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Article

Students’ Difficulties with Problem Posing in Early Childhood Education

by
Enrique Carmona Medeiro
1,* and
Nuria Climent
2
1
Department of Didactics of Mathematics, Faculty of Educational Sciences, University of Cadiz, 11510 Cádiz, Spain
2
Department of Integrated Didactics, Center for Research in Contemporary Thought and Innovation for Social Development (COIDESO), University of Huelva, 21071 Huelva, Spain
*
Author to whom correspondence should be addressed.
Educ. Sci. 2025, 15(5), 630; https://doi.org/10.3390/educsci15050630
Submission received: 7 March 2025 / Revised: 25 April 2025 / Accepted: 17 May 2025 / Published: 21 May 2025
(This article belongs to the Section Early Childhood Education)
Browse Figure
Versions Notes

Abstract

:
This study focuses on exploring students’ emerging difficulties during a problem posing task in an early childhood education classroom (4–5 years old). Through a single case study, and by considering the problems posed as an indicator of the students’ mathematical performance level, the difficulties inherent in the process of posing problems based on a given situation are characterised. The results reveal the existence of difficulties both in the exploration phase of the problem situation and in posing the problems themselves. This shows that the main difficulties are related to a lack of accuracy in language, which is typical of this stage, and to the notion of a non-mathematical problem that prevails in the students.

1. Introduction

Mathematical problem posing (hereafter MPP) has emerged as a recent area of interest in mathematics education (Aktaş, 2022; Cai & Hwang, 2020). It is considered a key element to foster the development of mathematical thinking (Cai et al., 2022; Singer et al., 2015). This growing interest is explained both by the importance that curricula attribute to it (Cai, 2022; Li et al., 2022; NCTM, 2003), and by the benefits it brings to mathematics teaching and learning processes (Cai & Hwang, 2023; Lee et al., 2018; Leikin & Elgrably, 2019; Li et al., 2022).
Research into MPP has been carried out at different educational levels, including primary (Bonotto & Santo, 2015; Possamai & Allevato, 2024), secondary (Cai et al., 2023; Krawitz et al., 2024), and university education (Baumanns & Rott, 2024; Montes et al., 2024). However, early childhood education (henceforth, ECE), the educational stage for children under the age of 6, remains largely under-explored (Carmona-Medeiro et al., 2024; van Bommel et al., 2024).
Despite the progress made in research, there is a clear need to build a more robust theory to understand the general phenomenon of MPP (Baumanns & Rott, 2021; Cai et al., 2015). One of the lines of research that requires attention from the research community concerns how to implement and manage MPP tasks to help students learn how to pose mathematical problems (Carmona-Medeiro & Climent, 2024; English, 2020; Li et al., 2022). In this regard, Cai (2022) points out three challenges teachers encounter during the implementation of MPP tasks that need to be addressed through empirical studies: (i) how to manage problems of a non-mathematical nature posed by students, (ii) how to manage problems students pose that are not related to the objectives of the task, and (iii) how to manage situations inherent to the open nature of MPP tasks.
This study aims to contribute to filling the existing gap about ECE students’ difficulties in problem posing processes. It lays the foundations for future research and didactic proposals that seek to integrate MPP into the classroom at this educational stage. In doing so, it addresses both the processes involved in the task implementation, through the analysis of students’ difficulties, and the products generated, through the characterisation of the problems posed. The context is a lesson with 4–5-year-olds, in which the problem situation is presented as a complex poster showing daily life in a neighbourhood. Given the paucity of studies conducted on ECE, there is no evidence as to whether the challenges outlined by Cai (2022) can be applied to this educational stage. It is understood that characterising problems posed by students is an indicator of both their level of performance and of the difficulties they face (Xu et al., 2020). Therefore, exploring students’ difficulties is useful to understand the challenges an ECE teacher faces during task implementation. This study aims to contribute to filling the existing gap about ECE students’ difficulties in problem posing processes. It lays the foundations for future research and didactic proposals that seek to integrate MPP into the classroom at this educational stage.

2. Theoretical Framework

2.1. Problem Posing

The research literature in mathematics education reflects a wide variety of approaches and definitions of what constitutes MPP (Baumanns & Rott, 2021; Cai, 2022). In the case of students, and from a broad perspective, MPP can be understood as performing two activities: (i) posing a problem based on a given situation, or (ii) posing a problem by modifying an existing problem (Silver, 1994). Henceforth, the focus of this study will be on MPP based on a given situation. Taking into account the level of openness the task grants to the problem poser, Baumanns and Rott (2020, 2021) distinguish between structured and unstructured MPP tasks. MPP tasks are structured when students are asked to pose problems based on a previously solved problem by modifying some of its structural elements. Tasks are unstructured when students pose problems based on a given situation that may or may not provide detailed information. Thus, MPP tasks form a spectrum in which both the restrictions and the information provided increase, and whose extremes are unstructured and structured tasks. This study focuses on unstructured tasks.
The defining framework assumed (Cai, 2022; Baumanns & Rott, 2020) implies that the activity of formulating necessarily leads to the construction of a problem that requires a solution. When recent studies within the framework of problem solving refer to mathematical problems, they do so to describe non-routine problems (Baumanns & Rott, 2020). However, it is important to note that the terms “task” and “problem” are used in a broad sense in the framework of problem posing, and an MPP task can hence lead to the construction of problems at any point along the routine/non-routine spectrum.
MPP tasks have been conceptualised in the literature as consisting of two fundamental elements: the problem situation and the prompt (Cai & Hwang, 2023; Cai et al., 2022). The problem situation sets the context and provides the background information for problem posing. Depending on whether the problem situation is based on real-world or mathematical references, it concerns a real-world or a mathematical context. The information provided by the problem situation can be qualitative or quantitative, and may include words, pictures, graphs, patterns, tables, and mathematical expressions (Cai & Hwang, 2023). The prompt guides the problem posing process by delimiting the expectations of the task. For the same problem situation, there can be numerous types of prompts, which may include a reference to the number of problems, the difficulty, the person the problem is intended for, or a combination of those prompts.
An MPP task can become more or less structured by making modifications in the problem situation and/or the prompt (Carmona-Medeiro et al., 2024). The choice of problem situation, and therefore of the context and the amount of information provided, influences the nature of the problems students can generate and the complexity of the mathematical work to be activated (Montes et al., 2024). Different studies have shown that the type of prompt has a direct impact on the characteristics of the problems students pose (Cai et al., 2023).
We know that the cognitive processes involved in problem posing have their own specific nature and, therefore, cannot be effectively described using traditional phase models of problem solving (Cai & Rott, 2024; Pelczer & Gamboa, 2009). Although there is still no general descriptive model of phases for problem posing comparable to Pólya’s four-step problem-solving model (1985), several studies have attempted to characterise its phases (Baumanns & Rott, 2022; Cai, 2022; Cruz, 2006; Koichu & Kontorovich, 2013; Pelczer & Gamboa, 2009; L. Zhang et al., 2022).
In particular, L. Zhang et al. (2022) propose that the processes involved in problem posing can be framed in three phases: (i) understanding the task, (ii) constructing the problem, and (iii) expressing the problem. Given the nature of this study, focused on identifying students’ difficulties during a problem posing task, attention is placed on two of these phases: the students’ understanding of the task and the oral expression of the posed problem.
The understanding of the task phase constitutes a key moment in the problem posing process, as it does not merely involve interpreting instructions but requires an active exploration of the problem situation to identify its mathematical potential. According to Baumanns and Rott (2022), this initial phase, which they call situation analysis, is crucial for generating mathematically meaningful problems, as it allows students to establish connections between the elements present in the problem situation and the underlying mathematical structures. Moreover, L. Zhang et al. (2022) note that cognitive conflicts arising during this phase act as creative triggers in the process of posing problems. Other studies (Cai & Rott, 2024; Crespo & Sinclair, 2008) highlight the importance of students becoming familiar with the task in order to recognise its limitations and possibilities, concluding that a prior and thorough exploration of the problem situation positively impacts the quality of the problems posed. However, identifying the opportunities the task offers, considering its conditions, constraints, and mathematical potential, is neither immediate nor self-evident, but requires deliberate interpretation and selection by the students (Pelczer & Gamboa, 2009). In the case of early childhood education, several studies (Martín-Díaz et al., 2020; Carmona-Medeiro et al., 2024) reveal that exploring the problem situation is essential for the viability of the task.
The problem expression phase, on the other hand, requires students to organise their ideas and articulate the language needed to shape the problem they have devised clearly and precisely (Baumanns & Rott, 2022). Crespo and Sinclair (2008) point out that the ability to express mathematical problems not only reflects understanding of the content, but also the ability to communicate mathematical ideas effectively. In the case of students under the age of six, studies indicate that limitations inherent to the development of literacy and linguistic skills (Carmona-Medeiro et al., 2024; van Bommel et al., 2024) affect the written or oral formulation of the problem. Thus, the expression phase becomes particularly sensitive in early childhood education, where communication difficulties may influence the quality of the problem posed, regardless of the students’ level of mathematical understanding.

2.2. Characteristics of the Problems Posed: An Indicator of Students’ Difficulties

MPP is a cognitively demanding activity (Cai & Hwang, 2023; Silver, 1994). Unlike problem solving, MPP, in general, is an activity that is far removed from the usual classroom habits (Chen & Cai, 2020; Singer et al., 2013). Its integration in the classroom is a challenge for both teachers and students, as it involves a disruption of prevailing rules and expectations of teaching and learning mathematics (Cai et al., 2015). To make progress in understanding how to integrate MPP in the classroom, it is necessary to understand which factors influence its implementation (Cai et al., 2015; Carmona-Medeiro & Climent, 2024). Although studies focused on ECE indicate the feasibility of introducing MPP (Carmona-Medeiro et al., 2024; Fosse et al., 2020; Martín-Díaz et al., 2020; van Bommel et al., 2024), little is known about the difficulties experienced by ECE students during MPP. The study conducted by Carmona-Medeiro and Climent (2024) on understanding how a teacher manages an MPP task in an ECE classroom (4–5 years) can be considered a direct antecedent of this study. The study reveals that most of the students’ difficulties involving teacher scaffolding are related to the students’ prevailing non-mathematical meaning of problems.
It should be noted that, at this age, children are in the process of learning how to read and write, for which reason the problem situation should provide information in a register other than the written one. Therefore, the choice of the register to be used to provide the problem situation is not random. Obtaining information from the students may represent a potential difficulty linked to the different registers that can be used for the choice of the problem situation. Most of the studies on MPP in ECE (Carmona-Medeiro et al., 2024; Martín-Díaz et al., 2020; van Bommel et al., 2024) provide the problem situation using registers that encourage manipulation and/or visualisation. It is known that manipulative and visual experiences are essential as a basis for verbal and cognitive development at these ages (Smith et al., 2011). However, little is known about the difficulties students experience in obtaining and communicating information from the problem situation.
Analysing problems posed by students is not only an avenue for assessing mathematical performance, it can also be an indicator of their emerging difficulties (Cai et al., 2015). Multiple approaches that analyse problems posed by students (Cai & Hwang, 2023; Carrillo & Cruz, 2016; Leavy & Hourigan, 2024; Palmér & van Bommel, 2020) have been considered in this study. Adaptations were made because of the peculiarities of children. Thus, with the focus on children’s difficulties, three variables are considered to characterise the problems posed: (i) mathematical nature; (ii) incomplete problems; and (iii) inconsistent problems.
The “mathematical nature” variable is in line with the difficulty Cai (2022) mentions, and refers to whether the problem posed is set in a framework in which the mathematical concepts or procedures necessary to solve the (mathematical or non-mathematical) problem are located. The “incomplete problems” variable refers to the absence of some of the structural elements of the problem posed (information and/or question) (Carmona-Medeiro & Climent, 2024). Finally, the “inconsistent problems” variable addresses both the problems posed in which there is no logical relationship between the information and the question, and those that are unrelated to the current problem situation and/or prompt (Cai, 2022).
In light of recent studies (Baumanns & Rott, 2022; Cai & Rott, 2024), it is relevant to distinguish between the processes involved in MPP and the products resulting from it. The former refers to the cognitive, linguistic, and interactional actions students perform throughout the problem posing activity, whereas the latter concerns the structure and mathematical quality of the posed problems. This study considers both dimensions: the analysis of the posed problems and the emerging difficulties identified during the activity, each offering a complementary perspective on students’ engagement with MPP.

3. Methodology

This study uses the single case design as a methodological approach (Yin, 2009), considering the implementation of an MPP task as the unit of analysis. It is an intrinsic case study (Stake, 1995) that consists of a group of students in a problem posing lesson in an ECE classroom. It allowed for exploring and characterising students’ emerging difficulties during problem posing. The interest of the case lies in: (i) the uniqueness of the sample, 4–5-year-old children; (ii) the distance, in relation to the usual classroom habits, with respect to MPP based on a given situation; (iii) the lack of research on the difficulties of students under the age of six.
This paper explores the emerging difficulties of students related to posing mathematical problems based on a given situation. The study seeks to characterise the problems posed with the aim of highlighting difficulties linked both to the mathematical activity of MPP and to the idiosyncrasies of the educational stage. In line with recent research (Baumanns & Rott, 2022; Cai & Rott, 2024), the analysis considers both the processes students engage in during the task, through the observation of difficulties as they unfold, and the products they generate, through the problems they manage to pose. This dual perspective enables a more comprehensive understanding of their engagement with MPP.

3.1. Context and Participants

The classroom analysed in this study involves a teacher, referred to as Elena (pseudonym), and 20 children aged between 4 and 5 in the second year of ECE in a public school in a small town in south-western Spain. At the time of the study, Elena had 15 years of experience and was well acquainted with the problem-based learning (PBL) approach, as she was an active member of a collaborative research group with a particular interest in the reflection on and implementation of learning situations related to PBL. Although the PBL approach was common practice in the classroom, MPP based on a given situation was a practice that was far removed from the usual classroom habits.

3.2. Description of the Lesson

The starting point of the MPP task implemented is a problem situation provided by means of a poster (Figure 1). The problem situation presented in the poster represents daily life in a neighbourhood. The information the poster provides includes individuals (pedestrians, workers, a policeman, vendors, customers, drivers, animals, etc.), objects (vehicles, clothes, fruit, urban infrastructure, safety elements, elements of nature), actions (buying and selling, moving around, directing traffic, tree pruning, etc.), and spaces (shops, houses, urban spaces, green areas, etc.).
The activity begins with gathering the students in an assembly in which Elena reminds them of the basic rules of participation and tells them a brief story about the Fields medal to contextualise the mathematical activity. The poster, initially covered with a cloth, is used as a resource to arouse curiosity in the students. Once the poster has been uncovered, Elena gives the instructions for the activity and guides the students in identifying and recognising the elements that make up the poster. The students participate spontaneously, naming the aspects of the poster that are interesting or familiar to them. After the initial exploration of the problem situation, Elena returns to the instructions, stressing that the aim of the task is to invent problems based on the poster.
Throughout the session, although the problem situation remains unchanged, the teacher makes several modifications to the prompt. The original task, referred to as Task 0, thus evolves because of the modifications the teacher makes to the prompt, giving rise to eight tasks. Table 1 provides the tasks shown according to the type of prompt used.
With Elena’s help, the students’ contributions progressively evolve into a coherent story, together with an appropriate question. The task runs for approximately an hour, and once most of the students have had a chance to pose a problem, the activity is considered completed.

3.3. Data Collection and Analysis

As the students are in the process of learning how to read and write, posing problems is done orally, and video recording was chosen as an instrument to collect data. It allows collecting both verbal and gestural data from the students. The session analysed was designed by the teacher with the support of a collaborative research group in which she participates together with teachers from other educational levels and educators in Didactics of Mathematics, including one of the authors of this paper. The implementation of the session is observed in a non-participant manner. It is video recorded, placing a camera in the back of the classroom so as not to interfere with the classroom activity.
In order to analyse the students’ emerging difficulties during the exploration phase of the problem situation, coding was performed freely, without using any instrument or resorting to a priori categories other than the perspective provided by the theoretical framework. This allowed us to group the students’ difficulties and construct emergent categories following an iterative coding process using the constant comparative method (Bryman, 2001). The emergent analysis of the data followed the grounded theory approach, a method commonly used in qualitative research (Charmaz, 2008). During the coding process, through grouping by units of meaning, different categories and indicators emerged. They were refined during and after the subsequent analysis (Strauss & Corbin, 1990), following the meaning condensation approach proposed by Kvale (1996). It was validated through investigator triangulation (Flick, 2007). The detailed analysis up to the first elicitation of categories was performed by the first author of this paper within the framework of his doctoral thesis, and was then discussed with the second author. The analysis of the students’ actions led to the identification of two emergent indicators linked to the students’ difficulties in exploring the problem situation (Table 2).
The problems formulated by the students were transcribed and subjected to an analysis of the three variables described: mathematical nature (yes/no), incomplete problems (absence of information/absence of question), and inconsistent problems (there is no logical relationship between the information and the question/the problem posed is not related to the problem situation and/or current prompt). This led to the identification of three other difficulties, this time associated with the problem posing phase itself (Table 3).

4. Results

The problem posing session takes place in three phases: (i) presentation and contextualisation of the activity, (ii) exploration of the problem situation, and (iii) problem posing. Table 4 shows the students’ difficulties in the last two phases, as Elena, the teacher, plays the main role in the first phase.
The analysis of the students’ actions performed on the information units allowed identifying five indicators (D1–D5) linked to the difficulties the students experienced during the problem posing task. The first two indicators (D1 and D2) refer to the difficulties the students experience during the exploration of the problem situation, while the remaining indicators (D3–D5) refer directly to the difficulties they experience during the problem posing process. The frequency of occurrences (ni) in which each type of difficulty was identified is presented below (Table 4):

4.1. Students’ Difficulties During the Exploration of the Problem Situation

Difficulties related to the recognition of the elements that make up the problem situation (D1) occur in three situations: (i) difficulty in distinguishing between objects that have some similarity in form or function (ni = 6); (ii) difficulty in recognising the specific meanings of road signs (ni = 2); and (iii) difficulty in recognising the gender of people (ni = 2). Some examples are shown below:
631P“I see a drawing of a child”
72P“Here is another one” [the student says there is another traffic light while she points to a streetlamp]
49P“I see a red circle”
198P“A woman with a small bag…”
The first two examples correspond to the difficulty in distinguishing between objects that have a certain similarity in form or function (i). In the first case, when the student confuses an advertisement with a drawing, the difficulty comes from not understanding the communicative intention of the advertisement. In the second case, when the student confuses a streetlamp with a traffic light, the difficulty arises from not identifying the functional difference (one illuminates, the other regulates traffic). In the third example, there is a difficulty in recognising the specific meanings of traffic signs (ii). In this case, the student sees the “no entry” sign only as a red circle without understanding its specific meaning. The last example shows the difficulty in recognising the gender of people (iii). The student confuses an older man with a woman, showing that, at this age, they do not yet have a clear outline of the subtle physical differences between men and women. Factors such as clothing, hairstyle, or posture may influence their perceptions.
Difficulties related to naming or communicating the elements of the problem situation (D2) are manifested in the lack of accuracy in the use of language. Some examples are given below:
13P“I see a truck cutting a tree.”
25P“It is not allowed there.”
40P“It is a post…for the light.”
62P“He is telling him he cannot pass.”
149P“Because it can blow very hard.”
501P“There are four.”
551P“There are none.”
50P“There, there” [pointing to the poster] [Joint reply]
19P“Down”
20P“On the left”
In the first case, the student attributes the action of cutting the branch to the truck instead of to the worker. The child observes the crane truck and the worker with a saw, but does not notice that the truck only holds the platform. In the second case, the student expresses the impossibility of performing an action without specifying it. In the third case, the student describes the object by looking at one of its parts and its function, but does not know the appropriate word for the object (“streetlamp”). In the fourth and fifth examples, the student expresses an action without specifying who performs it. In the sixth and seventh cases, the student expresses a quantity of measurement without specifying which number and/or unit of measurement the quantity of measurement refers to. The last three examples correspond to students’ difficulties in communicating the position of an element on the poster. This difficulty is significant because of its frequency of occurrence (ni = 24). The first example corresponds to the use of gestural language to communicate the position (ni = 11), while the remaining examples correspond to the use of relative positions (“up-down, right-left”), which are insufficient to locate the element on the poster (ni = 11).

4.2. Students’ Difficulties During the Problem Posing Process

The following section deals with the students’ emerging difficulties with regard to each of the three variables used.

4.2.1. Posing Non-Mathematical Problems

The analysis of the units of information related to indicator D3 led to the identification of three emergent descriptors (D3-1, D3-2, and D3-3). These descriptors are presented below (Table 5) together with their frequency of occurrence (ni):
The problems posed by the students are successively oriented towards non-mathematical situations, showing that, at this age, the notion of a mathematical problem is still incipient, and the notion of a personal problem prevails. In the first two cases (D3-1 and D3-2), the logic underlying the notion of personal problems implies it is not necessary to provide a question or information of a mathematical nature. Some examples of non-mathematical problems posed by students are provided below. The first three examples correspond to situations in which the students orient by posing the problem towards accidents or catastrophes (D3-1), while the remaining ones correspond to situations where the students orient by posing the problem towards situations that make it difficult or impossible to achieve an end (D3-2):
79P“A woman walking around with a bag… she runs away and gets hit by a car.”
161P“The firemen over there… want to get out of the fire station and they’re going to damage that corner because they’re going to hit it with the ladder.”
211P“What if the man who is cutting the tree drops a branch and it accidentally hits the head of the man sitting?”
156P“A woman walking around with a bag… She has to get home and doesn’t know the way.”
123P“This woman wants to go somewhere … but the policeman won’t let her pass… because the road is blocked. So she can’t get through.”
281P“A woman with a dog wants to go shopping for clothes. She wants to buy a waistcoat, but she doesn’t like it.”
In the last case (D3-3), although the students provide partial information of a mathematical nature and/or a question, the problems seem to be oriented towards exploring reasons or motives behind an action:
123P“The girl has two loaves of bread, why did she buy two loaves of bread?”
146P“The paper is flying away… Why is that boy’s paper flying away? [pointing to the sign in the pharmacy window]”
156P“A woman with a dog wants to go shopping for clothes. What does she want to buy?”
In all cases, the students identify an action or situation and formulate a question that invites a connection between the action and a reasonable reason (cause and effect relationship). The evolution from Task 0 to the subsequent tasks was a response to the students’ emerging difficulties (D3-1, D3-2, and D3-3) and the need to create a viable environment for problem posing. The teacher’s strategy focused on modifying the level of structuring of the initial task (free prompt) by progressively increasing both the constraints and the information provided through new prompts (structured prompts) that guided the tasks toward more structured situations. The modifications made to the initial task (Task 0) had an impact on the problems posed by the students, significantly reducing the emerging difficulties (D3-1, D3-2, and D3-3).

4.2.2. Posing Incomplete Problems

The analysis of the units of information related to indicator D4 led to the identification of two emergent descriptors (D4-1 and D4-2). These descriptors are presented below (Table 6) together with their frequency of occurrence (ni):
Throughout the session, numerous situations are identified in which it is clear that the problems the students pose are incomplete, either because the information about the problem is incomplete, because data are missing (D4-1), or because there is no question that requires mathematical thinking (D4-2). In general, these are partial advances in posing problems.
Incompleteness of information in the problem (D4-1) occurs when the students make partial progress in formulating the information of the problem. In all cases where incompleteness of information is identified, the absence of a question (D4-2) is simultaneously observed. Some examples in which it is clear that the students have difficulties in formulating a mathematical problem (information and question) and in which their interventions reflect only partial progress in the development of the information of the problem are provided below:
214P“That man wants to go to all the shops over there…”
270P“A woman with a dog… wants to go shopping for clothes…”
424P“That mum wants to buy fruit for her son.”
491P“There is only one apple, one pear, one orange, and one grape.”
The absence of a question in the problem posed (D4-2), in addition to being observed in cases where only a partial advance of the information of the problem is produced (ni = 33), occurs in situations where the information of the problem is incomplete (ni = 15). Some examples are shown below:
137P“The woman bought two loaves of bread and wanted to buy three…because there were…no more loaves of bread in the shop.”
427P“That mum wants to buy fruit for her son. She wants an apple, a pear, and a grape.”
474P“That window has the shape of a square.”
608P“A woman with a dog wants to buy a hat. The hat costs 10€ and the woman only has 1€.”
In all cases, the students give the information about the problem, providing a story (buying and selling), and/or data relating to mathematical quantities (discrete quantities or shapes). However, they do not ask any questions. The various modifications made by the teacher to the initial task (Task 0) had no impact on the emerging difficulties (D5-1 and D5-2).

4.2.3. Posing Inconsistent Problems

The analysis of the units of information related to the problems the students pose led to the identification of two descriptors (D5-1 and D5-2) linked to the inconsistency of the problems (D5). These descriptors are presented below (Table 7), together with their frequency of occurrence (ni):
The absence of a logical relationship between the information and the problem question (D5-1) occurs when the students make partial progress in the development of the information of the problem and formulate a question that is inconsistent with the story elements developed:
248P“A woman goes to buy bread…How many cars are there?”
Throughout the session, although the problem situation remains unchanged, the teacher makes several modifications to the prompt. These modifications to the original task (Task 0) represent a difficulty for the students. This shows that, although the teacher modifies the prompt and considers the previous tasks as completed, the students continue to initiate the posing of problems in relation to previous prompts (D5-2). This was observed, for instance, during Task 2, where the prompt is given by the instruction to pose a problem based on a story that has already started:
118T“…Let’s help Adrian with the first part of the problem: a woman goes to buy bread …. Can anyone think of any questions or stories we need to consider?”
The students start posing a problem that is related to previous prompts:
123P“The car goes…”

5. Conclusions

The results obtained in this study provide evidence of the difficulties experienced by ECE students (4–5 years) in posing mathematical problems based on a given situation. Five key indicators of difficulty were identified and characterised. Two of the difficulties occur during the exploration phase of the problem situation: difficulty in recognising the elements of the problem situation (D1), and difficulty in naming and communicating these elements (D2). The remaining three occur during the formulation phase: posing problems of a non-mathematical nature (D3), posing incomplete problems (D4), and posing inconsistent problems (D5).
With regard to the exploration phase of the problem situation, the data show that students find it difficult to accurately communicate the elements (objects, subjects, actions, spaces, etc.) they observe in the problem situation (D2). This difficulty can be explained by the fact that, at this age, they are still developing their language skills and, therefore, still consolidating their vocabulary, their ability to categorise, and their accuracy in the use of language. Previous research (Cai, 2022; van Bommel et al., 2024) pointed out that the representational register used in providing the problem situation may influence the students’ understanding. The findings of this study suggest that providing the problem situation by means of a complex picture may be a significant factor for exploring difficulties. The difficulties observed during this phase (D1 and D2) stress the need to include the exploration phase of the problem situation as an inherent phase of problem posing (L. Zhang et al., 2022) in ECE, as indicated in previous studies by Carmona-Medeiro et al. (2024) and Martín-Díaz et al. (2020).
In the problem posing phase, it was observed that a significant number of students posed problems of a non-mathematical nature (D3). This difficulty has been reported in research on other educational levels, and has been identified as one of the challenges that need to be investigated in task management (Cai, 2022). In addition to confirming that this difficulty is also prevalent in ECE, the study characterises the problems posed, which enables the analysis of children’s logic in greater depth. The tendency of students to pose non-mathematical problems reinforces the idea that the notion of personal problem prevails at this age (Carmona-Medeiro & Climent, 2024). When posing problems is oriented towards accidents or catastrophes (D3-1), or towards situations that make it difficult or impossible to achieve an end (D3-2), the prevailing children’s logic does not consider it necessary to use information of a mathematical nature or a question to be answered (Carmona-Medeiro & Climent, 2024). When problem posing is oriented towards enquiring into motivations or reasons for an action (D3-3), the underlying logic implies that the question can respond to a natural logic, and its solution can be invented (Ayllón et al., 2010; Carmona-Medeiro & Climent, 2024).
Another relevant finding is the high number of incomplete problems (D4). This suggests that the students have difficulties in including all the structural elements of a problem (Malaspina, 2021). In most cases, initial formulations were fragmentary and represented only a partial advance in the development of the information on the problem (D4-1), and therefore did not include any questions. Even when students develop a story, the formulation of the question is not clear. The data show the need for teacher scaffolding to complete the story/information, or to formulate a question. All the mathematical problems the students pose have gone through previous formulations in which some of the structural elements of the problem are not addressed. They evolved thanks to the pedagogical help of the teacher, essentially through questions, and to the collaboration of the students. This difficulty has been pointed out in previous studies on MPP tasks at early ages, where the role of the teacher in the co-construction of problems together with students (Palmér & van Bommel, 2020), the establishment of mathematical demands (Carmona-Medeiro & Climent, 2024), or the modification of the variables of the task (problem situation and/or prompt) (Carmona-Medeiro et al., 2024) to encourage the students’ progress in problem posing are highlighted.
Finally, the low number of inconsistent problems (D5) indicates that most students managed to maintain a certain degree of coherence between the information and the question. This suggests that, despite the difficulties, students have an intuitive understanding of the basic structure of a problem (Lowrie, 2002), although they still require support to make mathematical relationships explicit in their statements.
The modifications made to the task, aimed at increasing its level of structuring, proved useful for redirecting students’ problem posing towards mathematically meaningful situations. However, these adjustments had no impact on students’ difficulties in formulating complete problems. This suggests that, while increasing task structuring can promote the mathematical nature of the posed problems, overcoming challenges related to the inclusion of all the structural elements of a problem requires additional interventions. In this regard, we concur with Jastrzębowska (2023) and Chang (2007) that problem posing is viable within an environment of dialogue, emotional support, and progressive scaffolding, where teacher mediation plays a key role in consolidating the necessary competencies. Furthermore, Lowrie’s (2002) study demonstrates that first-grade children (6 years old) are capable of generating mathematically meaningful problems when provided with a supportive and stimulating environment, reinforcing the importance of pedagogical context in fostering problem-posing abilities from early ages.
Our analysis also suggests that starting from a non-structured situation accompanied by a free prompt can be a powerful didactic strategy when the aim is to access and understand students’ prevailing notions of what constitutes a problem.
This study identifies and characterises the emerging difficulties of ECE students during the activity of posing problems based on a given situation. It is believed that the indicators and descriptors provided in this study may be useful in other research on problem posing in ECE. Although further studies on how to implement and manage MPP tasks are needed (Cai, 2022; Carmona-Medeiro & Climent, 2024; English, 2020; Li et al., 2022), this study could provide teachers with guidance on the obstacles and opportunities inherent in teaching and learning through MPP (Cai et al., 2015; H. Zhang & Cai, 2021).

Author Contributions

Conceptualization, E.C.M. and N.C.; Methodology, E.C.M.; Validation, E.C.M. and N.C.; Formal analysis, E.C.M.; Investigation, E.C.M.; Resources, E.C.M.; Data curation, E.C.M.; Writing—original draft, E.C.M.; Writing—review & editing, E.C.M. and N.C.; Visualization, E.C.M.; Supervision, N.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Department of Didactics, Faculty of Educational Sciences, University of Cadiz, grant number N/A.

Institutional Review Board Statement

Ethical review and approval were waived for this study, since the study focused on observations and analyses without direct interventions that could pose a risk to participants, and considering the national regulations in force in Spain.

Informed Consent Statement

Informed consent was obtained from all subjects involved in the study.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Acknowledgments

To the projects PID2021–122180OB-I00 “financed by MCIN/AEI/10.13039/501100011033, Spain, and “FSE Investing in your future””. To the project ProyExcel_00297 of the Regional Ministry of Economic Transformation, Industry, Knowledge, and Universities of Andalusia. To the teams of the researches groups DPD (HUM-462) of the University of Cádiz and DESYM (HUM-168) of the University of Huelva. To the Research Centre COIDESO of the University of Huelva.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:
MPPMathematical problem posing
ECEEarly childhood education

Note

1
The transcription records are labelled as follows: the number of each turn; the speaker (T for teacher and P for pupil); the transcription of each speaker’s utterance; accompanying gestures in square brackets.

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Education 15 00630 g001
Figure 1. Poster shown to the students for mathematical problem posing.
Figure 1. Poster shown to the students for mathematical problem posing.
Education 15 00630 g001
Table 1. Types of tasks according to the prompt used.
Table 1. Types of tasks according to the prompt used.
PromptTasks
Posing a problem related to the posterTask 0
Posing a problem related to the poster different to the previous onesTask 1
Posing a problem in continuing a storyTasks 2 and 7
Posing a problem based on a questionTask 3
Posing a problem using a wordTasks 4 and 8
Posing a problem using a quantityTask 5
Posing a problem based on a specific part of the posterTask 6
Table 2. Emergent indicators (D1 and D2) related to students’ difficulties in exploring the problem situation.
Table 2. Emergent indicators (D1 and D2) related to students’ difficulties in exploring the problem situation.
Indicators
D1. Difficulty in recognising the elements of the problem situation
D2. Difficulty in naming/communicating the elements of the problem situation
Table 3. Characterisation of problems posed by the students related to difficulties (D3, D4, and D5) during problem posing.
Table 3. Characterisation of problems posed by the students related to difficulties (D3, D4, and D5) during problem posing.
Indicators
D3. Posing problems of a non-mathematical nature
D4. Posing incomplete problems
D5. Posing inconsistent problems
Table 4. Emergent indicators linked to students’ difficulties (D1–D9) during the problem posing task, together with their frequency of occurrence (ni).
Table 4. Emergent indicators linked to students’ difficulties (D1–D9) during the problem posing task, together with their frequency of occurrence (ni).
Indicatorsni
D1. Difficulty in recognising the elements of the problem situation10
D2. Difficulty in naming/communicating the elements of the problem situation39
D3. Posing problems of a non-mathematical nature32
D4. Posing incomplete problems81
D5. Posing inconsistent problems6
Table 5. Emergent descriptors linked to posing non-mathematical problems (D4-1 and D4-2), together with their frequency of occurrence (ni).
Table 5. Emergent descriptors linked to posing non-mathematical problems (D4-1 and D4-2), together with their frequency of occurrence (ni).
Descriptorsni
D3-1. Orienting problem posing towards accidents or catastrophes11
D3-2. Orienting problem posing towards situations that make it difficult or impossible to achieve an end17
D3-3. Orienting problem posing towards enquiring into the motivations or reasons for an action4
Table 6. Emergent descriptors linked to posing incomplete problems (D4-1 and D4-2), together with their frequency of occurrence (ni).
Table 6. Emergent descriptors linked to posing incomplete problems (D4-1 and D4-2), together with their frequency of occurrence (ni).
Descriptorsni
D4-1. Incompleteness of information in the problem posed33
D4-2. Absence of a question in the problem posed48
Table 7. Emergent descriptors linked to posing inconsistent problems (D5-1, D5-2 and D5-3), together with their frequency of occurrence (ni).
Table 7. Emergent descriptors linked to posing inconsistent problems (D5-1, D5-2 and D5-3), together with their frequency of occurrence (ni).
Descriptorsni
D5-1. There is no logical relationship between the information and the question 1
D5-2. Initiate posing a problem that is not related to the current prompt5
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Carmona Medeiro, E.; Climent, N. Students’ Difficulties with Problem Posing in Early Childhood Education. Educ. Sci. 2025, 15, 630. https://doi.org/10.3390/educsci15050630

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Carmona Medeiro E, Climent N. Students’ Difficulties with Problem Posing in Early Childhood Education. Education Sciences. 2025; 15(5):630. https://doi.org/10.3390/educsci15050630

Chicago/Turabian Style

Carmona Medeiro, Enrique, and Nuria Climent. 2025. "Students’ Difficulties with Problem Posing in Early Childhood Education" Education Sciences 15, no. 5: 630. https://doi.org/10.3390/educsci15050630

APA Style

Carmona Medeiro, E., & Climent, N. (2025). Students’ Difficulties with Problem Posing in Early Childhood Education. Education Sciences, 15(5), 630. https://doi.org/10.3390/educsci15050630

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