Coordinating Mathematical Structure and Pedagogical Appropriateness: Preschool Teachers’ Problem Posing Across Differently Structured Tasks
This qualitative study of 15 preschool teachers reveals that while they value pedagogical appropriateness, their ability to coordinate these demands with explicit mathematical structures in problem-posing tasks varies significantly, performing best in structured formats and struggling to create coherent mathematical problems in open-ended contexts.
Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer
In the earliest years of life, mathematics is not a set of abstract rules to be memorized, but a way of making sense of the world. It lives in the rhythm of a song, the arrangement of blocks on a rug, and the sharing of snacks at a table. For young children, a mathematical problem is not a question on a worksheet; it is a situation that invites them to notice a gap, compare two things, or figure out how to make something fair. This is the realm of problem posing: the act of creating these situations rather than just solving them. While adults often assume that simply placing a child in a playful setting with toys is enough to spark mathematical thinking, researchers have long wondered if the adults guiding these moments truly understand how to shape a simple activity into a genuine puzzle. The challenge lies in balancing two needs: the situation must be fun and accessible to a small child, yet it must also contain a clear mathematical relationship that requires the child to think, not just follow an instruction.
A recent study set out to examine how preschool teachers navigate this delicate balance. The researchers gathered fifteen experienced teachers from Turkey and asked them to create math problems for young children. The teachers were first interviewed to understand what they believed about teaching math and what they said they did in their classrooms. Then, they were given a specific task: to look at two different pictures and invent three types of problems for each. The first picture showed a pile of colorful building blocks, and the second showed a family shopping in a grocery store. For each picture, the teachers had to create a "structured" problem where the answer was very clear, a "semi-structured" problem where the child had some choices, and an "open" problem where the teacher had to decide everything from the question to the goal. This process generated ninety different problem situations, which the researchers then analyzed to see if they were truly mathematical puzzles or just activities.
The findings revealed a clear gap between what teachers said they valued and what they actually produced when asked to design a problem from scratch. In their interviews, the teachers spoke passionately about the importance of play, using real-life situations, and letting children participate actively. They described a classroom full of stories, games, and hands-on materials. When they created the problems, they successfully captured this spirit. Almost every situation they invented was developmentally appropriate and connected to the child's world. However, when the researchers looked closely at the mathematical structure of these problems, a pattern emerged. When the teachers were given a highly structured prompt, they produced excellent, clear problems. But as the prompts became more open and left more decisions to the teachers, the quality of the problems became more variable.
The study found that out of the ninety problems created, sixty-eight were considered "sufficient," meaning they had a clear mathematical goal and enough information for a child to solve them. Sixteen were "insufficient," lacking a clear requirement or enough detail, and six were not problems at all. Crucially, all six of the non-problems appeared in the open-ended tasks. In these instances, the teachers had created engaging scenarios—like asking a child to play with blocks or go shopping—but they had failed to turn those scenarios into a specific mathematical challenge. The child was invited to act, but not to reason about a specific quantity or relationship. The teachers had successfully made the situation fun, but they had not made it mathematically necessary.
This suggests that creating a good math problem for a young child is a complex skill that goes beyond knowing how to play or knowing math facts. It requires the teacher to coordinate three things at once: a context the child understands, a specific mathematical relationship to explore, and a clear requirement that makes the child need to solve it. The study showed that when the task structure was removed, many teachers struggled to supply the missing mathematical piece. They could easily describe a story or a game, but they found it difficult to ensure that the story or game actually demanded a mathematical solution. The open tasks did not automatically produce better or more creative problems; instead, they simply placed the full burden of design on the teacher, revealing who could successfully weave the math into the play and who could not.
The researchers also noticed that the two pictures led to different kinds of mathematical thinking, even though the overall success rate was the same. The picture of the building blocks naturally led teachers to create problems about patterns, colors, and spatial arrangements. The picture of the grocery store led to problems about counting, sharing, and money. This indicates that the environment shapes what a teacher notices, but the teacher's skill determines whether that notice becomes a solvable problem. A grocery store scene could easily become just a story about buying food, unless the teacher specifically frames it as a question about how many apples are needed or how to split a bill.
Ultimately, the study suggests that professional training for preschool teachers needs to focus on this specific coordination. Teachers already possess strong pedagogical resources; they know how to engage children and make learning meaningful. What they need is practice in making the mathematical structure of a problem explicit without losing the child's interest. The researchers propose that teachers would benefit from workshops where they can compare different versions of the same problem, revise weak prompts into strong ones, and discuss why a certain question works while another does not. By practicing the act of designing these situations, teachers can learn to ensure that every playful moment they create is also a moment of genuine mathematical discovery. The goal is not to replace play with drills, but to ensure that play itself is structured in a way that invites the child to think.
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