approaches to reach their students (Toh & Lui, 2014). In this subsection, we show that problem-posing activities were also embedded in such unconventional innovative approaches to teaching mathematics through the use of comics.
5.2.1 Unconventional approaches for mathematics instruction
Based on existing educational literature discussing the positive impact of comics on educational goals (e.g., Cho, 2012; Tilley, 2008), an approach of using comics for teaching mathematics for low-progress learners was conceptualized (Toh et al., 2016). An alternative teaching package for selected mathematics topics from Grades 7 and 8 was developed by the research team, taking into account the pedagogical principles focusing on low-progress learners (see Toh et al., 2016, for a description of the comics instructional package). Being an alternative teaching package (that is, alternative to the usual textbook or conventional instructional material), the comics package was endowed with all essential expositions, practice questions, and activities required for the teaching and learning of the topics.
5.2.2 Problem-posing activities in the comics instructional package and their enactment
The development of the comics instructional package gave the research team the opportunity to include activities that are not typically found in existing textbooks but were deemed to be pedagogically sound by the research team and the participating teachers. Although the package was designed with low-progress learners in mind, it features a wide range of questions with varying levels of difficulty. Notably, activities that allowed students to create their own questions, in addition to answering the provided questions, were also integrated into the comics teaching package, further distinguishing the content from the existing textbooks (see Figures 3, 4, and 5 for three samples of such activities).
Sample of a problem-posing task integrated with data collection and tabulation in a statistics lesson.
Sample of an activity of data interpretation with problem posing.
Sample of a problem-posing task in the form of telling a story based on a given graph plot.
Problem-posing-type activities (as shown in Figures 3, 4, and 5) were incorporated in the comics instructional material. During the lesson enactment, the teachers not only used the problem-posing tasks as part of the classwork, they further creatively adapted the problem-posing tasks to engage students in role play to encourage them to “tell a story” (Toh et al., 2018). In particular, the graph in Figure 5 became an opportunity for students to tell a story after giving an interpretation of the meaning of the axes and the numerical values. The study reported in Toh et al. (2018) highlighted how getting students to create their own questions was challenging for most low-progress learners, as they were not accustomed to formulating their own problems. To support these learners, teachers provided significant scaffolding throughout the problem-posing process during the lesson enactment. Nevertheless, such activities were found to be useful for the students, as they addressed both their cognitive and affective needs.
6. The importance of problem posing in the curriculum
Problem posing, in contrast to problem-solving which is the heart of the Singapore mathematics curriculum, is rarely mentioned in the curriculum documents. It is thus not surprising that problem posing is not much emphasized in the local printed textbooks, which include only minimal problem-posing activities as discussed above. Without official inclusion of problem posing in the official syllabus documents, it is likely that problem posing will only attract sporadic attention from the local education community.
From the collection of policy documents and publications on problem posing that we surveyed in this study, it is apparent that educators are generally convinced of the benefits of teaching problem posing in the mathematics classroom for primary, secondary, and even undergraduate levels, although problem posing might not have caught the attention of policy makers. There is also evidence that problem posing has been enacted in mathematics classrooms, though not in a widespread manner nor conducted under the label of problem posing. We also note from our survey of existing literature that the topics that were selected for problem-posing activities were limited: Most studies on problem posing focused primarily on arithmetic and word problems, with only one each in school geometry and algebra at the secondary level. Despite the lack of widespread problem-posing practice, we strongly believe that problem posing could be useful for other processes that are stressed in the Singapore mathematics curriculum.
6.1 Problem posing and other aspects of the Singapore mathematics curriculum
Among the many features of the Singapore mathematics curriculum, we discuss three of these features that illustrate the importance of developing mathematical problem-posing dispositions among students: (1) mathematical investigation, (2) mathematical modeling, and (3) computational thinking.
6.1.1 Mathematical investigation
Mathematical investigation is a common practice used in Singapore mathematics classrooms to develop students’ mathematical thinking (Ministry of Education, 2023, 2024). Unlike problem-solving which usually has a clear goal, mathematical investigation usually has ill-defined goals (Frobisher, 1994). As an illustration, an open-ended investigative task is as follows: “A rectangle has an area of 36 cm2. Investigate its perimeter” (Toh, 2010). In such an open investigative task, students are expected to pose their own problems or to generate patterns based on a given stem (Toh, 2010; Yeo, 2012). Thus, the richness of the investigative activity is highly dependent on the students’ problem-posing ability followed by their problem-solving ability in solving the problem(s) they pose.
The problem-posing activity that accompanies an open investigative task is important because, according to Bastow et al. (1986), “the subsequent solution of problems generated by the exploration highlights such mathematical processes as organizing, … justifying and explaining. More generally, it provides opportunities for the development of independent mathematical thinking in learners” (p. 1). Students with good problem-posing dispositions will thus be ready to handle mathematical investigative tasks well.
6.1.2 Mathematical modeling
Mathematical modeling (currently labelled as modeling and application in Singapore mathematics curriculum documents under the process arm of the pentagon framework in Figure 1) has been recognized as an important process of the Singapore mathematical curriculum framework since 2007. The importance of problem posing in mathematical modeling has been recognized by researchers (e.g., Bonotto, 2010; Cai & Hwang, 2021). In performing mathematical modeling, students should first be required to pose their own problems and, in this way, they transit from the real world to a problem statement which they can mathematize (Stillman, 2015). In fact, this is reflected in the four-phase cycle of mathematical modeling proposed by the Ministry of Education (2012), the first phase of which (“formulating”) involves understanding the real-world problem situation followed by selecting key variables, forming relationships between variables, and making sound assumptions about the real-world context. This again shows that it is important for students engaging in mathematical modeling to have a good problem-posing disposition.
6.1.3 Computational thinking
The most recent feature included in the Singapore mathematics curriculum is the introduction of computational thinking into the curriculum. Computational thinking has been described as the thought process involved in formulating problems and developing approaches to solving them in a manner that can be implemented with a computer (Wing, 2006). Computational thinking requires an individual to understand the structure of the problem. To be able to understand the structure of a problem, modified problem-posing (that is, creating a new problem by transforming an existing problem) is important (Fukui et al., 2024; Fukui & Sasaki, 2022). The four key processes of computational thinking are decomposition, generalization, abstraction, and algorithmic thinking. Researchers such as Fukui and Sasaki (2022) have demonstrated how the key processes of computational thinking reinforce, and are in turn reinforced by, the elements of problem posing (Table 6). Thus, a good problem-posing disposition is important for students engaging in computational thinking.
The concepts of computational thinking and related activities that could be interpreted as problem-posing activities (Fukui et al., 2024).
7. Conclusion
Mathematical problem posing has not received much attention in Singapore as reflected by its almost total absence in official curriculum documents and the very low number of activities in printed textbooks endorsed by the Singapore Ministry of Education. Many explicit studies on problem posing in Singapore were carried out by academics as local exploratory studies rather than large-scale research tailored to the demands of the local mathematics education community. This is reflected by the very low number of peer-reviewed journal articles published on problem posing. However, there are some traces of the enactment of problem posing and students’ problem-posing attainments in the Singapore mathematics classroom, although the mathematics topics proposed for problem posing are limited.
We believe that this lack of emphasis in the official mathematics curriculum explains the scarcity of attention toward problem posing in Singapore and consequently, the lack of empirical “success stories” associated with problem-posing implementation. This further makes it a challenge for problem posing to become a widespread practice in Singapore mathematics classrooms beyond mere sporadic research studies. We believe that an official mention of the term problem posing, rather than a subtle embedding within the curriculum document, is essential to popularize problem posing.
Mathematicians such as Polya believe that problem posing is a natural progression of problem-solving; the latter is central to the Singapore mathematics curriculum. One message that we obtained from the articles we surveyed in this study is that problem posing contains a set of skills that might be different from those of problem-solving. More importantly, this skill set is not restricted to the elite few but applicable to the entire spectrum of students independent of their mathematics achievement. Even though few official studies on problem posing were found, some educators were found to be engaging students in problem-posing activities under the label of problem-solving and other innovative teaching approaches (without the official label of “problem posing”).
We strongly believe that engaging students in problem posing should be emphasized in the Singapore mathematics curriculum just as the importance of engaging students in problem-solving has always been stressed. With the current emphasis of The Singapore Teaching Practice (Academy of Singapore teachers, n.d.) for classrooms to engage in more subject disciplines, there has been an increased push in getting students to go beyond just problem-solving to develop their sense of inquiry in real-world contexts. This sense of inquiry could be facilitated by an individual's problem-posing disposition. In addition, the Singapore Mathematics Curriculum Framework (Ministry of Education, 2012, p. 14) clearly listed the critical roles of thinking skills and metacognition within the mathematical problem-solving framework. These skills can be engendered through problem-posing activities given that problem posing requires students to go beyond using necessary concepts to link them coherently together to formulate a problem (Chua, 2023).
With the Singapore mathematics curriculum's increasing emphasis on learning experience (Ministry of Education, 2023, 2024), we believe that problem posing could also be presented as a crucial learning experience or learning process for mathematics students to experience during their career as students. We agree with the curriculum development and design tradition that getting students to experience problem posing as part of their “learning experience” (e.g., Tyler, 1949) is as important as engaging students to experience problem-solving or acquiring mathematics concepts in their classroom instruction.
Another compelling reason for problem posing to be stressed within the mathematics curriculum documents is that problem posing could be instrumental for other newly introduced initiatives such as computational thinking, mathematical modeling, and mathematical investigation. These problem-solving-related initiatives could provide valuable contextualized opportunities to engage students in problem posing, whereas generic skills in problem posing could enhance students’ progress toward the new initiatives.
It is common knowledge among educators in Singapore (in fact, internationally) that assessment drives the mode of lesson delivery. Without fixing problem posing as a component for assessment, it could be challenging to get schoolteachers to engage their students substantially in problem posing. A similar challenge was faced in mathematical problem-solving in the Singapore mathematics classrooms more than a decade ago. Consequently, Toh et al. (2011b) devised a rubric for assessing students’ problem-solving in which both the products and processes of problem-solving are assessed. For problem posing to become a common practice in the mathematics classroom, we propose that a similar holistic assessment scheme be developed for problem posing. The development of such assessment schemes could begin by building on Yeap and Kaur’s (1999) work of coding students’ responses and Toh et al.’s (2011b) problem-solving assessment rubric.
There is little to no evidence, based on our survey of the existing literature on problem posing, to suggest the existence of collaboration between researchers and practitioners on problem posing in Singapore mathematics classrooms. In our review of local education literature, long-term researcher-practitioner collaborative efforts in enacting problem posing was not found. We believe that a long-term partnership is essential for a practice to become a mainstream practice in the curriculum. In other words, we share the view of other researchers such as Lemke and Sabelli (2008) that a productive partnership resulting in a sustainable innovative practice requires ongoing cultivation, commitment, and re-commitment, and this entire process takes much time.
For a practice to gain widespread adoption among classroom teachers, it is also crucial to translate theory into workable classroom strategies. However, there is a notable lack of local publications addressing the translation of research on problem posing into practical strategies in the classroom. Drawing on Schoenfeld’s (1985a; 1985b) emphasis on the importance of translating theories of problem-solving into classroom practice, we argue that similar efforts are also essential for integrating problem posing into authentic classroom practices.
Footnotes
Contributorship
The first author provided the first draft. Both authors conducted the review of the articles and revised the first draft. Both authors contributed to this article.
Declaration of conflicting interests
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The authors received no financial support for the research, authorship, and/or publication of this article.
Informed consent
This is a review article. No consent is required as no empirical studies were carried out.
