Integrating problem posing into school mathematics

Abstract

For problem posing to play a more central role in mathematics classrooms, teachers must have access to resources for problem-posing activities. The integration of problem posing into school mathematics can be evaluated through the lens of three levels of curriculum: the planned curriculum, the intended curriculum, and the implemented curriculum. This article discusses how different countries are trying to integrate problem posing into school mathematics at the three different levels. This article provides recommendations to integrate problem posing into school mathematics to present a low barrier to entry for teachers and students, which can then serve as a springboard to solve the problem of problem posing in school mathematics.

Although historically problem solving has been more central than problem posing in school mathematics and mathematics education research, over the past several decades curriculum reforms in many countries around the world have begun to raise the profile of problem posing at different educational levels. For example, in both China and the United States, curriculum standards have clearly spelled out the need for problem posing in school mathematics. This has been reflective in part of a growing recognition that problem-posing activities can promote students’ conceptual understanding, foster their ability to reason and communicate mathematically, and capture their interest and curiosity (Cai, 2025; Cai et al., 2015; Chinese Ministry of Education, 2022; National Council of Teachers of Mathematics [NCTM, 1989, 1991]; Ran et al., in press). It is thus natural to consider how problem posing can be integrated into mathematics curriculum and instruction.

For problem posing to play a more central role in mathematics classrooms, teachers must have access to resources for problem-posing activities. In particular, mathematics curriculum materials should feature a good representation of problem-posing activities. It is important to have problem-posing activities in the curriculum materials that teachers regularly use, as curriculum can be a powerful agent for instructional change (Ball & Cohen, 1996; Cai & Howson, 2013). Thus, the significance of including productive and robust problem-posing activities in curriculum standards and materials should not be overlooked. The purpose of this special issue is to understand how different countries integrate problem posing in their curriculum standards and/or curriculum materials as well as instruction.

The integration of problem posing into school mathematics can be evaluated through the lens of three levels of curriculum: the planned curriculum, the intended curriculum, and the implemented curriculum (Cai & Hwang, 2021; adapted from Cai & Howson, 2013; Lloyd et al., 2017). The planned curriculum ¹ refers to the inclusion of problem posing within policy documents and official curriculum standards. The intended curriculum concerns the representation of problem posing in mathematics textbooks and other instructional materials such as written teaching materials. The implemented curriculum describes how teachers adapt and deliver instruction to integrate problem posing in the classroom. Table 1 delineates which type(s) of curriculum are addressed by each paper in this special issue.

Table 1.

Types of curriculum addressed by each paper in the special issue.

Papers	Country	Planned	Intended	Implemented
Baumanns and Rott	Germany	√	√
Bokhove	England, USA and Singapore		√
Cai et al.	China	√
Muirhead et al.	USA		√	√
Pang and Lee	South Korea	√	√
Toh and Chua	Singapore	√	√	√
Vistro-Yu et al.	Philippines		√

1. Planned curriculum

Across the studies, the planned curriculum shows a wide range of attention to problem posing. The Chinese 2022 mathematics standards stand out by explicitly and extensively embedding problem posing as a key element, tying it to students’ cognitive development and their ability to connect mathematics with real-world situations (Cai et al., 2025). In contrast, Singapore's official curriculum documents almost entirely omit problem posing, showing near total silence on the topic (Toh & Chua, 2025). Similarly, German educational standards rarely mention problem posing, and, when they do, it is narrowly associated with mathematical modeling rather than broader instructional aims (Baumanns & Rott, 2025). Bokhove (2025) also found mathematical problem posing to be inconsistently represented across the national curricula of England, Singapore, and the United States: Whereas Singapore includes some references to student-initiated questioning, England and the United States were found to largely omit explicit mentions of mathematical problem posing. Thus, overall, the planned curriculum varies significantly by country, from highly explicit and central (China and United States) to limited mention (Singapore) to completely absent (England and Germany).

2. Intended curriculum

At the level of intended curriculum, the picture becomes more nuanced. In South Korea, revisions over time have increasingly emphasized problem posing, resulting in broader, more systematic curricular frameworks—though textbook tasks remain somewhat narrow in focus (Pang & Lee, 2025). The alignment between curriculum intentions and textbook content has improved over time, but problem-posing tasks still heavily focus on certain content areas and favor structured over open-ended tasks (Pang & Lee, 2025). In the Philippines, although the curriculum does not overtly mention problem posing, researchers identified what they call “curriculum opportunities” where problem posing could be implicitly integrated, suggesting an indirect but hopeful space for development (Vistro-Yu et al., 2025). In Singapore, some problem-posing activities were found in early primary grades and sporadically at the secondary level, but overall opportunities remain minimal (Toh & Chua, 2025). A comparative analysis across England, the United States, and Singapore revealed that language choices around problem posing differ, with England and the United States reflecting a stronger intended emphasis than Singapore (Bokhove, 2025). In general, although some countries signal a space for problem posing, the degree of intended support remains uneven and often weak. Finally, in Germany, textbook analysis revealed that problem posing is rare, concentrated in Numbers and Operations, and primarily structured and routine (Baumanns & Rott, 2025). The findings from this special issue suggest that there has not been much change in the past 10 years regarding the inclusion of problem posing in school textbooks. In fact, about 10 years ago, Cai and Jiang (2017) analyzed problem posing in Chinese and U.S. elementary school mathematics textbooks, finding that only a small proportion (about 3% or less) of the tasks in these textbooks included problem posing.

3. Implemented curriculum

The findings of papers studying the implemented curriculum show that problem posing is inconsistently realized at this level. In Singapore, evidence of students’ ability to pose problems has emerged mainly through isolated case studies rather than systemic practice (Toh & Chua, 2025). Case studies from the Problem-Posing Based Learning (P-PBL) Project demonstrate that with teacher initiative, meaningful integration of problem posing into middle school classrooms is possible, fostering greater student engagement and agency (Muirhead et al., 2025). Overall, although small-scale successes exist, widespread implementation of rich, diverse problem-posing practices across contexts is not yet a reality.

4. Taking action toward integrating problem posing into school mathematics

Because current curriculum materials do not incorporate significant and consistent experiences with problem posing for students, it is unreasonable to expect that problem posing will spontaneously be given much attention in classrooms. Although it may take much longer for textbooks to include a significant number of problem-posing tasks, we should not wait until they do so to implement problem posing in classrooms. Having said that, we should also not shift the burden to teachers given that they already face multiple demands on their time and attention, preventing them from devoting large amounts of time preparing for significant changes to their lessons. Ideally, problem posing would be introduced through small, incremental changes that would be accessible to teachers and students while offering the promise of rich returns in student learning (Cai & Hwang, 2021). Fortunately, researchers have been exploring ways to implement problem posing in classrooms. In this section, we discuss several actions taken to integrate problem posing into school mathematics.

4.1 Developing problem-posing tasks

As the papers in this special issue show, current mathematics textbooks in different countries only include a very small proportion of problem-posing tasks. There is thus a need to develop problem-posing tasks to be implemented in classrooms to realize the promise of problem posing for students’ learning and development. As indicated elsewhere (Cai, 2022; Cai & Hwang, 2023), a problem-posing task includes a situation and a prompt. Different situations can use the same prompt, and the same situation can use different prompts. Teachers can simply convert existing problem-solving tasks in current textbooks into problem-posing tasks. Cai (2022) suggested different ways to develop problem-posing tasks based on such existing problem-solving textbook tasks, such as by taking away the question in a problem-solving task and replacing it with a problem-posing prompt.

Consider the following problem-solving example. John's fifth-grade class did a survey of the number of stuffed animals each student has. Figure 1 shows the number of students with different numbers of stuffed animals. (1) How many students have six stuffed animals? (2) What is the total number of stuffed animals that students in John's class have? (3) What is the average number of stuffed animals students in John's class have?

Figure 1.

Number of stuffed animals.

For this problem-solving task, we can delete the three questions and instead ask students to “Pose three different mathematical problems that can be solved using the information in the graph.”

Readers may refer to Cai (2022) and Cai and Hwang (2023) for additional details about developing problem-posing tasks. In addition, Cai and Hwang (2021) made recommendations to better integrate problem posing into the school mathematics curriculum: (a) empowering teachers as curriculum redesigners to reshape existing curriculum materials in simple ways that create learning opportunities for mathematical problem posing, (b) enhancing existing curricula with additional problem-posing tasks that include support in the form of sample posed problems, and (c) encouraging students to pose problems with different levels of complexity. These three recommendations are practical and feasible based on evidence from teacher professional development focused on problem posing for teaching mathematics (Cai & Hwang, 2021).

4.2 Supporting teachers to learn to teach through problem posing

One effective method for implementing problem posing in the classroom is to develop teaching cases that other teachers can use to implement problem posing in their instruction. Some researchers have already tried to develop teaching cases as well as analyze existing teaching cases. For example, Zhang and Cai (2021) analyzed 22 teaching cases using problem posing to teach mathematics. Their analysis not only showed what teaching cases can look like but also the various components of teaching mathematics through problem posing. Meanwhile, Hwang et al. (2025) presented findings from a P-PBL project that supports teachers to implement problem posing in the classroom. The study was especially designed to address the fact that textbooks continue to include very few opportunities for problem posing. These researchers have been working to support teachers to integrate problem posing into classroom instruction, drawing on textbooks as a resource. They found that, for a given curriculum, problem posing catalyzes changes in teachers’ classroom instruction, but care must be taken to connect such changes to textbook lesson learning goals.

In addition, the field of mathematics education has been developing P-PBL instructional models to illustrate how problem posing can be used in the classroom, including the steps as well as the research involved in such P-PBL instructional processes (Cai, 2022). Figure 2 shows a P-PBL instructional model developed by Cai (2022). A detailed discussion can be found in Cai (2022, 2025). This model not only provides a guide for research on P-PBL but also provides a guide for implementing P-PBL in classrooms, including handling students’ responses to problem-posing tasks (Ran et al., 2025).

Figure 2.

The problem-posing based learning instructional model (Cai, 2022).

5. A call for solving the problem of problem posing in school mathematics

Despite 30 years of calls to integrate problem posing into school mathematics, there has been little substantive progress, as the set of papers in this special issue shows. Although problem posing is a part of policy documents, it has largely not penetrated the level of the intended curriculum—textbooks and other curriculum materials—much less the level of the implemented (or attained) curriculum. Problem-posing activities simply are not included consistently or purposefully in current mathematics curricular materials (Cai & Jiang, 2017).

We believe that a different approach is needed to solve this problem. Researchers have offered three recommendations for incorporating mathematical problem posing into school mathematics (Cai & Hwang, 2021). These recommendations are designed to present a low barrier to entry for teachers and students. They require only minor changes to common mathematics classroom activities and curriculum materials, and thus teachers (and curriculum developers) can adopt them without substantially increasing the burdens on teachers’ planning and instructional time.

Although many teachers may have little experience with using problem-posing activities in the mathematics classroom, problem posing offers enticing benefits in terms of its potential for deepening students’ engagement with mathematics and gaining a better understanding of students’ mathematical thinking. The relatively minimal investment in professional development that would be needed to help teachers gain confidence in using problem posing in mathematics instruction would be well spent (Cai, 2022; Muirhead et al., 2025). Problem-posing teaching cases and supporting teachers’ teaching empowers teachers to implement problem posing in their classrooms (Hwang et al., 2025) despite the relatively small number of problem-posing tasks in school mathematics textbooks.

This special issue includes papers on problem posing for only a select number of countries due to space limitations. As such, the ideas presented in this special issue are constrained in this regard, and we hope that in the future scholars from other countries can discuss their respective countries’ problem posing in school mathematics. In the meantime, ideas shared in this special issue could serve as useful references for other countries that want to integrate problem posing into their school mathematics. As indicated by Silver and Li (2025), there are many possible directions in which to explore the integration of problem posing into school mathematics. I sincerely hope this special issue could serve as a springboard for such efforts to solve the problem of problem posing in school mathematics.