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22 pages, 2696 KB

Open AccessArticle

How Children See Geometric Shapes: Eye-Movement Evidence of Developing Structural Reasoning

by Maryam Zolfaghari, Karl Wesley Kosko and Hadi Rahmati

Educ. Sci. 2026, 16(3), 377; https://doi.org/10.3390/educsci16030377 - 2 Mar 2026

Viewed by 713

This study investigates how young children’s geometric reasoning develops through the act of drawing, examining how their eye movements, actions, and verbal explanations interact to reveal emerging structural awareness. Grounded in a developmental framework of structural reasoning, the study extends this model from [...] Read more.

This study investigates how young children’s geometric reasoning develops through the act of drawing, examining how their eye movements, actions, and verbal explanations interact to reveal emerging structural awareness. Grounded in a developmental framework of structural reasoning, the study extends this model from static visual products to the dynamic processes involved in constructing geometric figures. Using an exploratory qualitative case study design, three children (ages 5.5–7.5) completed line, circle, and rectangle drawing tasks while their gaze and actions were recorded using mobile eye-tracking. Gaze data, video recordings, drawing product, and verbal responses were synchronized and analyzed frame by frame to examine gaze–action coordination. Analysis revealed a progression from partial structural awareness, where gaze remained embedded in action, to structural awareness, where gaze projected multiple steps ahead to coordinate global shape structure. Between these, an intermediate, process-oriented phase was identified, characterized by alternating gaze-in-activity and anticipatory fixations supporting local planning. These fine-grained gaze patterns reveal micro-level transitions in geometric reasoning that are not observable from final drawings alone. The study refines current models of geometric development by revealing how perceptual, representational, and embodied processes dynamically integrate during drawing, offering a more nuanced understanding of early structural reasoning and its implications for teaching geometry. Full article

(This article belongs to the Special Issue Exploring Mathematical Thinking in Early Childhood Education)

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26 pages, 678 KB

Open AccessArticle

The Enhancement of Number Sense Through the Interactive Reading of Mathematical Stories in Kindergarten

by Maryam Ghaith Almulhim and Taro Fujita

Educ. Sci. 2026, 16(2), 296; https://doi.org/10.3390/educsci16020296 - 11 Feb 2026

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Abstract

Developing children’s number sense is an important aspect of early mathematical education and has been the focus of multiple studies targeting the kindergarten stage. We investigated the impact of reading mathematical stories on the number sense of kindergarten children. A small-scale intervention study [...] Read more.

Developing children’s number sense is an important aspect of early mathematical education and has been the focus of multiple studies targeting the kindergarten stage. We investigated the impact of reading mathematical stories on the number sense of kindergarten children. A small-scale intervention study was conducted with 46 kindergarten children aged 5–7 years. The study employed a non-equivalent quasi-experimental design involving comparison and intervention groups. The intervention involved eight mathematical stories presented in interactive reading environments during their class storytime. Therefore, both the books and the interactive reading style were considered core components of the intervention in this study. A pretest, posttest, and delayed test measured the children’s number sense, and the resulting data was analysed with ANCOVA. The results showed the intervention to have a promising effect on their number sense: the experimental group significantly outperformed the control group on both the posttest and delayed test. We consider it important that teachers be encouraged to make the maximum use of kindergarten storytelling sessions to further children’s early mathematical understanding. With acknowledgement of the limited sample size and its implications for the statistical generalisability of the findings, this study should be regarded as an exploratory investigation that can inform and encourage future large-scale research. In addition, the findings offer meaningful pedagogical implications that may support teachers and curriculum designers in early childhood education and provide valuable insights into the potential effects of reading mathematical stories interactively with children in kindergarten in authentic classroom contexts. Full article

(This article belongs to the Special Issue Exploring Mathematical Thinking in Early Childhood Education)

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19 pages, 476 KB

Open AccessArticle

Dialogues in Play: Conversational AI and Early Mathematical Thinking

by Shaoru Annie Zeng

Educ. Sci. 2025, 15(11), 1516; https://doi.org/10.3390/educsci15111516 - 10 Nov 2025

Cited by 3 | Viewed by 1883

Abstract

As conversational artificial intelligence (CAI), including smart speakers, social robots, and dialogic learning apps, becomes increasingly present in children’s lives, its potential to support early mathematical thinking warrants closer attention. While existing research largely concentrates on literacy and language development, the role of [...] Read more.

As conversational artificial intelligence (CAI), including smart speakers, social robots, and dialogic learning apps, becomes increasingly present in children’s lives, its potential to support early mathematical thinking warrants closer attention. While existing research largely concentrates on literacy and language development, the role of CAI in early numeracy remains underexplored. This paper examines how voice-based CAI might contribute to informal mathematical thinking in early childhood. Adopting a conceptual lens, this paper synthesises existing theory and research to examine the potential roles of CAI in early mathematical learning. Grounded in sociocultural theory and dialogic pedagogy, this paper positions CAI as a potential mediator of early mathematical thinking through responsive dialogue. Four interrelated dimensions (child agency, cognitive scaffolding, mathematical language quality, and responsiveness and timing) are identified as a conceptual lens for evaluating how dialogue-based interactions with CAI may support or constrain young children’s mathematical thinking. Rather than framing CAI as a direct teaching tool, this paper explores its potential role as a dialogic partner in play-based numeracy and inquiry. The framework contributes to early mathematics education by highlighting both the promise and the limitations of CAI, offering guidance for research, technology design, and educational practice. It underscores the need for intentional, ethically informed integration of CAI that approximates the qualities of human dialogue while acknowledging current constraints in real-world use. Full article

(This article belongs to the Special Issue Exploring Mathematical Thinking in Early Childhood Education)

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23 pages, 753 KB

Open AccessArticle

Analysis of Preschool Teachers’ Dialogue with Children During Combinatorial Activities

by Tatjana Hodnik and Adrijana Mastnak

Educ. Sci. 2025, 15(11), 1489; https://doi.org/10.3390/educsci15111489 - 5 Nov 2025

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Abstract

This article deals with the quality of dialogue between preschool teachers and children during mathematical activities. Thoughtful and attentive dialogue captures children’s attention and allows them to reason, express thoughts and disagreements, and ask questions. The aim of the study is to determine [...] Read more.

This article deals with the quality of dialogue between preschool teachers and children during mathematical activities. Thoughtful and attentive dialogue captures children’s attention and allows them to reason, express thoughts and disagreements, and ask questions. The aim of the study is to determine how well preschool teachers are able to utilize the potential of a combinatorial problem to ask questions and promote children’s mathematical thinking. Video recordings of eleven preschool teachers formed the basis of the study. Qualitative and quantitative analysis of the transcribed recordings provided an insight into the quality of the dialogues. The results of the study show that the preschool teachers mainly focused on combinatorics during the dialogue, although they also frequently asked the children questions about numbers, as well as questions from a non-mathematical context. In the analyzed dialogues, there was a lack of focusing questions, problem-solving questions, and questions requiring the children to justify their explanations. Unlike other studies that mainly investigate children’s thinking when solving selected combinatorial problems, the present study highlights the role of the teacher as the main actor in the process of children’s mathematics learning. Full article

(This article belongs to the Special Issue Exploring Mathematical Thinking in Early Childhood Education)

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21 pages, 711 KB

Open AccessArticle

The Role of Spatial Reasoning in Growing and Spatial-Repeating Patterns in First and Second Graders’ Structural Development of Mathematics

by Joanne T. Mulligan

Educ. Sci. 2025, 15(11), 1479; https://doi.org/10.3390/educsci15111479 - 3 Nov 2025

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Abstract

A cross-sectional qualitative study examined how different pattern types accounted for wide variation in children’s Awareness of Mathematical Pattern and Structure (AMPS), illustrating how spatial and patterning skills are interrelated. An interpretive descriptive analysis of responses was conducted for two interview-based Growing Square [...] Read more.

A cross-sectional qualitative study examined how different pattern types accounted for wide variation in children’s Awareness of Mathematical Pattern and Structure (AMPS), illustrating how spatial and patterning skills are interrelated. An interpretive descriptive analysis of responses was conducted for two interview-based Growing Square Array (GA) and Spatial-Repeating Pattern (SP) tasks with 405 children from Grade 1 (n = 189) and Grade 2 (n = 216). Analysis of developmental levels of AMPS indicated that advanced multiplicative spatial structures were employed in GA patterns for 20% of Grade 1 and 35% of Grade 2 children, respectively. Responses to SP tasks extended beyond the ‘unit of repeat’ to the use of dynamic visualization, and orientation and transformation skills for 60% of children. Responses showing advanced structural features moderately increased at Grade 2 for both tasks. Micro-level analyses of illustrations of interview responses and from a Pattern Construction task, drawn from five case studies, revealed how multiplicative and transformation skills were utilized in forming repetitions and growing patterns in complex and novel ways. While the ‘unit of repeat’ is fundamental, the integration of more complex multi-dimensional patterning with spatial concepts can re-focus learning and pedagogy on establishing interrelationships between patterning and spatial concepts, and broader mathematical knowledge. Full article

(This article belongs to the Special Issue Exploring Mathematical Thinking in Early Childhood Education)

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33 pages, 3248 KB

Open AccessArticle

Playing Gebeta in Preschool: Informal Pathways to Early Numeracy Through Directionality and Bundling

by Oliver Thiel

Educ. Sci. 2025, 15(10), 1365; https://doi.org/10.3390/educsci15101365 - 14 Oct 2025

Cited by 1 | Viewed by 1965

Abstract

Preschool children develop essential mathematical concepts through play, yet little is known about how traditional board games can support this process. This small-scale microgenetic case study investigates how children unfamiliar with the Ethiopian game Gebeta (a mancala-type game) learn to play the game [...] Read more.

Preschool children develop essential mathematical concepts through play, yet little is known about how traditional board games can support this process. This small-scale microgenetic case study investigates how children unfamiliar with the Ethiopian game Gebeta (a mancala-type game) learn to play the game and what mathematical competencies emerge. Video observations were conducted with 5-year-olds in Norway as they engaged in repeated play sessions. Event logs and transcripts were analysed with a focus on children’s developing strategies and difficulties. The results show that children quickly understood the basic rules but faced challenges with directionality (i.e., maintaining the correct direction of play) and differentiating between different types of game moves. Over time, they demonstrated evident progress in following the rules. They began to treat groups of counters as units, moving from one-by-one counting to bundling and unitising. These developments align with Bishop’s fundamental activities locating and counting. We conclude that Gebeta provides a playful context in which children can practice directionality and bundling, both of which are crucial for avoiding common errors in later school mathematics, such as off-by-one counting errors and misunderstandings of the number system. The study suggests that Gebeta can be introduced in early childhood settings without simplifying the rules, providing an engaging resource for early mathematics education. Full article

(This article belongs to the Special Issue Exploring Mathematical Thinking in Early Childhood Education)

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28 pages, 2046 KB

Open AccessArticle

Virtual Versus Physical Number Line Training for 6-Year-Olds: Similar Learning Outcomes, Different Pathways

by Eva-Maria Ternblad, Maybi Morell Ruiz and Sonja Holmer

Educ. Sci. 2025, 15(10), 1350; https://doi.org/10.3390/educsci15101350 - 11 Oct 2025

Cited by 1 | Viewed by 867

Abstract

According to previous research, young children’s numeracy skills may be scaffolded by practicing on the number line. A number line estimation task (NLET) is often conducted with pen and paper, while linear number games are often implemented on a computer or a tablet. [...] Read more.

According to previous research, young children’s numeracy skills may be scaffolded by practicing on the number line. A number line estimation task (NLET) is often conducted with pen and paper, while linear number games are often implemented on a computer or a tablet. If and how the format—physical or digital—influences the accuracy of the estimations is, however, not well-known. If regarding NLET performance as dependent on specific strategies and hypothesizing that these strategies may be affected by the material used, we may also assume that different materials may either support or hinder children’s learning. In this paper, we explore whether training with a physical versus a virtual NLET game will affect children’s strategies when solving NLETs, and if these strategies relate to the accuracy of the estimations. Sixty-two 6-year-old children played an NLET game (virtual or physical) for three sessions, being scaffolded and guided by a researcher. NLET performance was measured by pre- and post-tests, as well as during the intervention. The results show that even if the condition did not significantly affect the children’s overall numeracy skills, the children in the physical condition did express more advanced strategies during the intervention. These strategies, in turn, predicted NLET performance. Full article

(This article belongs to the Special Issue Exploring Mathematical Thinking in Early Childhood Education)

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25 pages, 5460 KB

Open AccessArticle

Supporting Pre-School Children’s Number Learning Through Embodied Representation

by Jennifer Way and Katherin Cartwright

Educ. Sci. 2025, 15(9), 1170; https://doi.org/10.3390/educsci15091170 - 7 Sep 2025

Cited by 1 | Viewed by 3307

Abstract

Situated within the Embodied Learning in Early Mathematics and Science project in Australia, this paper explores the relationships between ‘embodied activities’ used by a preschool teacher and the children’s development of number sense over six months. Using an instrumental case study approach, qualitative [...] Read more.

Situated within the Embodied Learning in Early Mathematics and Science project in Australia, this paper explores the relationships between ‘embodied activities’ used by a preschool teacher and the children’s development of number sense over six months. Using an instrumental case study approach, qualitative data from multiple sources including self-reported data from the teacher, activity descriptions, two task-based interviews with nine children, and number-knowledge data extracted from a mathematics achievement assessment (pretest and post-test) was analysed. Pattern searching techniques across text and video revealed connections between the embodied activities implemented by the teacher and the children’s development of subitizing and counting skills, mathematical drawing, and number magnitude knowledge. We propose that attending to specific aspects of children’s physical development, particularly finger dexterity and drawing skills, in experiences that focus on representing number concepts, can support their development of number sense. Full article

(This article belongs to the Special Issue Exploring Mathematical Thinking in Early Childhood Education)

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