Abstract
Keywords
1. Introduction
In mathematics education, problem-posing-based learning (P-PBL) is the practice of using problem-posing tasks to create broadly accessible opportunities for students to achieve cognitive and noncognitive mathematical learning goals. When students undertake problem posing, they engage in sensemaking (Cai, 2022; Christou et al., 2005; Cifarelli & Cai, 2005), build connections between mathematics and their own prior experiences and knowledge (Bonotto, 2010; Walkington & Bernacki, 2015), and exercise agency as creative doers of mathematics (Akay & Boz, 2010; Silver, 1997). Recognizing the potential for P-PBL to have a positive impact on student learning (Ran et al., in press), decades of policy-guiding curriculum documents have promoted its inclusion in school mathematics. For example, the National Council of Teachers of Mathematics’ (NCTM, 1989) highly influential
In the face of such enduring recommendations in curriculum framework documents for more problem posing in school mathematics and teachers’ positive views of problem posing, it is unfortunate that research has revealed a dearth of problem posing in widely used curriculum materials in the United States. Only a very small proportion of problem-posing activities are included in popular elementary and middle school mathematics textbooks (Cai & Jiang, 2017; Silver, 2013), and what is included is not developed in a consistent way across topics and grade levels. Because teachers are at the heart of implementing any educational innovation or improvement of ideas, this means there is a critical need to support teachers to integrate problem posing into their instruction and to investigate how to support them to implement P-PBL effectively in their classrooms. One natural way to support teachers to work around the limited support for problem posing in published curricula is to help them reshape existing problem-solving tasks in simple ways that transform lessons to create learning opportunities with problem posing. Indeed, this is one of the ways that our P-PBL research project is working to support teachers to teach mathematics through problem posing (Cai, 2022; Cai et al., 2023; Han et al., 2024, Hwang et al., 2025).
The study presented here is part of our larger P-PBL research project. The present focus is to share and examine two cases that illustrate how teachers are learning to teach mathematics through problem posing. More specifically, we address the following research question: How do teachers implement problem posing in the context of a U.S. middle school mathematics curriculum? In this paper, we begin by exploring the role of problem posing in U.S. mathematics curricula. Then, we give an overview of the larger P-PBL Project—its work and initial findings—and focus on two specific cases of teachers in the project.
2. Problem posing in U.S. curricula
Mathematics curriculum is commonly conceptualized in four main levels: the planned curriculum, the intended curriculum, the implemented curriculum, and the attained curriculum (e.g., Cai & Hwang, 2021; Lloyd et al., 2017). It should be noted that the intended curriculum refers to educational system-level (e.g., national, district, or school level) expectations for mathematics learning, including both written textbooks and curriculum standards (Husen, 1967; Lloyd et al., 2017; Travers, 1992). To examine specific problem-posing activities in textbooks, in this paper, we distinguish between textbooks and curriculum standards and refer to curriculum standards as planned curriculum and written textbooks as intended curriculum, as we have done elsewhere (e.g., Cai & Hwang, 2021). These different levels of curriculum refer to different perspectives from which a curriculum can be examined. In what follows, we examine the role of problem posing in U.S. curricula at the planned and intended curriculum levels. Then, we examine preexisting research on teachers learning to teach through problem posing.
2.1 The planned curriculum
Planned curriculum refers to national standards documents as opposed to instructional materials or textbooks which indicate intended curriculum. NCTM's (1989)
In both the curriculum and teaching standards, problem posing is conceived of in three ways: as a cognitive activity, as a learning goal, and as an instructional approach (Cai et al., 2023). Problem posing as a cognitive activity refers to the thinking processes that are entailed in posing mathematical problems. Problem posing as a learning goal refers to the development of problem-posing skills. In this conception, the goal is to help individuals, including students and teachers, increase their capacity to pose high-quality mathematical problems. Problem posing as an instructional approach refers to the idea of teaching mathematics through problem posing. Although developing problem-posing skills may be a subsidiary goal in this kind of instruction, the emphasis is on engaging students in problem-posing tasks and activities to help them achieve other cognitive and noncognitive learning goals (Cai et al., 2023). In the P-PBL Project, we have focused on supporting and studying problem posing as an instructional approach.
2.2 The intended curriculum
The intended curriculum is defined in multiple ways by scholars but is generally conceived of as the official written plan outlining learning objectives, content, materials, and assessments, developed by educational policymakers and experts to guide instruction. Cai and Hwang (2021) specifically defined problem posing at the intended curriculum level as “problem posing activities and learning opportunities in published curriculum materials” (p. 1404). In this paper, when we talk about intended curriculum, we refer to instructional materials or textbooks. The alignment between policy requirements and actual curriculum materials presents an important area for investigation, particularly in the case of problem posing. Given the consistent inclusion of problem posing in decades of standards documents (NCTM, 1989, 1991, 2000, 2020), it is important that curriculum materials consistently offer opportunities for students to engage in problem-posing activities in which they are the ones who formulate problems. However, research indicates that textbooks include relatively few problem-posing tasks, with inconsistent distribution across grade levels and content areas (Cai & Jiang, 2017; Singer et al., 2015).
Research has also shown that the instructional materials that make up the intended curriculum can significantly influence teachers’ instructional decisions and modulate students’ opportunities to learn, eventually influencing students’ learning outcomes (Remillard & Heck, 2014). Given this observation, the limited presence of explicit problem-posing tasks in many curriculum materials, despite their recognized importance in policy documents, raises questions about how effectively current mathematics curricula can support the development of students’ problem-posing abilities. Ultimately, the support for students’ problem posing depends on teachers’ instructional decisions. Remillard and Heck (2014) proposed the notion of the teacher-intended curriculum to describe how teachers may draw on the intended curriculum (e.g., their textbook) and other resources to envision and plan their instruction.
2.3 Teachers learning to teach mathematics through problem posing
Because teaching mathematics through problem posing has not historically been common practice in the United States (and other countries), teachers who wish to use problem posing as an instructional approach must develop their own knowledge of and beliefs about problem posing (Cai & Hwang, 2021). Regarding teachers’ knowledge, there have been some studies on teachers’ learning of how to pose mathematical problems. Researchers first focused on training teachers on how to pose their own mathematical problems. Some studies explored further as they moved on to teaching teachers how to pose problems for students to solve or how to create problem-posing tasks for students to pose mathematical problems. Later studies asked teachers to plan their lessons around P-PBL (e.g., Gonzales, 1994). More recently, Ellerton (2013) designed a teacher education curriculum that integrates more problem-posing activities so that teachers actually experience problem posing as learners. This work identified the importance of providing opportunities for teachers to discuss their posed problems with others so that they can notice any errors or flaws in their posed problems and then revise their problems accordingly after these discussions.
Research has also shown that teachers’ problem-posing activities (e.g., identifying the attributes of given problems and finding alternatives for these attributes) improve their development of different forms of mathematical knowledge, such as the definitions of and connections between different mathematical concepts (Lavy & Shriki, 2010). In addition to teachers posing problems, researchers have found some positive impacts of teachers’ involvement in planning their lessons using instructional models that integrate problem solving and problem posing. For example, Gonzales (1994) examined teachers’ learning of how to pose problems by extending Polya's four-step problem-solving process to a five-step problem-solving/problem-posing process, with the added fifth step being “posing a related problem.” In this fifth step, preservice teachers (PSTs) created their own mathematical problems in various ways (e.g., reversing knowns and unknowns, changing the given values of the data, or changing the real-life context of the given problem-solving task). Beyond simply posing problems, the PSTs were asked to plan lessons based on the five-step process. Gonzalez found that these experiences increased PSTs’ knowledge regarding differentiating lessons depending on grade level and mathematical strengths and increased their insights into teachers as facilitators of mathematical knowledge.
With respect to teachers’ beliefs about and perspectives on problem posing, research has shown that teachers may consider problem posing to be a venue for supporting students’ mathematical sensemaking (e.g., Christou et al., 2005; Han et al., 2024). For example, Han et al. (2024) explored 15 U.S. middle school mathematics teachers’ views about problem-posing tasks. They found that the teachers considered how problem posing could be a useful tool for supporting students’ sensemaking, their learning of mathematical content and processes, and their development of identities as creative doers of mathematics. However, many teachers have had little to no experience with using problem posing as a task for students (see, e.g., Cai et al., 2020) and thus have not had opportunities to develop informed perspectives about problem posing as an approach to mathematics instruction.
Yet, teachers’ beliefs about both mathematics learning and instructional practices influence how they utilize curriculum and implement instructional practices (Lloyd, 2002; Wilkins, 2008). Thus, it is important to understand the nature of teachers’ perspectives on problem posing and the development of their beliefs as they gain experience with problem posing and incorporating problem posing into their instruction. For example, Barlow and Cates (2006) found that when a group of U.S. teachers implemented a form of P-PBL in their classrooms, they developed productive beliefs about mathematics, mathematics teaching, and engaging students in problem posing. The teachers expressed appreciation for incorporating problem posing in instruction, and they believed that problem posing could promote students’ higher order thinking, deepen their mathematical understanding, and help them develop ownership of the mathematics they were learning in addition to potentially being used as a tool for assessing students’ understanding. These perspectives accord with findings from Cai and Hwang's (2021) study with teachers in China in which teachers reported increased confidence in teaching through problem posing in classrooms after attending professional development workshops on problem posing. Many of those teachers expressed that problem posing can be helpful for increasing students’ engagement with mathematics and can enable students to value learning mathematics. As another example, Li et al. (2020) found that teachers’ participation in workshops on problem posing—including its nature, significance, practical experience, and lesson design—led to improvements in problem quality and shifts in beliefs about P-PBL's advantages and challenges.
In this paper, we draw on the work of two teachers in the P-PBL Project to examine the details of two cases of teacher-intended curriculum where the teachers wove together their
3. The P-PBL Project: Two cases
3.1 The P-PBL Project
This study is part of our larger P-PBL Project. This 4-year, longitudinal project has two major goals. The first goal is to support teachers to teach mathematics through engaging their students in mathematical problem posing. Through ongoing partnership and collaboration, a networked improvement community of teachers and researchers integrate problem posing into daily mathematics instruction and continuously improve the quality of P-PBL through iterative task and lesson design. The second goal of the project is to longitudinally investigate the promise of supporting teachers to teach with P-PBL for teachers’ instructional practice and students’ learning.
To achieve these two goals, this project is supporting teachers to integrate problem posing into their daily teaching, from developing individual problem-posing instructional tasks to designing entire problem-posing lessons to writing teaching cases that describe their implementations of problem posing within the tasks and lessons of the IM curriculum. These tasks and lessons, many of which build directly upon opportunities included in the teachers’ existing IM materials, engage students in problem-posing activities that support their reasoning and sensemaking as they learn mathematics.
We chose for the P-PBL Project a research site where teachers and students are using the open-access IM middle school curriculum in the United States. This choice was based on two reasons. The first reason is that IM already includes some problem-posing tasks. Although the number of such tasks is relatively small, teachers still face challenges implementing them in the United States. This allows us to study how to support teachers to implement the existing problem-posing tasks in the curriculum materials. The second reason is that IM is open access, which means that there exists both the freedom and ample opportunities to develop problem-posing tasks based on the existing tasks in the textbook.
The IM curriculum, which is used in a geographically and economically diverse collection of school districts across the United States, employs a problem-based learning approach, where students actively engage with rich, real-world problems that encourage exploration, reasoning, and collaboration. Problem-based learning supports deep conceptual understanding by positioning students as problem solvers who construct their own knowledge through inquiry, discussion, and application of mathematical principles. As such, problem-based learning lays a rich foundation necessary for problem posing. By incorporating features such as open-ended tasks and Mathematical Language Routines (MLRs; Zwiers et al., 2017), including “Co-Craft Questions and Problems” (MLR #5), the IM curriculum encourages students to actively formulate and refine mathematical questions. The Co-Craft Questions routine involves showing students only part of a task in the curriculum materials (i.e., omitting the question or problem) and then prompting them to generate questions based on the given information. A stated purpose of the Co-Craft Questions routine is to “create space for students to produce the language of mathematical questions themselves” (Zwiers et al., 2017, p. 14). This routine therefore synergizes well with efforts to teach mathematics through problem posing. That said, the routine is positioned within the IM curriculum materials as a support for English language learners, and thus it is not always perceived by teachers as a mechanism for engaging all students with problem posing. 2 Moreover, an analysis of the Grade 6 IM curriculum reveals that among its nine units containing 147 lessons, 18 lessons across six of those units include Co-Craft Questions activities. This represents 12.2% of the total number of lessons in the Grade 6 IM curriculum.
In the IM curriculum, many lessons do begin with contexts or scenarios that encourage students to identify mathematical relationships or challenges. For example, students might be asked to consider “what-if” scenarios or extend a problem to new situations, which naturally incorporates problem posing as part of problem solving. These sensemaking activities are a natural segue to problem posing, and often the instructional routine “Notice and Wonder” (a routine in which the teacher or textbook presents students with a situation or artifact such as a photograph) asks students to respond with something they notice and something they wonder about the situation or artifact. This routine is incorporated to foster students’ inquisitiveness about mathematical ideas (Rumack & Huinker, 2019). Lessons also often begin with exploratory contexts designed to prompt students to uncover relationships and construct their own problems, promoting both agency and intellectual curiosity. Collaborative tasks further enhance this process by facilitating dialogic interactions that generate new questions and perspectives. However, like the Co-Craft Questions routine, the Notice and Wonder routine is not positioned in the IM curriculum as an explicit mathematical problem-posing opportunity, and students’ questions may often be nonmathematical.
3.2 Teaching with problem posing in the P-PBL Project
In this section, we present two cases from the P-PBL Project that illustrate how project teachers have been implementing problem posing in the context of their middle school mathematics curriculum. In both cases, the teachers transform activities in the IM curriculum from problem-solving tasks to problem-posing tasks using a technique akin to the Co-Craft Questions routine. The case of Ms. Garrett 3 shows how she uses problem posing as an entry point to reach the mathematics of the lesson while building her students’ mathematical agency. The case of Ms. Brown illustrates her use of her students’ posed problems to both engage them in the mathematics of the lesson and to better understand her students’ mathematical thinking. Both cases illustrate problem posing as an instructional approach.
3.2.1 The case of Ms. Garrett: Sensemaking and co-craft questions as an entry point
Ms. Garrett has been teaching for 14 years. She started as a paraprofessional working with one student during that student's 4 years of high school. Ms. Garrett enjoyed teaching and decided to pursue her teaching degree; she has been teaching middle school mathematics for the past 10 years. Her current school has been identified as needing improvement, with fewer than 10% of students meeting proficiency on the state mathematics exam. Ms. Garrett has been using the IM curriculum.
Ms. Garrett has been working to support her students in breaking old habits learned through I Do-We Do-You Do, encouraging students to think and reason and make their thinking public. The “I Do-We Do-You Do” routine refers to a structured teaching approach where the teacher first demonstrates a concept (“I Do”), then guides students through a similar task collaboratively (“We Do”), and finally allows students to independently apply the concept on their own (“You Do”), essentially gradually releasing responsibility to the learner. Instead of the “I Do-We Do-You Do” routine, she has been infusing problem posing into her lessons daily to support her students in owning their learning and seeing themselves as doers of mathematics. Through the routine, Ms. Garrett creates opportunities for students to develop confidence in their own agency as learners through problem posing. Ms. Garrett often uses questions her students pose on quizzes, homework, and warm-ups in her class. She ensures that her students have opportunities to answer some of their questions to build student agency.
The lesson described here (IM Grade 7, Unit 2, Lesson 12) focuses on the constant of proportionality and interpreting graphs of proportional relationships. This lesson builds on previous lessons where students developed an understanding of the concept of the constant of proportionality and how it appears in a graph. The new mathematical idea in this lesson involves comparing two different proportional relationships by graphing their corresponding lines on the same axes. The students compare the two relationships visually by comparing the steepness of the lines. They also make the connection that comparing the steepness of lines is equivalent to comparing the numerical constants of proportionality. More specifically, the steeper line has the greater constant of proportionality (“steeper” = “greater constant of proportionality”). Ms. Garrett embraced the teacher-facing learning goals:
Create and interpret graphs that show two different proportional relationships on the same axes and Generalize (orally and in writing) that when two different proportional relationships are graphed on the same axes, the steeper line has the greater constant of proportionality (Kendall Hunt, 2024).
In the previous school year, Ms. Garrett had facilitated Activity 12.2 (see Figure 1) in this lesson as it was described in her materials; she put students in groups of two or three and provided them with the necessary resources to complete the table and graph and to answer the given problem.
IM grade 7 unit 2 lesson 12 activity 12.2 (Kendall Hunt, 2024).
Figure 2 represents four hypothetical students’ graphs based on the activity in Figure 1. As can be seen in Figure 2 below, the graph for Mai's journey does not reflect a proportional relationship between the distance from the ticket booth and elapsed time. The students in Ms. Garret's classroom experienced difficulty filling out the table for Mai because of the 10-s delay in Mai's start time. Students were unsure what to include as the start time when Mai's distance was zero. The discussion included students providing options such as 0, 1, 5, and 10 meters. The class discussed the rest of the description of Mai's journey and reasoned that her steady pace could be mapped back to 10 s using a steady rate. Although this was an important moment in the students’ learning, nonlinear equations were not the focus of this lesson. The IM materials Ms. Garrett used introduce proportional relationships by building on the Grade 6 idea of equivalent ratios. Their Grade 7 study of proportional relationships prepares the way to study linear functions in Grade 8. Because proportional relationships were the focus of this lesson, Ms. Garrett decided not to spend much time on this aspect of the task and pushed students to investigate the graph of the four hypothetical characters’ journeys.
Race to the bumper cars (Kendall Hunt, 2024).
This year, Ms. Garrett hoped to create better opportunities for her students to connect the different features of a graph with parts of the situation it represents. In particular, she wanted them to attach meaning to any point on a graph and interpret the meaning of the distance when the time is 1 s as both the constant of proportionality of the relationship and the person's speed in the context in meters per second. She redesigned activity 12.2 by inserting the problem-posing prompt shown in Figure 3. This follows the pattern of the Co-Craft Questions routine, in which students are presented with a problem-posing prompt after seeing the given information in a problem but before they see the actual question or problem-solving prompt.
Ms. Garrett's problem-posing prompts for IM activity 12.2.
Having provided this opening, Ms. Garrett allowed her students to make sense of the problem situation and the underlying mathematical ideas. Her students came up with various problems that provided Ms. Garrett insight into what they understood about the problem, how they were thinking about the relationships, and their readiness to engage with the key mathematical concepts.
Several students indicated that they expected Ms. Garrett to ask them questions that could be answered based on reading the graph:
How long did it take for Lin and Mai to get to the bumper cars? How long did it take to get to the bumper cars? How fast are they going? How many steps did Mai and Diego take to get to the bumper cars? How did it take compared to Tyler?
Several students asked questions tangential to the task but not about the proportional relationships:
How much did the tickets cost? How many people are in line? How many seats are there? What angle will the bumper car be?
A few students included information they had previously learned about proportional relationships:
Who got there first? Why is Mai the only one without a COP [constant of proportionality]? Is Mai a proportion [sic]? What is the multiplant [sic]? How much faster was Lin than Diego? Did Diego and Mai travel at the same speed? How do you know?
Ms. Garrett was pleased that her students could generate so many different questions and even some more complex problem-solving tasks. She made the decision to ask for “questions” that she, as a teacher, might ask because that would allow every student in her class to have access to the task. She had learned that student access is the key to deeper student thinking, and she wanted to design her problem-posing task in a way that would ensure that all her students would have the opportunity to make sense of the mathematics in the problem situation and share their thinking.
3.2.2 The case of Ms. Brown: Infusing problem posing into a textbook task
Ms. Brown is a Grade 8 teacher in the same school as Ms. Garrett. She also uses the IM curriculum, and she chose to infuse problem posing into a lesson on function graphs (Grade 8 Unit 5 Lesson 5; Kendall Hunt, 2024) by converting a problem-solving task into a problem-posing task. The lesson has two main mathematics learning goals:
Describe (orally and in writing) a graph of a function as “increasing” or “decreasing” over an interval and explain (orally) the reasoning. Interpret (orally and in writing) a graph of temperature as a function of time, using language such as “input” and “output” (Kendall Hunt, 2024).
To achieve these goals, the lesson consists of four major tasks.
5.2. Time and temperature task (8.5.5: More graphs of functions; Kendall Hunt, 2024).
To infuse problem posing into her instruction, she decided to modify a task focusing on problem solving into a task offering problem-posing opportunities to students. In doing so, she took out the original questions paired with the problem situation and presented the problem situation only (Hwang et al., 2025). Note, however, that she did not discard the original questions. Rather, she saved them to inform her pedagogical evaluation of the students’ posed problems and how best to use them in the lesson.
Ms. Brown presented the problem situation (Figure 5) on the board and said to the students, “Based on the graph, create as many questions as you can.” In pairs, students worked quietly for a couple of minutes, sharing some of their problems with each other while Ms. Brown circulated. Then, she asked the students to pause what they were doing and to share what they had generated so far. The students shared their own questions, such as “Is it a function?” “What time was the temperature the ‘hot’ time?” “What's the function?” and “It's not straight, why not straight?”
IM activity 8.8.5.2 as presented in Ms. Brown's class.
Ms. Brown valued the questions the students shared, and she wanted students to come up with more mathematical questions. She asked students to continue writing down more problems related to the graph. This time, students worked more interactively as they talked to their peers when creating problems. After a couple of minutes, some students volunteered to share their problems, including but not limited to “What's the
Ms. Brown directed the students to work individually on the six questions related to the graph originally included in the textbook version of the activity (see Figure 4). After working individually, the students discussed their answers with their partners. Then, Ms. Brown asked volunteers to share their answers to the first three questions. For the latter three questions, she led discussions regarding the answers to the questions with several students’ contributions. She covered key ideas that she intended to implement, such as describing the graph as “increasing” or “decreasing” over a certain interval, interpreting the graph of temperature as a function of time, using precise language such as “input” and “output,” comparing the relative height of the graph, and identifying another time that is at the same temperature.
After this problem-solving activity, Ms. Brown once again asked the students to create their own questions that someone else could answer using the given graph. She gave about 2 min for students to create their own questions individually and then instructed students on what they would do with the questions. Ms. Brown used an interesting method for sharing posed problems: She asked a student to share their problem and to pick another student to answer it. Then, the student who answered would share their problem and ask yet another student to answer it. She continued this chain until all the students had a chance to share their problems and have them answered. Among the questions the students shared are the following:
Finding the temperature at an exact time, such as “What's the temperature at 1 pm?” or “Why the graph ends at 12?” Finding the time(s) when the temperature is at a certain temperature, such as “What time does the graph reach 58?” or “What two times was the temperature 54?” Finding the lowest or highest temperature, such as “What's the temperature when it's the lowest?” Finding the interval(s) when the temperature increases or decreases, such as “At what time does the temperature decrease?” “Does the temperature increase or decrease later the day after sunset?” Finding the temperature difference between certain times, such as “How much is it increasing between 2 o’clock and 3 o’clock?” “What's the temperature difference between 5 and 10?” or “What's the temperature difference between 7 and 12?”
Many of these questions were quite simple to answer, requiring only a simple reading of temperature(s) from the graph or computing one- or two-step operations such as addition or subtraction. For example, an answer to one of the problems, “What time does the graph reach 58?” is sometime between 4 and 5, like 4:45 pm, and between 7 and 8, like 7:10 pm. Another example is “How much is it increasing between 2 o’clock and 3 o’clock?” An answer to this problem may be the difference between the temperatures at 3 p.m. and 2 p.m., which is about 2.6 degrees. For about 10 min, the class shared their own questions.
After everyone shared their own questions and answered their classmates’ questions, Ms. Brown acknowledged her students’ efforts to create their own problems and to try to answer their classmates’ problems. Although she did not explicitly provide a summary of what the students’ questions were about, the class touched on all the key ideas by solving the six given questions attached to the graph. Moreover, through the two opportunities for problem posing—before and after problem solving—they made sense of the graph and explored both similar and different mathematical problems that could be answered using the graph. In addition, they were able to hear and see what their peers thought about the same graph. Through her method of sharing posed problems with the class, Ms. Brown learned how each of her students was making sense of the graph of a function. This guided her to make instructional adjustments to meet individual students’ needs. The students also had an opportunity to learn from their classmates, especially having an opportunity to see the diversity of problems that their peers formulated, helping them to expand their thinking about the possible types of problems they could generate.
4. Discussion
The cases of Ms. Garrett and Ms. Brown provide two illustrations of how middle school teachers in the P-PBL Project are implementing problem posing in the context of their existing curriculum materials. In both cases, the teachers actively drew on both their curriculum materials (IM) and what they had learned about problem posing through the P-PBL Project to fashion a teacher-intended curriculum (Remillard & Heck, 2014). By modifying textbook problem-solving tasks, they hoped to create new opportunities for their students to learn mathematical concepts and to develop mathematical agency through posing their own problems. In this section, we look across the two cases to highlight some common threads.
4.1 Fostering and assessing students’ mathematical thinking and sensemaking
Both Ms. Garrett and Ms. Brown made use of problem posing to help their students engage with the mathematics of their respective lessons by making sense of the problem situations (Cai & Hwang, 2023; Christou et al., 2005). In the case of Ms. Garrett, the goal was for students to be able to explicitly make sense of the graph's features in terms of the physical situation and to connect their existing understanding of the constant of proportionality with both the distance when the time is 1 s and the person's speed in the context in meters per second. By letting her students take time to explore the situation and pose problems, they were able to notice several mathematical features and start activating their relevant prior knowledge (Bonotto, 2010; cf. Krawitz & Schukajlow, 2018), such as knowledge about speed, time, and proportional relationships between them. In the case of Ms. Brown, by providing the problem-posing opportunity before problem solving, her task modification gave all her students a chance to make sense of and ask questions about the time and temperature graph before adding the additional cognitive load of answering the textbook's set of questions. Because problem posing is an activity with a low barrier to entry (Silber & Cai, 2021), it increases the likelihood that students can engage with the problem situation and begin making sense of it without the need to also parse the demands of solving a given problem.
In addition, both Ms. Garrett and Ms. Brown gained useful information about their students’ mathematical thinking through the problems they posed. The students’ problems quickly revealed to the teachers who had noticed the mathematical core ideas of the lesson, who might be focusing on nonmathematical features, and who was beginning to make connections between the mathematical ideas of the lesson and their existing knowledge. Thus, Ms. Garrett and Ms. Brown both employed problem posing as a mechanism to support formative assessment that would provide them with useful information to inform their pedagogical decision making (Kwek, 2015).
4.2 Handling students’ posed problems
When students pose their own mathematical problems, it is incumbent on the teacher to handle those problems (Cai, 2022). That is, teachers must make instructional decisions about how and when students will examine, discuss, and even solve their posed problems. Moreover, because students may pose many more problems than there is time in the lesson to address them all, teachers must give some consideration to which problems to select for attention. As described in their cases above, Ms. Garrett and Ms. Brown each created opportunities for their students to solve at least some of their posed problems. For example, in her daily practice of asking her students to pose problems, Ms. Garrett let them know that some of the posed problems would appear on future homework assignments, quizzes, or tests. This both serves to motivate students to pose problems that Ms. Garrett might want to select and confers significance to the activity of problem posing. The students can see that their problem posing is not just another classroom activity—it is not problem posing just for the sake of posing problems.
Ms. Brown's approach was somewhat different. She did not ask her students to solve the problems they posed in the first part of the lesson, although she did make a point of explicitly valuing them. At this point in her lesson, her intention with the problem-posing activity was to encourage students’ sensemaking about the given graph. However, when she asked the students to pose problems again after they had solved the textbook problems, she had them solve each other's problems in a chain of posing and solving. This routine for handling the students’ posed problems reinforces the students’ belief that their actions matter—their agency as posers of problems shapes the mathematics that the class engages with (Barlow & Cates, 2006).
4.3 Student agency and engagement
The teachers in the P-PBL Project are very aware that their students face deep-rooted academic challenges and that mathematics is often a frustrating experience for them. Thus, they are sensitive both to the potential for problem posing to increase student engagement and to how they can position their students through problem-posing activities to support their agency in the mathematics classroom. Indeed, the 2 years leading up to the cases presented above were especially challenging for getting students to share their thinking. Along with many other negative effects of the pandemic on students’ psychological and social well-being (Viner et al., 2022), the pandemic had severely curtailed their students’ engagement with school. At the point when these lessons were taught, students had yet to return to their pre-pandemic level of socialization and engagement in Ms. Garrett and Ms. Brown's school.
Encouragingly, since she has started using problem posing in her classroom, Ms. Garrett has noticed that her students are more willing to engage verbally and share their ideas. The shifts in sensemaking that her students have engaged in have changed the dynamic of the classroom in positive ways. Her students are less hesitant, more enthusiastic, less afraid to take risks, and more frequent participants. Their posed problems frequently anticipate the mathematical goals of the lesson, with students sometimes posing the same problems the textbook will pose later in the lesson. As with Ms. Brown's posing/solving routine, this allows Ms. Garrett to share agency and ownership with her students with respect to the direction of the mathematics of the lesson (Barlow & Cates, 2006). Her students are pointing the way to the mathematics rather than the book (or Ms. Garrett) telling them where to go.
5. Concluding thoughts and directions for future research
One of the primary goals of the P-PBL Project is to support teachers to teach mathematics through engaging their students in mathematical problem posing. In this project, we have provided regular professional learning opportunities and workshops in which teachers learn about problem posing, instructional routines for implementing problem posing in their classrooms, opportunities for problem posing within their curriculum materials such as Co-Craft Questions, and ways to modify tasks and lessons in their curriculum materials to incorporate problem posing (Cai & Hwang, 2021). In addition, teachers have opportunities to collaborate on designing problem-posing tasks and to rehearse with each other how they will implement those tasks in their classrooms.
Working together with middle school teachers around the IM curriculum materials in the United States, we have explored ways to build on those curriculum materials to more effectively incorporate problem posing into mathematics lessons in ways that support students’ sensemaking, conceptual understanding, agency, and engagement with mathematics. In particular, in this paper, we have shared two cases drawn from the classroom practice of our project teachers that illustrate how the Co-Craft Questions instructional routine may be used effectively to create rich cognitive and noncognitive learning opportunities for students. However, this routine is certainly not the only mechanism for incorporating problem posing into mathematics instruction. Future research could explore in detail other promising mechanisms for including problem posing in lessons. For example, in the case of Ms. Garrett, we note that she built problem-posing opportunities into multiple places in her lessons (e.g., “exit tickets”) where the problems students pose may ultimately end up on homework, quizzes, or tests. Research is needed to investigate how such strategies and other routines such as Notice and Wonder may work to influence students’ learning outcomes.
Moreover, research is needed to further investigate the implementation and effectiveness of P-PBL in classrooms. Under what circumstances and with what students do implementations of P-PBL fulfill the promise of higher quality learning opportunities? To support such work, research is also needed to establish and verify models for problem-posing processes (Cai & Rott, 2024), both in terms of the cognitive and affective mechanisms of problem posing. In particular, such models would ultimately serve to inform our understanding of how problem posing can encourage students’ sensemaking, their development of conceptual understanding, and their engagement with and beliefs about mathematics.
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