The Effect of Visual Reasoning on Arithmetic Word Problem Solving

Abstract

First-grade students often encounter challenges in understanding and solving arithmetic word problems due to their limited reading comprehension abilities. Despite these difficulties, students may employ arbitrary strategies, such as combining numbers based on specific keywords, even if they lack a full understanding of the problems. Research suggests that effective mathematical reasoning involves the use of visual mental representations during the problem-solving process. To address this, some studies have explored methods to enhance students’ comprehension of word problems. Building on this, the current study explores the impact of first-grade pupils creating visual representations of problem situations on their comprehension and the number of correct solutions. In a typical math class, 45 first graders received a paper-and-pencil task, and, in a visual context, they solved similar problems after reading and illustrating the situation. The findings reveal that while most participants correctly represented the problem situations through drawing, about half struggled to determine the numeric solutions. Nevertheless, the visual context led to an increase in the number of correct problem solutions compared to the normal context, suggesting the potential benefits of incorporating visual representations in enhancing comprehension and problem-solving skills.

1. Introduction

Problem solving holds a pivotal role in the primary school mathematics curriculum, aiming to bridge abstract mathematical concepts with real-world applications [1,2]. Building upon the foundational work of researchers [3,4,5], the conceptual knowledge essential for tackling elementary addition or subtraction word problems is articulated in terms of semantic relations within the quantitative information embedded in the problem text. These relations involve the cognitive processes of comparison, combination, and change.

Extensive research spanning the last few decades has been dedicated to unraveling the intricacies of the errors children encounter during the problem-solving journey. A nuanced understanding of problem solving as a multifaceted process, encompassing phases like comprehending and defining the problem situation, constructing a mathematical model, executing computational procedures, interpreting numeric outcomes, and evaluating the model, has been outlined [6,7,8]. Scholars [6,7,8] have postulated that faulty representation of the problem situation or the inappropriate selection of mathematical operations when determining the unknown element are common sources of errors among children. Additionally, authors [9,10] emphasize that a lack of comprehension of abstract mathematical linguistic forms embedded in the problem text can impede the creation of mental representations, consequently affecting the problem-solving process. Moreover, [3] contends that inadequate conceptual and procedural knowledge specific to certain problem situations contributes to poor performance in problem solving.

Noteworthy observations by [6,11] indicate that children often neglect real-world knowledge when approaching word problems. This tendency is linked to certain features of instructional practices, such as the prevalence of stereotyped word problems and the premature imposition of a formal approach to arithmetic problem solving.

The challenge intensifies for primary school pupils, who, in the early stages of learning to read, grapple with comprehending word problems. Despite a lack of understanding, some pupils initiate the problem-solving process by resorting to arbitrary strategies, randomly combining numbers present in the problem into mathematical operations guided by specific keywords, like “more” for addition and “less” for subtraction [11]. This phenomenon underscores the need for targeted interventions to enhance students’ conceptual understanding and foster a more informed approach to mathematical problem solving.

The gradual achievement of text comprehension is linked to the creation of mental images that simulate the described action or aspect within a sentence [12,13]. Specifically, understanding a word problem involves the formation of mental representations that encapsulate the intricacies of the problem situation conveyed by the textual content. Building on insights from mathematical reasoning research, it becomes apparent that mental representations of abstract mathematical concepts often take on a visual form, drawing upon one’s visually sensed experiences [14].

Recent research underscores the critical role of visualization in mathematical problem solving [15,16]. Mathematical visualization is conceptualized as the process of constructing visual representations that elucidate the relations between quantitative data presented in the problem text [17,18,19]. These visual representations can manifest either internally, as mental images derived from various real-life experiences, or externally, encompassing paper-and-pencil drawings, models, diagrams, physical manipulatives, or computer-simulated tools. Scholars such as [17,20] articulate visualization as a multifaceted construct, encompassing the ability, process, and product of creating, interpreting, using, and reflecting upon pictures, images, and diagrams. Whether in the mind or on paper or through technological means, visualization serves as a powerful tool for describing and communicating information, fostering thinking, and enhancing the understanding of new ideas.

Visual reasoning in mathematics, as described by [21], is characterized as a process that involves expressing verbal information through concrete visual representations, effectively illustrating the relationships between mathematical expressions and concepts. By promoting the utilization of visual reasoning in the learning of mathematics, comprehension is translated into one’s ability to leverage the provided information to effectively solve problems [22]. This emphasizes the integral role of visualization not only in understanding mathematical concepts but also in applying that understanding to practical problem-solving scenarios.

Recent endeavors to enhance pupils’ problem-solving abilities have witnessed diverse approaches aimed at fostering a deeper understanding of mathematical relations within word problems. In a notable study, [23] achieved significant success in improving the problem-solving performance of elementary school pupils by incorporating a hands-on approach. The method involved physical manipulation of objects to recreate the problem situation, facilitating the formation of accurate mental representations pertaining to the relations between quantitative information embedded in the problem.

Another line of inquiry, undertaken by [8], delved into the impact of representational illustrations accompanying word problems on the problem-solving process. The expectation was that such illustrations would assist pupils in mentally visualizing the situation, leading to more realistic problem-solving outcomes by leveraging everyday life knowledge. However, contrary to expectations, the findings did not reveal a positive effect on realistic problem solving.

In a more recent contribution, [24] highlighted the significance of visual approaches through examples of mathematical tasks that pose challenges and offer multiple solutions. The emphasis on visual strategies underscores the growing acknowledgment of the role of visualization in enhancing mathematical comprehension and problem solving. Recognizing the pivotal role of teachers in shaping the quality of mathematical instruction, [24] focused on pre-service teachers, arguing that teacher education programs should expose them to experiences mirroring those they will later facilitate for their students. This exposure is seen as instrumental in enhancing their understanding of the challenges associated with visualization.

Building on these insights, [25,26] investigated the utilization of dynamic visualization based on instruction, aiming to narrow the emotional and cognitive gaps between high- and low-performing individuals. The results of these studies make meaningful contributions to both theoretical and methodological aspects, offering practical insights into the effects of instruction-based dynamic visualization on the academic performance of mathematics students in both emotional and cognitive domains.

However, despite these advancements, a literature gap exists regarding the specific impact of visual reasoning, particularly drawing, on the problem-solving processes of first-grade students in the context of mathematical word problems. This study aims to fill this gap by exploring the effectiveness of instructing students to create visual representations through drawing in improving comprehension and problem-solving outcomes.

2. Materials and Methods

The study investigates the connection between visual reasoning, problem-solving processes, and cognitive dimensions in the context of mathematical word problems among first-grade students. Focusing on part-whole relations between sets of elements, integral to the first-grade mathematics curriculum, the research explores the impact of visual reasoning on students’ proficiency in problem solving, particularly involving addition and subtraction operations.

The rationale is grounded in the theoretical proposition that integrating visual reasoning methodologies into learning processes may act as a cognitive scaffold, fostering the development of accurate mental representations of mathematical relations. This is anticipated to contribute to enhanced comprehension and increased accuracy in solving mathematical word problems. The study aligns with educational psychology frameworks emphasizing the significance of visual aids in enhancing cognitive processes and understanding abstract mathematical concepts.

• Hypotheses:

H1: Enhancement of Comprehension through Visual Reasoning: The first hypothesis suggests that the incorporation of visual reasoning, specifically through drawing, serves as a cognitive scaffold for first-grade students, facilitating a deeper comprehension of mathematical word problems. Visual representations are posited as cognitive tools aiding in the formation of accurate mental images of problem relations.

H2: Correlation between Visual Representations and Correct Solutions: Building upon the first hypothesis, the second anticipates a positive correlation between creating accurate visual representations through drawing and subsequent accuracy in problem solutions. Students who are adept at translating mental images into precise visual representations are expected to exhibit an increased number of correct solutions, highlighting visual reasoning as a mediator between comprehension and problem-solving proficiency.

H3: Visual Proficiency among Students with Average Reading Comprehension Abilities: The third hypothesis targets the intersectionality between visual reasoning and reading comprehension abilities. It postulates that students with average reading comprehension abilities can create precise visual representations, potentially compensating for variations in reading comprehension abilities and influencing accurate problem solutions.

In summary, we propose to investigate and describe the correlation between students’ reading comprehension abilities and their visual and mathematical reasoning performances.

We expect that visual reasoning will help pupils form accurate mental representations of the mathematical relations in the problem, improving their comprehension of the problem situation and increasing the number of correct problem solutions afterwards, raising the following questions: can first-grade students’ comprehension of word problems be improved by instructing them to create visual representations of the problem situation through drawing? Does the number of correct problem solutions increase when students create accurate visual representations of the problem situation through drawing? Do students with average reading comprehension abilities create precise visual representations of the problem situation through drawing and subsequently determine correct problem solutions?

We hypothesized that asking first-grade pupils to create visual representations of the problem situation by drawing will improve comprehension, determining an increased number of correct problem solutions. Also, we predicted that pupils with average reading comprehension abilities would create correct visual representations of the problem situation leading them to perform the appropriate operations to determine the correct problem solution.

These hypotheses form the theoretical framework guiding the investigation into the dynamics of visual reasoning, reading comprehension, and problem-solving processes in first-grade mathematics education.

2.1. Participants

A total of 45 first-grade pupils, comprising 22 boys and 23 girls, actively participated in this experimental study, with their ages ranging between 7 and 8 years (mean age 7.13). The participants were selected from two distinct first-grade classes within the same urban primary school situated in Cluj-Napoca, Romania.

Prior to the commencement of the experiment, a comprehensive assessment of each pupil’s mathematical performances and reading comprehension abilities (RCA) was undertaken. This assessment was conducted globally by the respective teacher, who completed an individual form for each participant. The individual mathematical abilities (IMP) were assessed on a scale from 1 (poor) to 5 (high): 5 (27 participants, including 15 boys and 12 girls), 4 (10 participants, comprising 2 boys and 8 girls), 3 (6 participants, with 4 boys and 2 girls), and 2 (2 participants, encompassing 1 male and 1 female). The scoring for reading comprehension abilities ranged on a scale from 1 (poor) to 5 (high), with the following distribution: 1 (5 participants, involving 3 boys and 2 girls), 2 (3 participants, with 1 boy and 2 girls), 3 (8 participants, including 3 boys and 2 girls), 4 (16 participants, featuring 8 boys and 8 girls), and 5 (13 participants, comprising 7 boys and 6 girls). The validation of the grading scale was determined to have a Cronbach’s Alpha of 0.741 > 0.7.

The detailed breakdown of the distribution of mathematical and reading comprehension abilities among the participants is presented in

Table 1. Distribution of mathematical and reading comprehension abilities ranges towards the sample.

Gender			Age				Individual Mathematical Abilities (IMP)				Reading Comprehension Abilities (RCA)
Participants			Participants				Participants				Participants
Total	Boys	Girls	Years	Total	Boys	Girls	Score	Total	Boys	Girls	Score	Total	Boys	Girls
45	22	23	7	39	19	20	5	27	15	12	5	13	7	6
			8	6	3	3	4	10	2	8	4	16	8	8
							3	6	4	2	3	8	3	2
							2	2	1	1	2	3	1	2
							1	0	0	0	1	5	3	2

. This tabulated representation provides a clear overview of the diverse capabilities within the sample, accounting for gender-specific differences and the varying levels of proficiency in both mathematical and reading comprehension domains. The categorization ensures a nuanced understanding of the participants’ baseline competencies, laying the foundation for a rigorous investigation into the impact of visual reasoning on their problem-solving processes.

2.2. Method

The study employed a within-subjects design, with participants being individually tested in both normal and visual contexts. The problems presented in both contexts aimed to assess, compare and combine semantic relations between sets of objects, aligning with the study’s overarching objectives.

In the first stage of the experiment, hereafter referred to as the normal context, pupils engaged in an individual paper-and-pencil task during a standard mathematics class. They were presented with the word problem: “Radu has 3 pencils, and Tudor has 4 more pencils than Radu. How many pencils do children have altogether?” The teacher read the problem aloud once, and pupils were instructed to read it one more time, individually, and solve it independently, recording the solution procedure and answer on paper. During the task, the teacher provided no problem-solving suggestions to pupils. They were only advised to solve problems in the usual way as they did in their class work. Once all the pupils had completed their tasks, the teacher collected the tests and explained that the results would be discussed a few days later. There were no discussions on how pupils should address the problem, or which were the correct solutions they should provide.

In the second stage of the experiment, referred to as the visual context, another word problem was introduced during a regular mathematics class: “5 frogs are sitting on a water lily leaf, and 3 fewer frogs are sitting on the leaf nearby. How many frogs are sitting on the lily leaves altogether?” The problem was written on the board and read aloud by the teacher, as in normal context. Pupils were then instructed to read the problem one more time, independently and visually illustrate the problem situation through drawing, and subsequently perform the mathematical operations to determine the numerical solution on the back of the page and provide an answer to the problem’s question. Pupils were instructed to carefully read the problem and illustrate it, following the information from the problem statement. The teacher offered no suggestions or advice on how pupils should address the problem-solving process.

The collected data were analyzed using appropriate statistical methods to examine the impact of visual reasoning on comprehension and problem-solving abilities in mathematical word problems. The within-subjects design allowed for a detailed exploration of individual differences, contributing to the robustness of the study’s findings.

3. Results

The results section outlines distinct categories of problem solutions observed in both normal context (NC) and visual context (VC) among first-grade students. These categories include Correct Problem Solution (CPS), Solution Error (SE), and No Answer (N/A) in the normal context. In the visual context, pupils’ drawings are categorized into Correct Visual Representation (CVR) and Representation Error (RE), with further distinctions made for contextual and structural aspects.

In the examination of problem-solving processes, a comprehensive taxonomy was developed to categorize responses within both a normal context (NC) and a visual context (VC). Three distinct categories of problem solutions emerged:

Correct Problem Solution (CPS): Final CPS (fCPS): Participants achieved a correct problem solution by exclusively performing the final addition operation (e.g., 5 + 2 = 7). This category specifically captures instances where the participants directly combined the given elements to ascertain the total, denoted as final CPS (fCPS). Complete CPS (cCPS): This subcategory encapsulates correct problem solutions achieved through a two-step process. Participants engaged in both subtraction (e.g., 5 − 3 = 2, determining the second set of frogs) and addition (e.g., 5 + 2 = 7, combining both sets of frogs). The designation complete CPS (cCPS) signifies the holistic understanding and execution of the problem-solving process.

Solution Error (SE): Partial Solution Error (pSE): Participants in this category exclusively performed the subtraction operation (e.g., 5 − 3 = 2) but failed to progress to the subsequent addition step. Alternatively, participants provided a numeric solution differing from the correct problem solution. This classification, termed partial solution error (pSE), denotes an incomplete or inaccurate problem-solving attempt.

No Answer (N/A): Participants who fell into this category did not provide any numerical solution or operational steps in response to the problem. The absence of a formulated answer is denoted as No Answer (N/A).

This categorization scheme allows for a nuanced analysis of the diverse ways in which first-grade students approach and respond to mathematical word problems, providing valuable insights into the specific nature of their cognitive processes and problem-solving strategies.

Within the visual context, the analysis of pupils’ drawings involved a systematic categorization to discern and interpret the diverse visual representations. The categorization was structured as follows:

Correct Visual Representation (CVR): Structural Correct Visual Representations (sCVR): Drawings falling into this category precisely depicted the numeric information embedded in the problem, effectively illustrating the relations between the two sets of elements (e.g., groups of five frogs and two frogs). This subcategory, labeled structural correct visual representations (sCVR), captured instances where pupils accurately conveyed the essential elements essential for solving the problem.

Contextual Correct Visual Representations (cCVR): Some pupils extended their drawings beyond the structural elements to incorporate contextual details, such as lily leaves, flowers, trees, and even additional frog families. While these contextual details enriched the visual representation, they were deemed as having no substantive relevance to the solving process. Thus, drawings falling into this subcategory were coded as contextual correct visual representations (cCVR).

Representation Error (RE): Contextual Representation Error (cRE): Drawings characterized by inaccuracies in illustrating the sets of elements needed for determining the total value were classified as representation errors (RE). Within this category, if the errors involved contextual visual details unrelated to the problem-solving process, the drawings were specifically coded as contextual representation errors (cRE).

This meticulous categorization framework contributes to a precise analysis of how pupils visually approach and interpret mathematical word problems. By distinguishing between accurate representations, additional contextual elements, and specific types of errors, the study gains valuable insights into the connection between visual reasoning, problem comprehension, and the quality of visual representations in the learning process.

Table 2. Descriptive analyses and correlations.

Variable	Mean	Minimum	Maximum	Std. Deviation	r	Correlation Variable
IMP	3.9778	1.00	5.00	1.33976	0.325 *	NC
RCA	3.6444	1.00	5.00	1.28197	0.408 *
NC	2.5500	2.00	4.00	0.87560	0.359 *	VC
sCVR	1.7381	1.00	2.00	0.44500	−0.345 *	sCVR
cCVR	1.5476	1.00	3.00	0.63255
VC	2.8529	1.00	4.00	0.98880

Note: * significant at p < 0.05.

presents the descriptive statistics and correlation coefficients for key variables, shedding light on the relationships between individual mathematical performance (IMP), reading comprehension abilities (RCA), and specific outcomes in both the normal context (NC) and visual context (VC) of problem solving.

The correlations suggest nuanced associations between IMP, RCA, and specific outcomes in both NC and VC. Positive correlations indicate a potential interdependence between variables, while negative correlations, as observed with sCVR, prompt further investigation into the dynamics of structural correct visual representations in the context of mathematical problem solving.

In normal context, we assumed that pupils’ understanding level of problem situations was associated with the amount of correct problem solutions (CPS) they provided. Solving problems in a visual context revealed an increased comprehension level of the problem situation, reflected by the amount of correct visual representations (CVR), which impacted the problem-solving process and the quality of problem solutions. Results obtained in a visual context underlined a significant correlation (p = 0.044 < 0.05) between the problem solutions determined in a normal context and the problem solutions determined in visual context (

Table 3. Correlation between CPS in a normal context (NC) and CPS in a visual context (VC).

		Correlation	p
Pair 1	NC and VC	0.359	0.044

). Findings evidenced significantly improved problem solutions (CPS) when pupils solved the problem in a visual context compared to problem solutions determined in a normal context. Pupils with higher RCA and IMP levels who determined CPS in a normal context maintained their performance solving the problem in a visual context. About a third of pupils that provided pSE in a normal context, most of them with very good IMP and medium RCA, determined CPS in a visual context, after creating correct visual representations (CVS) of the problem situation.

Analysis of the existing associations between the level of personal skills and abilities involved in the problem-solving process, such as reading comprehension abilities (RCA) and individual mathematical performance (IMP) assessed by the teacher, and the quality of visual representations and problem solutions determined by pupils in a normal context and a visual context was determined via an ANOVA test (

Table 4. ANOVA test.

Dependent Variable	Factor	F	p
IMP	sCVR	8.667	0.005
	cCVR	5.108	0.011
RCA	sCVR	6.336	0.016
	cCVR	7.340	0.002
	NC	4.635	0.016

).

A statistically significant relationship was observed between individual mathematical performance (IMP) and the accuracy of visual representations. Pupils assessed with a very good IMP exhibited significantly higher correctness in both structural correct visual representations (sCVR) (p = 0.005 < 0.05) and contextual correct visual representations (cCVR) (p = 0.011 < 0.05). This emphasizes the integral role of individual mathematical proficiency in generating precise visual depictions of problem-solving scenarios.

Further, a statistically significant relationship emerged between pupils’ reading comprehension abilities (RCA) and the quality of visual representations in the visual context. Pupils with higher RCA, reaching a level of 5, demonstrated a substantial correlation with both sCVR (p = 0.016 < 0.05) and cCVR (p = 0.002 < 0.05). This indicates that enhanced reading comprehension abilities contribute positively to the creation of correct visual representations, encompassing both structural and contextual aspects.

The ANOVA test highlighted a statistically significant relationship between the normal context (NC) and certain problem-solving outcomes (F = 4.635, p = 0.016 < 0.05). This suggests that the way pupils approached problem solving in the normal context had an impact on subsequent outcomes, emphasizing the importance of considering different problem-solving scenarios.

4. Discussion

The examination of participants’ problem-solving processes in the visual context reveals intriguing nuances that extend beyond a straightforward correlation between correct visual representations (CVR) and subsequent correct problem solutions (CPS). Despite a heightened percentage of participants with CVR, a notable one-third faced challenges in determining CPS in the visual context.

An in-depth analysis of this subset of participants with CVR who provided a solution error (SE) exposes a distinctive pattern in their problem-solving approach. These participants demonstrated difficulty in executing the requisite mathematical operations, relying on a simplistic combination of numbers through subtraction, as prompted by the keyword “less” in the problem. This suggests a potential limitation in their ability to translate visual representations into accurate mathematical solutions, raising questions about the extent of their mathematical comprehension.

The correlation between pupils’ reading comprehension abilities and the quality of visual representations in the visual context underscores the notion that a higher quality of understanding of the problem situation corresponds to more accurate visual representations. However, a crucial insight emerges—comprehension alone does not guarantee the ability to determine the numeric solution. Some first-grade participants exhibited challenges in grasping the task or the question posed by the problem, hindering their capacity to associate and execute the necessary mathematical operations.

Further complexity arises when considering participants who successfully rendered the problem situation through drawing but faltered in providing the correct problem solution in the visual context. This subset was prompted to reflect on their drawings and offer an answer to the question directly. Initially, these participants resorted to a subtraction approach (5-2) prompted by the keyword “less”. However, through repeated guidance, they eventually recognized the potential of leveraging their drawings to arrive at the correct numeric solution. This process highlights the significance of scaffolding in guiding students to effectively utilize visual aids in problem solving.

Drawing parallels with scholarly works [1,11] that discuss the suspension of logical thought, some participants in our experiment seemed constrained by the formal arithmetic conventions learned in school. This observation underscores the importance of adopting a more creative approach to teaching word problems, moving beyond stereotyped methodologies that rigidly adhere to predefined steps.

This study augments existing scholarship by addressing specific lacunae in the literature related to early-grade mathematical problem solving. Unlike previous research that has primarily focused on establishing correlations between visual representations and problem-solving outcomes, our investigation ventures into the nuanced challenges that impede certain participants from translating accurate visual depictions into correct problem solutions. In doing so, we shed light on complexities that extend beyond conventional correlations.

Furthermore, the study pioneers in exploring the relationship between reading comprehension abilities and the subsequent determination of mathematical solutions. While past literature has acknowledged the significance of comprehension, our findings highlight instances where a strong comprehension does not guarantee successful problem solution determination, revealing a notable gap in current discourse.

Moreover, this research unveils a less-explored facet concerning the impact of formal arithmetic education on problem-solving approaches. Specifically, it identifies a subgroup of participants whose reliance on conventional subtraction approaches signals potential limitations in transcending the formal arithmetic bounds imparted in school. This observation underscores a crucial gap in the understanding of how formal education may shape problem-solving strategies among early-grade students.

Lastly, our research provides valuable insights into the effectiveness of scaffolding techniques in guiding students who initially struggle to connect visual representations with problem solutions. This nuanced exploration of scaffolding in the context of mathematical problem solving contributes specific and tangible insights to an underexplored aspect of the literature.

In line with the findings of [27], our study underscores the significance of integrating visual reasoning methodologies into learning processes as a cognitive scaffold. The authors of [27] emphasize the importance of mathematical representations in solving non-routine problems, suggesting that the association between generating representations and creating inferences imposes a high cognitive demand on learners. Moreover, [27] highlights the necessity of allowing students adequate time to develop their mental models of given problems. These insights align with our study’s emphasis on cognitive scaffolding and underline the role of time in facilitating deeper engagement with visual reasoning tasks. The mention of representational resources in [27] offers practical guidance on implementing visual reasoning methodologies effectively. This highlights the complexity of learning through visual reasoning and underlines the necessary conditions for its success, including the strategic use of various representational resources to manage complexity.

5. Conclusions

In conclusion, this study contributes significantly to the extant literature on early-grade mathematical problem solving by delving into specific intricacies that previous research has not extensively addressed. Our findings underscore the importance of scrutinizing the translation of visual reasoning into correct problem solutions, revealing that a mere correlation between visual representations and outcomes does not encapsulate the diverse challenges encountered by young learners.

The observed discrepancy between participants’ ability to create accurate visual representations and their subsequent failure to determine correct problem solutions aligns with previous studies emphasizing the nuanced nature of early-grade mathematical cognition [28,29,30,31,32]. While past research has acknowledged the importance of visual representation, our study brings to light the necessity of nuanced analysis to unravel the factors inhibiting successful translation of these representations into accurate problem solutions.

Moreover, the identified subgroup of participants who struggled to move beyond formal arithmetic approaches echoes concerns raised by previous scholars [28,29]. This observation points to a potential gap in the alignment between formal arithmetic education and the diverse problem-solving strategies exhibited by students. Our results contribute a specific context to this ongoing discourse, suggesting the need for a more comprehensive examination of the impact of formal education on the problem-solving repertoire of early-grade students.

Furthermore, the study advocates for a paradigm shift in educational practices, echoing the sentiments of scholars who emphasize the need for creativity in approaching mathematical problem-solving tasks [30,31]. The limitations observed when participants adhere strictly to stereotyped methodologies signal the importance of fostering flexibility in problem-solving approaches, aligning with the broader discourse on pedagogical strategies in early-grade mathematics.

Lastly, our research extends the conversation on scaffolding techniques by demonstrating their effectiveness in guiding students initially struggling to connect visual representations with problem solutions. This aligns with previous studies emphasizing the utility of scaffolding in supporting learners [33,34,35,36,37]. Our specific insights contribute to this ongoing discussion by illustrating how scaffolding can be tailored to address the unique challenges encountered by young students in the domain of mathematical problem solving.

In essence, this study’s exploration of early-grade mathematical problem solving expands the boundaries of current understanding and underscores the need for targeted interventions and instructional adjustments to enhance the efficacy of educational practices in this critical developmental stage.

While our research has provided valuable insights into the impact of visual reasoning on problem-solving abilities within this specific cohort, caution is warranted in extrapolating these findings to broader populations. The limited sample size may restrict the extent to which our results can be applied to other settings or populations, emphasizing the need for further research with larger and more diverse samples to validate and extend our findings.

Exploring the cognitive processes involved in the transition from visual representations to formal arithmetic during early-grade mathematical problem solving constitutes a critical area for future research. A longitudinal approach could be adopted to trace the developmental trajectory of visual reasoning skills, offering valuable insights into the sustained impact of early interventions on later mathematical abilities. Moreover, investigating individual differences in students’ responses to visual reasoning interventions would allow for tailored instructional strategies, promoting inclusivity in educational practices.

Incorporating technology-enhanced interventions to enhance visual reasoning and mathematical problem-solving skills opens new avenues for research, presenting innovative approaches that align with the demands of contemporary classrooms. Additionally, future studies could delve into the effectiveness of professional development programs for teachers, focusing on seamlessly integrating visual reasoning strategies into the curriculum. This avenue holds promise for advancing pedagogical practices and ensuring teachers are well-equipped to nurture students’ mathematical problem-solving skills.

A cross-cultural perspective on visual reasoning strategies presents another intriguing research direction. Examining potential variations across different cultural contexts enriches our understanding of whether visual reasoning skills are universal or influenced by cultural factors. This research could contribute to the development of culturally sensitive educational interventions and resources.

These suggested research directions collectively aim to deepen our understanding of the relationship between visual reasoning and early-grade mathematical problem solving. By addressing these avenues, future research endeavors can refine educational practices, providing valuable insights for educators, policymakers, and researchers seeking to optimize learning outcomes in mathematics.