Fostering Reflection and Attention to Enhance Struggling Students’ Mathematical Problem Solving—A Case Study

Abstract

Research has shown that attention plays a crucial role in developing mathematical problem-solving skills, particularly for students who struggle with non-routine tasks. Even basic operations require shifts in attention, underscoring the deep connection between attention and mathematical cognition. Attentional strategies are observable and can be developed with targeted scaffolding. This study aimed to enhance high school students’ attentional engagement in problem-solving through a structured intervention. Over an academic year, twelve struggling students in Grades 11 and 12 participated in three one-on-one sessions with a researcher, receiving focused instruction. These sessions encouraged reflection and attention by using the “CCRSRC” model: Connections (identifying similarity connections among the problems presented); Choice (the student deciding which problem to solve); Reflection (explaining the choice); Solving (an attempt is made); Repetition (repeating steps 1–4 as often as wished); and Choice (to end the repetition and move on). Mason’s theory of shifts of attention was used to examine learners’ attentional development. This article provides a detailed analysis of one intervention case, offering insight into how CCRSRC actions serve as catalysts for fostering learner attention. In addition to describing and characterizing a single case, the article summarizes the attention data of all learners involved in the individual intervention.

1. Introduction

Research in cognitive psychology emphasizes that working memory capacity and attentional control are crucial for solving math problems, as they enable learners to focus on relevant information and avoid distractions (Wiley & Jarosz, 2012). In arithmetic tasks, understanding numbers is closely linked to attentional processes (Masson & Pesenti, 2016). Although various cognitive and metacognitive frameworks have been proposed to support struggling learners, many interventions are general and lack structured, repeatable methods tailored to individual learning situations. More importantly, few studies provide micro-analytic insights into how specific teacher–student interactions influence attentional and cognitive shifts during problem-solving—an omission that limits our understanding of learning as it happens in real-time. Despite evidence connecting attention and math performance, the impact of cognitive frameworks on students with low attentional control remains under-explored (Cai et al., 2024). Experimental research supports this, showing that attentional patterns in numerical problem solving are affected not only by the task features but also by framing effects (Masson & Pesenti, 2016; Masson et al., 2024). Studies such as those by A. Mason and Singh (2010) show that explicit metacognitive scaffolding—particularly in peer-supported environments—can significantly improve students’ ability to monitor and adapt their problem-solving strategies.

To address this gap, the present study uses the CCRSRC model (Ovadiya, 2025), an intervention framework designed to promote mathematical problem solving through six structured phases that enhance attentional focus: Connections—building similarity connections (SCs) among presented problems, defined as the cognitive process of recognizing structural relationships and common mathematical features that facilitate transfer of solution strategies across problems (Ovadiya, 2025; Hiebert & Carpenter, 1992); Choice—selecting a problem to solve; Reflection—explaining the reason for that choice (reflection is an action that builds new knowledge (Ovadiya, 2024); Solving—attempting a solution; Repetition—optionally repeating steps 1–4; and Choice—deciding to stop and move forward. The model draws on insights from cognitive load theory, attention and reflection research, self-regulated learning, and mathematics-specific heuristics. It guides learners through a cyclical process aimed at problem solving (Ovadiya, 2025). This study applies the CCRSRC (Connection, Choice, Reflection, Solving, Repetition, Choice) model in a one-on-one intervention setting and conducts a detailed discourse analysis to examine how the CCRSRC model supports the learner’s attentional development for problem solving.

The analysis is guided by J. Mason’s (2010) theory of attentional shifts, which sees mathematical learning as dependent on learners’ ability to shift attention across different task aspects, such as from surface features to structural elements or from individual procedures to overarching strategies (J. Mason, 1989, 2010). These “delicate shifts of attention,” as Mason describes them, are rarely spontaneous in learners struggling with the material and often require carefully crafted instructional cues. These ideas support the CCRSRC model, which emphasizes not only generating solutions but also regulating attention and encouraging reflection as key parts of problem-solving. This study responds to recent calls for research that bridges theory and classroom practice through detailed micro-analysis of instructional discourse. As Erath et al. (2021, p. 458) state: “We need more studies that provide qualitative or quantitative results on the impacts of design principles or teaching practices across a range of classrooms, or that examine how changes in these design principles or teaching practices can impact students’ learning of mathematics.” By documenting how the CCRSRC model is implemented in a one-on-one setting with a struggling high school student, this study aims to shed light on the attentional processes that support productive mathematical engagement. This research aimed to define, identify and characterize through microanalysis the promotion and improvement of problem solving among struggling high school students by encouraging their reflection and attention, through the CCRSRC intervention.

2. Theoretical Background

2.1. Learning Mathematics and Individual Differences

Today, solid mathematical skills are considered essential. However, some students with learning disabilities/difficulties in mathematics (LDM) tend to fall behind their peers early in their educational journey (U.S. Department of Education, 2022, as cited in Xin et al., 2023) and, unfortunately, continue to struggle as they progress into secondary school (Carcoba Falomir, 2019, as cited in Xin et al., 2023), despite efforts to provide equal access to learning resources and high standards for all students. The term “LDM” is used here, as proposed by Xin et al. (2023), rather than mathematical disabilities or difficulties (MD or MLD), because MD “seems to indicate that the disability, or deficit, is fixed within the individual student” (Xin et al., 2022, p. viii). In fact, the mathematical performance of students with LDM has declined since 2020 (U.S. Department of Education, 2022, as cited in Xin et al., 2023). Jitendra et al.’s (2018) meta-analysis showed that struggling students do not receive sufficient instructional time focused on higher-level mathematics, despite its importance.

For more than a decade, curriculum standards have emphasized conceptual understanding and higher-order thinking in areas such as problem solving, mathematical modeling, and algebra readiness (CCSSI, 2012, as cited in Xin et al., 2023). These standards can make mathematics particularly challenging for students with LDM. However, there is evidence that one-on-one interventions can significantly enhance students’ mathematical knowledge and skills as well as improve their self-perception of their problem-solving (PS) abilities (Hord et al., 2018).

Understanding the cognitive mechanisms underlying individual differences in mathematics learning is essential for designing effective interventions. Cognitive load theory provides a framework for understanding how students with LDM process mathematical information. The theory focuses on the ‘element interactivity effect,’ which becomes critical when learners must simultaneously process multiple interacting elements—a common challenge in mathematical problem solving (Chen et al., 2023; Sweller, 2024). Students with LDM often struggle with high element interactivity, as their working memory capacity may be overwhelmed when attempting to coordinate multiple problem components simultaneously.

Cognitive flexibility represents another crucial factor in addressing individual differences in mathematics learning. This ability to switch between different mental tasks and adopt strategies to adapt to changing environments plays a key role in mathematical performance (Castro-Sánchez et al., 2019; Orakcı, 2021). Matthews (2018) found that adolescents’ cognitive flexibility significantly predicts their mathematics performance and learning autonomy. Importantly, cognitive flexibility enables students to access and transfer knowledge from long-term memory. As Sweller notes, improving long-term memory allows students to recall past problem-solving situations and recognize structural similarities across problems that may appear superficially different.

Building on these theoretical insights, the current study employs an intervention designed to address both cognitive load management and cognitive flexibility development. The intervention helps learners enhance their long-term memory and mental flexibility through structured activities that encourage them to: observe problems systematically, identify similarities and differences across problem types, engage in reflective thinking about solution strategies, and make deliberate choices in their problem-solving actions. By fostering these interconnected cognitive processes, the intervention aims to support students with LDM in developing more robust and transferable problem-solving capabilities.

2.2. Why Struggling Students Have Difficulty in Problem Solving

Students who struggle with mathematical problem solving do not know how to develop generalized PS strategies, which makes each new problem seem unique to them. They are unskilled in problem analysis processes, meaning they cannot connect new knowledge about the topic, concepts, or algorithms (Hord et al., 2020). A meta-analysis of traits among weak mathematics students revealed common themes: low participation, limited relevant experience, lack of prior mathematical knowledge, misperceptions of necessary fundamental skills, language difficulties, and math anxiety (Lake et al., 2017). Additionally, they struggle to process, store, and integrate information when it is unfamiliar or complex (Barrouillet et al., 2007). Cognitive overload also affects them; they lose track of important information in their short-term working memory due to distractions from other parts of the problem or trying to focus on multiple pieces of information simultaneously (Swanson & Beebe-Frankenberger, 2004). Furthermore, their lack of experience in complex problem solving leads to continued poor achievement (Warshauer, 2015).

2.3. Rethinking P-S Teaching Strategies: Toward a Unified Pedagogical Approach

Research on problem-solving (P-S) instruction reveals a spectrum of approaches, yet also highlights a persistent gap in how these strategies are implemented for different learners. At one end of the spectrum, Schoenfeld (2022) proposes two well-supported instructional models: a scaffolded model, where heuristic strategies (HS)—defined as “rules of thumb for successful problem solving” that help individuals understand problems better or make progress toward solutions (Schoenfeld, 1985)—are explicitly taught and gradually withdrawn as problem complexity increases; and a diverse exposure model, where students engage with a range of problems with recurring strategy reinforcement. Both models center on the idea that students must eventually take ownership of strategy selection and transfer. However, studies by Russo and colleagues (Russo et al., 2020a, 2020b) challenge the equity of access assumed in such models. They observe that in practice, weaker students are often relegated to low-level tasks, with little opportunity to engage in the kind of high-order thinking that Schoenfeld’s models rely upon. This critique does not reject Schoenfeld’s framework, but rather exposes a gap between theoretical models and their actual implementation, particularly for struggling students. This gap is further illuminated by Gersten et al.’s (2009) meta-analysis of interventions for students with learning disabilities. Their five recommended components—explicit instruction, use of heuristics, verbalization of reasoning, visual representation, and varied example sequencing—echo the core mechanisms in Schoenfeld’s scaffolded model. Yet Gersten’s findings emphasize the need to retain support structures, suggesting that for some learners, strategic fading may need to be slower or even cyclical. Thus, we begin to see an emerging tension: how to sustain cognitive demand while ensuring accessibility. Lake et al. (2017) reinforce this need for accessibility, but from a broader pedagogical lens. Their meta-analysis identifies mentoring, active learning, peer support, and diagnostic feedback as effective for students with weak math performance. However, their findings largely position the teacher as a general mentor, without detailing the specific instructional moves needed to foster deep problem-solving. This is precisely where Lester (2013) and Lester and Cai (2016) contribute a crucial refinement. He articulates the teacher’s role in P-S not only as a task designer and responsive guide but also as a metacognitive agent—an external monitor, a facilitator of awareness, and a model problem solver. His framework offers a precise pedagogical articulation of the mentor role that Lake et al. conceptualize more broadly. Building on these researchers’ ideas, the Ovadiya (2021) synthesizes them into nine theory-based intervention principles, offering teachers actionable pathways to support students during problem solving while also attending to equity and differentiation. These principles may help bridge the implementation gap identified by Russo et al., operationalizing both Schoenfeld’s and Gersten’s models in ways that are responsive to student diversity. Finally, Shi et al. (2023) provide additional insight by exploring cognitive tools, such as mind mapping. Although created for younger learners, these tools have demonstrated potential in enhancing clarity, integration, and strategic thinking across STEM subjects. When integrated into the type of responsive, metacognitive instruction envisioned by Lester and operationalized by Ovadiya (2021), such tools can strengthen students’ ability to engage in authentic, high-level problem solving regardless of their initial skill level. In summary, these researchers share a common concern: how to teach problem-solving in ways that are both cognitively challenging and instructionally inclusive. While Schoenfeld provides structural models, Russo uncovers equity gaps; where Gersten and Lake offer scaffolds, Lester and Author define instructional precision; and where Shi et al. highlight enabling tools, the overall insight is clear: effective P-S instruction requires both pedagogical vision and responsive implementation. The current study attempts to implement these suggestions. Teaching heuristics strategies for problem solving

It has previously been shown that heuristic strategies, schema-based, and model-based problem solving can improve struggling students’ PS skills. Meta-analyses have found that problem structure representation and interventions based on models were the most effective strategies (Myers et al., 2022; Witzel et al., 2022).

According to Fuchs et al. (2021), providing students with worked examples can significantly enhance their performance on the SAT test, which involves solving model expression problems. However, LDM students may struggle to identify pertinent information in word problems. Explicit instruction can effectively address this issue and further improve their performance, and model-based problem solving, especially that which points out connections between mathematical concepts, can help students develop basic P-S skills (Xin et al., 2023).

Kaitera and Harmoinen (2022) suggested that teachers could foster P-S strategies through discussions and visual heuristics tools. Other studies have also shown the value of teaching heuristic strategies (Ovadiya, 2025; Fiorella & Mayer, 2016; Gersten et al., 2009; Trinchero & Sala, 2016). Learning heuristic strategies increases positivity toward mathematics (Wakhata et al., 2023). In this study, the researcher attempts to implement recommendations from previous studies that focus on developing strategies to promote mathematical problem solving among individuals who struggle. These efforts aim to create a general classroom implementation model and an individual approach for one-on-one intervention.

2.4. Attention and Mathematical Problem-Solving

A growing body of research highlights the vital role of attention in developing mathematical problem-solving (P-S) skills, especially for students who find non-routine tasks challenging. J. Mason’s (2010) theory of shifts of attention offers a conceptual framework for understanding how learners navigate mathematical activity through transitions between surface-level features and deep structural understanding. According to Mason, the ability to shift attention intentionally between actions, objects, properties, and relationships is a hallmark of mathematical fluency. Strohmaier et al. (2020) and Norqvist et al. (2025) employed eye-tracking to demonstrate that students tackling creative mathematical tasks exhibited more flexible and exploratory gaze behavior than those solving routine problems, thereby highlighting the cognitive demands of attentional control in open-ended mathematical contexts. Masson et al. (2024) similarly found that young learners exhibit operation-specific attentional shifts when solving arithmetic problems, suggesting that even foundational mathematical processes are closely tied to attentional dynamics. These studies affirm that attentional strategies are not only observable but also developable with appropriate scaffolding. Importantly, evidence from intervention-based studies suggests that attention is a trainable cognitive resource. Cai et al. (2024), working within the framework of PASS theory—a neurocognitive model that defines intelligence through four interrelated brain-based processes: Planning, Attention, Simultaneous processing, and Successive processing (Das et al., 1994)—reported that students with low attentional control showed significant gains in both attention and math performance following a structured intervention. Similarly, Doz et al. (2024) and Justicia-Galiano et al. (2017) found that visuospatial working memory—a key attentional mechanism—mediates the relationship between math anxiety and achievement. Both studies underscore the central role of attentional processes in mathematical learning. The Ovadiya’s (2025) study contributes to this line of research by demonstrating how structured interventions can explicitly cultivate attentional flexibility among struggling high school students, who often lack access to these strategies due to low expectations and limited exposure to high-level mathematical discourse. In alignment with this perspective, the Ovadiya (2025) developed the CCRSRC model, which structures the problem-solving process around six stages: building Connections, making a Choice, engaging in Reflection, attempting a Solution, engaging in Repetition, and concluding with a final Choice. This model operationalizes Mason’s theory by guiding students through a process that cultivates awareness of how their attention shifts during problem solving, especially as they progress from identifying superficial similarities to formulating algorithmic and heuristic strategies.

Through this lens, Johnson et al.’s (2021) concept of reasoned transfer—the ability to reorganize mathematical knowledge across contexts—can be seen as a product of deliberate attentional shifts fostered by instructional models like CCRSRC. Thus, the current study not only supports Mason’s theoretical model (2010) but also offers a concrete, research-based approach to its implementation in classroom and one-on-one intervention settings.

2.5. Theory for Data Analysis

“Shifts of attention” is a theory proposed by J. Mason (1989, 2008, 2010). He defines learning as a ‘transformation’ of attention (J. Mason, 2010), where ‘attention’ is not only what an individual attends to but also how they attend to the specific objects of attention. He defines five different levels of attending: holding wholes, which occurs before any distinctions are made. Here, the observer gazes at the whole, in general, and does not focus on particular details. Next comes discerning details, where attention is caught by a specific, prominent detail. Recognizing relationships, the next level, when specific connections are made between components, often occurs automatically, followed by perceiving properties, where attention now focuses on instantiations of properties. Finally comes reasoning based on perceived properties. At this point, the selected properties transform into the only basis for further reasoning. Based on Mason’s idea about the ‘transformation’ of attention, this author constructed a one-on-one intervention method termed ‘CCRSRC’ that contains meta-cognitive (such as reflection) and cognitive (including: choice, building connections, and solving) actions. Since the intervention process involved instructing the learner to engage in self-talk aloud, the student’s actions and statements can be analyzed through the lens of “shift of attention theory”.

It is essential to acknowledge that while Mason’s “Shifts of Attention” theory provides a helpful framework for interpreting Ronny’s problem-solving process, the assignment of specific attention levels is necessarily inferential. The theory describes internal cognitive transformations that may not be fully observable through external actions or verbalizations alone. Although Ronny’s statements and behaviors suggest movement through several attention levels, such as discerning details, recognizing relationships, and reflecting on strategies, definitive evidence of reasoning solely based on perceived properties (the highest attention level) is less clear. Therefore, interpretations should be considered plausible hypotheses rather than conclusive determinations.

To improve the accuracy of interpreting attention levels, the researcher created a “bank” of actions, words, sentences, and phrases used by students during one-on-one interviews conducted during the intervention. She linked the language and actions to the attention levels based on what seemed most appropriate and relevant for each level of attention. (See Table 1 below) and confirmed the accuracy through discussions with two colleagues, who sorted and matched language and actions to the “attention levels” until consistent results were achieved among all three testers.

Table 1. A Demonstration of the Analysis of Level of Attention (J. Mason, 2010).

Attention Level	Mason’s Definition	Examples from Mason’s Ideas	Example Statement from the Interview, Explanation Keywords/Actions Expressed
Holding wholes	The individual stares at the whole without focusing on particular details. Precedes the phase of making distinctions.	Looking at the whole expression without noticing details	‘There is a graph here.’ The student’s statement is uttered before reading the question as a result of the visual identification of a graph. Keywords: There is/are, I see, It looks like…, This seems…, I see the whole…, silence before speaking
Discerning details	Attention focuses on one particular detail that stands out from the rest of the elements.	Noticing that there is a minus sign.	‘I am not sure I remember the connection between radius and area of a circle or the geometric series in this question.’ The student is referring to Problem 2. Attention is on context between geometric series, area of a circle, and radius. In his opinion, geometric series are connected to a circle. This diverts his attention from understanding the problem. Concept from the question, shape from a graph, number. Keywords: Here is…, This part…, I notice…, There is a minus/sign/line/number…
Recognizing relationships	Discerning specific connections between specific elements. Often occurs spontaneously.	“This is related to what we did earlier.”	‘I think the vertex of a parabola determines the positive values’ In problem 7, the student has identified a specific relationship between a vertex and positive values in the function. Connection between data and strategy. Keywords: This goes with…, This is like…, It connects to…, If this, then that…, Same as before
Perceiving properties	Attention is focused on structured relationships as instantiations of properties.	This triangle is isosceles. The function is continuous here.	‘$4^{\mathrm{x}-1}+2^{\mathrm{x}-2}=68$. 4, 2, and 68 are all numbers that can be represented with 22n’. The student noticed that the bases of the number are related to the number 2 and expressed a significant connection. If… then… Keywords: It is [even/odd/symmetric/continuous]…, This shape is…, This must be equal/exact/opposite…
Reasoning based on perceived properties	Selected properties are attended to as the only basis for further reasoning.	Since the function is continuous, the Intermediate Value Theorem applies.	‘What percentage of cows yield more than 25 liters per day’. Ahh! I can draw a curve, enter the data ‘25 liters’, and calculate. It is simple!’ The student considers drawing a curve a very important step after reading the problem. She understood the procedure since she created it by plotting data on the graph. Drawing a curve. Keywords: Because… therefore…, Since it is [property], then…, So we can conclude…, That means…

3. Materials and Methods

3.1. The CCRSRC Intervention Process

This intervention model (Ovadiya, 2025) guides the learner to solve problems as they proceed through six steps: (1) Connections (identifying SCs-similarity connections among the problems presented, this offer selective attention.); (2) Choice (deciding which problem to solve); (3) Reflection (explaining the choice); (4) Solving (an attempt is made); (5) Repetition (repeating steps 1–4 as often as wished); (6) Choice (to end the repetition of solving the problems and move on).

This process can catalyze shifts of attention. Since the solver is guided to switch between different cognitive and metacognitive actions. The generic CCRSRC model for the present study is shown in Figure 1. Cognitive actions were in phases C-connections and C-choice, and a meta-cognitive process is constructed in phase R -reflection. It thus implements the recommendations of researchers (J. Mason, 2010, 2016; Silver, 1987) who suggest that P-S skills require developing cognitive and meta-cognitive P-S processes. The model allows personal choice, meaning that students can take responsibility for their decisions about the problems they choose to solve and about the strategies they applied (CC and RC stages).

Figure 1. Generic CCRSRC Flowchart (Ovadiya, 2025): Determined by Analyzing the Transcripts of the One-on-One Intervention Sessions. (F: Formulation; A: Algorithmic; H: Heuristic.). The problems are labeled with the letters A-B-C-D.

The strategy of the intervention was to present problems similar to those from class that could offer a sort of ‘trajectory’ based on previous mathematical knowledge, similar to the approaches suggested Colletti et al. (2023), where by a situation is contrived to foster acquisition of new knowledge by encouraging initiation of a search for a solution by thinking about similar ones. Thus, for the first session of five problems, the author first thought of potential SCs) to ensure such a process could occur and then added two problems each time to assess whether the students could build analogical SCs between them, even though the additions made the cluster of problems analogically distant.

3.2. Research Goals

To define, identify and characterize through microanalysis the promotion and improvement of problem solving among struggling high school students by encouraging their reflection and attention, through the CCRSRC intervention.

3.3. Research Question

How does the CCRSRC intervention promote problem solving among struggling high school students, as evidenced by shifts in attention and reflective thinking observed through microanalysis?

3.4. Participant Profiles

Twelve students participated in the study: six from Grade 11 and six from Grade 12. All participants were preparing for the Israeli matriculation examinations (Bagrut) at the three-point level, which represents the intermediate level of mathematical proficiency required for high school graduation. Participants were selected based on two primary criteria: (1) persistent low achievement on routine mathematics assessments, and (2) limited or absent use of heuristic strategies and systematic problem-solving approaches during preliminary problem-solving observations.

Participants presented diverse profiles of cognitive challenges that affected their mathematical problem-solving performance. Several students had been formally assessed and diagnosed with specific learning challenges:

Attention difficulties: Some participants had diagnosed attention-deficit disorders affecting sustained focus and concentration during problem solving

Memory difficulties: Several students exhibited diagnosed challenges with working memory and/or long-term memory retrieval

Executive function difficulties: Some participants showed diagnosed impairments in executive functions, including perseverance, self-monitoring, planning, and strategic control during problem solving

Information processing difficulties: Several students had diagnosed challenges with processing speed and information integration

Many participants presented with co-occurring difficulties across multiple domains. Despite these varied cognitive profiles, all participants shared the common characteristic of struggling with mathematical problem solving and demonstrating minimal strategic approach to novel mathematical tasks.

3.5. Research Population

Students with varying levels of difficulty in solving math problems were selected from Grade 11 (n = 15) and Grade 12 (n = 15) classes, in which the official curriculum of the Israeli Ministry of Education was taught and would later be assessed by three-point matriculation (Bagrut) examinations. All participants underwent a preliminary interview that included observing them as they solved problems to identify whether they used any heuristic strategies or other strategies. Those found to be poor in both were chosen for the interventions: six students from each class (A total of twelve). Each participant underwent three one-on-one intervention meetings (as in Simon et al., 2010) while also engaging in classroom learning where intervention principles (Ovadiya, 2021) were integrated into the teaching over the course of a year.

The sample of 12 participants yielded 36 in-depth longitudinal interviews, as each participant was interviewed three times, with each interview lasting between 2 and 4 h. This design resulted in 6–12 h of in-depth engagement per participant. Research demonstrates that qualitative studies commonly achieve saturation within 9–17 interviews (Hennink & Kaiser, 2022), and our 36 interviews substantially exceed this threshold. The longitudinal qualitative interview (LQI) design provides comprehensive temporal data, allowing for the exploration of processes and changes that cannot be captured in single cross-sectional interviews (Hermanowicz, 2013).

We have clarified that sample size assessment in qualitative research should consider not only the number of participants but also the depth and richness of data collected (Vasileiou et al., 2018). The longitudinal approach, with multiple extended interviews per participant, ensures substantial data richness and saturation of meaning. Additionally, we have noted that a complementary quantitative study employing an experimental design has been conducted and is currently under review for publication in a separate manuscript. The present qualitative study and the quantitative study serve complementary purposes within a broader research program: the qualitative findings provide depth and contextual understanding of how students experience and navigate the CCRSRC model, while the quantitative study addresses questions of effectiveness and generalizability.

3.6. Rationale for Selecting an Illustrative Example

This article presents an in-depth case study of one participant, Ronny (pseudonym), an 11th-grade student. Ronny was selected as the focal case because his developmental trajectory exemplifies one of the four distinct attentional development profiles identified in the broader study (Ovadiya, 2025).

Analysis of all twelve participants across the 36 intervention sessions revealed four profiles characterizing how students shifted their attentional focus to similarity connections (SCs) across three levels—formulation, algorithmic, and heuristic—over the course of the three intervention sessions:

• Tri-stage development: Progressing through all three levels—building SCs at the formulation level, transitioning to the algorithmic level, and finally developing heuristic-level connections
• Bi-stage development (formulation to algorithmic): Building SCs at the formulation level followed by transition to the algorithmic level; heuristic strategies were rarely or never developed
• Bi-stage development (algorithmic to heuristic): Formulation-level connections were rarely or never employed; focus shifted from algorithmic to heuristic levels
• Bi-stage development (formulation to heuristic): The algorithmic level was typically bypassed, with students moving directly from formulation to heuristic connections

Ronny represents the first profile (tri-stage development), demonstrating the fullest progression through all levels of attentional focus development. This complete developmental arc provides valuable insights into how the cyclical reflection processes embedded in the CCRSRC model promote sustained attentional focus and strategic problem-solving growth in students who struggle with mathematical problem solving.

The detailed microanalysis of Ronny’s discourse across three intervention sessions allows for close examination of moment-to-moment attentional shifts and the role of reflection in redirecting and sustaining focus. While Ronny serves as the focal case, all twelve participants successfully engaged with the CCRSRC model, openly shared their thinking processes during problem analysis, and exhibited similar patterns of attentional development during problem solving. A macro-level summary of attentional patterns across all participants is presented in Table 2.

Table 2. Relationship Between Formulation, Algorithmic, Heuristic (F-A-H) Connections, Attention Levels, and Solving Problems During the Three Interventions with 12 participants. Legend: A—Algorithmic, F—Formulistic, H—Heuristic; P—problem, (p)—Only partially solved; grayed areas indicate problems not offered that session. Attention levels: HW—holding wholes, DD—discerning details, RR—recognizing relationships, PP—perceiving properties, RPP—reasoning based on perceived properties. Example of how to read the table. First line: R1 represents what R did in the first session. She built four algorithms SCs to solve problems 2, 3, 4, and 5. Problems 6 to 9 were not offered. Her “attention levels” are identified as HW—holding wholes, DD—discerning details, RR—recognizing relationships, PP—perceived properties, PRR—reasoning based on perceived properties.

	Problem	P1	P2	P3	P4	P5	Attention Levels	P6	P7	Attention Levels	P8	P9	Attention Levels
Student–Session
R1			A	A	A	A	HW, DD, RR, PP, PRR
R2		A	A					A		HW, RR
R3									H		H	H	RR, PP, PRR
OS1			F(p)			F(p)	HW
OS2								A(p)	A(p)	HW
OS3											H(p)	H(p)	RR, PP, PRR
G1		F(p)			F(p)		HW
G2								A	A	HW, DD, RR, PP
G3									H			H	HW, DD, RR, PP, PRR
M1		A	A			H	HW, DD, RR, PP, PRR
M2								F	A(p)	HW, DD, PP
M3									A		A	H	HW, DD, RR, PP, PRR
L1					A	A	HW, DD, RR, PP, PRR
L2									A(p)	HW
L3					A						H	H	HW, DD, RR, PP, PRR
J1						F(p)	HW
J2				A(p)		A		A		HW, DD
J3									H		H	A	DD, RR, PP, PRR
B1		F(p)					HW, DD
B2			A		A				A	HW, DD, RR
B3											A(p)		DD, RR, PP, PRR
Y1			A				HW, DD
Y2						H		H		DD, RR, PP, PRR
Y3											H	H	DD, RR, PP, PRR
C1			F(p)				HW
C2						A			A	DD, PP, RR, PRR
C3											A		DD, RR, PP, PRR
RC1			A				HW
RC2			A					A	H	DD, PP, RRP
RC3									H		H	H	DD, RR, PP, PRR
O			A	A			HW, DD
O			A					H	A	DD, RR, PP, PRR
O									H		A	H	DD, RR, PP, PRR
A1			A(p)	A			HW, DD
A2			A					A	A	DD, RR, PP
A3									A		A	H	HW, DD, RR, PP, PRR

3.7. One-on-One Intervention—General Procedure

The intervention program lasted one academic year and was based on principles for teaching struggling students mathematics. These included processes to develop heuristic literacy, reduce memory load, teach heuristic P-S strategies, and encourage students to generalize the P-S ideas they were exposed to (Ovadiya, 2021).

The intervention one-on-one sessions took place at the end of November, the end of February, and the end of April. Each lasted between 60 and 140 min (depending on how many problems the student chose to solve). All were documented both in writing and on video.

During one-on-one interventions, students were given (5, 7, and 9) problems in three sessions, respectively. They were asked to build SCs between them, solve them, and construct other sets of SCs between the problems, while performing the actions of the CCRSRC model.

Problems used in each intervention session

The problems were routine, but the students in the intervention lacked the skills to solve them. Five problems were presented in the first session, seven in the second (comprising the original five plus two additional ones), and nine in the third (the seven from the second session plus two additional ones). This (5, 7, and 9) idea concept was inspired by Miller’s ‘The Magical Number Seven, Plus or Minus Two’ (Miller, 1956). All problems were similar to those solved in class, both in their external structure and internal structure, such as the mathematical concepts and algorithms required to solve them. Example problems are presented in Appendix A.

3.8. Qualitative Research Design: Rationale and Methodological Approach

Qualitative research is defined as “the study of the nature of phenomena,” including “their quality, different manifestations, the context in which they appear or the perspectives from which they can be perceived,” but explicitly excluding “their range, frequency and place in an objectively determined chain of cause and effect” (Busetto et al., 2020, p. 2). Unlike quantitative research, which is rooted in positivist philosophy emphasizing causality, generalizability, and replicability, qualitative research is especially appropriate for answering questions of “why” and “how” something is observed, assessing complex multi-component interventions, and focusing on intervention improvement rather than establishing causality (Busetto et al., 2020). Qualitative studies do not typically use control groups (Busetto et al., 2020, p. 11), as their purpose is fundamentally different from that of experimental designs. The present qualitative study was designed to explore participants’ lived experiences with the CCRSRC model and to understand the processes through which the model influences critical reading and reasoning. The research aim was to develop a rich, contextualized understanding of the phenomenon from participants’ perspectives—a goal for which control groups are neither methodologically appropriate nor epistemologically necessary.

We have also noted that the complementary quantitative study (under review) employs an experimental design that addresses questions of causal effectiveness and generalizability. Together, the qualitative and quantitative studies offer a comprehensive understanding of both the mechanisms through which the CCRSRC model operates and its measurable impact on student outcomes.

3.9. Reflexivity and Trustworthiness in Qualitative Research

Qualitative research relies on nuanced judgments that require researcher reflexivity. Unlike quantitative research, which strives to reveal truths as free as possible from researcher “bias,” qualitative research depends on subjectivity (Olmos-Vega et al., 2022). Reflexivity is defined as “a set of continuous, collaborative, and multifaceted practices through which researchers self-consciously critique, appraise, and evaluate how their subjectivity and context influence the research processes,” framing reflexivity as a way to embrace and value researchers’ subjectivity (Olmos-Vega et al., 2022).

Discourse analysis involves subjective interpretation by its very nature. However, when discourse analysis is anchored in a theoretical approach with appropriate methodological safeguards, it constitutes a rigorous methodology (Hagen, 2023). The present study employed rigorous methodological safeguards to ensure trustworthiness: the use of three independent raters represents peer debriefing, which “is regarded as one of a complement of techniques used to enhance the credibility and trustworthiness of qualitative research through the use of external peers, comparable to internal validity in quantitative research” (Nguyen & Spall, 2008).

Investigator triangulation, applied by involving several researchers in addressing organizational aspects of the study and the process of analysis, significantly enhances credibility. When data are analyzed independently by multiple researchers and interpretations are compared and discussed until the most suitable interpretation is found, this represents a rigorous approach to ensuring trustworthiness (Korstjens & Moser, 2018).

Rather than representing a limitation, the interpretive nature of discourse analysis coupled with collaborative verification measures constitutes a methodologically sound approach that aligns with established standards for rigorous qualitative research.

3.10. Data Analysis

To determine how thinking about problem-solving occurred, I looked for evidence of Mason’s five levels of attention (J. Mason, 2010) during the one-on-one intervention interviews. The first step of the analysis was to distinguish between the levels of attention. For this, I identified keywords corresponding to each level of the intervention transcript. By including the Connections Choice Reflection Solving Repetition Choice intervention actions in the transcript, it became possible to identify the level of attention the learner applied to each action. I created a table with common phrases, formulations, or actions typical of this level to code the interview transcripts. Two colleagues verified this analysis, and after they found evidence of Mason’s five levels of attention in the interventions video and transcripts, we discussed several transcripts and concurred on what would be considered suitable for each level. For each interview, we developed a profile detailing the participant’s trajectory of the levels of attention they exhibited. Through this micro-analysis, we gained insights into how the student shifted their focus and attention between cognitive and meta-cognitive actions while trying to solve the problem and implement CCRSCR actions.

Table 1 shows the attention levels observed in the interventions, as agreed upon by the researcher and two colleagues who helped analyze the data. The analysis of the transcripts from the one-on-one intervention was performed using a micro-analysis approach. The researcher (and) examined each word, sentence, method statement, and action taken by the student to determine the level of attention based on these elements. Table 1 below demonstrates how the researcher reviewed the transcripts of the discourse.

4. Results

4.1. Micro View: Ronny’s Intervention Sessions

To better illustrate the process, I present an analysis of one participant’s (Ronny’s) journey across the three intervention meetings, emphasizing her development in applying the CCRSRC actions and her navigation of different levels of attention throughout the problem-solving process. A summary of the analyses for all participants will be provided later, highlighting the relationships between implementing the problem-solving intervention model and shifts in attention. The researcher (Ovadiya, 2025) developed a method for categorizing connections based on specific characteristics. Connections directly derived from the problem’s phrasing are labeled Formulation (F); those relying on implicit algorithms are classified as Algorithmic (A); and those related to problem-solving strategies are defined as Heuristic (H). The quality of these connections can serve as an indicator of shifts between different attention levels, while the accompanying discourse offers additional evidence regarding the solver’s focus.

4.2. First Intervention Session: Solving Five Problems by CCRSRC Actions

Description of the CCRSRC Flowchart

Explanation of Figure 2

Figure 2. CCRSRC Flowchart of Ronny’s First Intervention Session. Shaded elements signify PS steps that were the Result of Building SCs. The red words describe the model’s actions.

Line 1: Ronny declined to build any SCs before attempting to solve any of the problems. She chose to begin with P1 by attempting to solve x − 1 + x − 2 = 68 (for: $4^{\mathrm{x}-1}+2^{\mathrm{x}-2}=68$), but when she finished, she did not know what to do with the x. The researcher directed her attention to the bases (clue), and she solved the problem.

Line 2: She repeated the process. She noted SCs between P1 and P3 (Connection). ‘Perhaps P3 is similar to P1 because both have an exponential equation’ (algorithmic connection). ‘If it is possible to solve for a, I might be able to reduce the expression and call it a’ (Reflection). She then chose to solve P3. She did not analyze P3 in depth, but she recognized that its external structure included exponential values (formulation connection) and proposed replacing the exponential expression with a, as she did when she solved P1.

While solving, she reflected that she did not need to implement the SC idea (Reflection): ‘Actually, I do not have to replace any part of the expression with a value such as a because I can simply find the solution for x in the exponential expression 2x = 3.’ Therefore, although she began to solve P3 using SCs, reflective thinking allowed her to recognize her error in choosing the ‘connection’ and replace one idea with (another) geometric mean (Choice).

Line 3: She repeated the process, and on her initiative to apply CCRSRC actions, she found a Similar Connections between P3 and P5: ‘Both problems can be solved by applying the geometric mean’ (algorithmic connection). She thus chose to solve P5 and managed the task independently.

Line 4: She repeated the process and noted the SCs between P2 and P5: ‘P2 is also similar to P5 because the geometric mean can also solve it’ (algorithmic connection). She chose to solve P2. However, she did not remember the formula for the area of a circle and gave up.

Line 5: She repeated the process and noted SC between P2 and P4: ‘P4 can be solved using the same formula as P2 for finding the ratio of progression’ (algorithmic connection). She chose to solve P4 independently while explaining to herself every stage. Line 6: She chose to resign.

Interpretation. The process was supposed to be based on building SCs between two or more problems, solving one, and then solving the second by implementing the SC. Ronny eventually built SCs between P5, P2, and P4 (using the geometric mean strategy), but this did not occur immediately. In fact, she did not build an effective SC between P1 and P3 and solved them independently of one another. However, after solving P3, she built an algorithmic connection (geometric mean strategy) and successfully applied it to P2, P4, and P5.

Initially, Ronny’s resources were meager, but by recognizing the algorithmic connection, she broadened her pool of resources. Although her poor general mathematical knowledge did not allow her to fully solve all the problems, it can be presumed that had it not been for the intervention aimed at building connections and guiding the choice of problem, her lack of mathematical knowledge would have been overwhelming, and Ronny would have failed to solve any of the problems. It seems that the SC process furnished a reflective tool that helped her organize her knowledge and resources to address the problems.

Suppose we utilize the shift of attention theory. In that case, we can say that at the beginning of the intervention, Ronny’s attention was associated with holding wholes because the structure of her attention was to stare at the entire problem without focusing on details. However, after solving P1, she progressed to discerning details: her attention was caught by a particular detail (‘Perhaps P3 is similar to P1 because both have an exponential equation’) that stood out from the rest of the elements in the problem. Then, she progressed to recognizing relationships (between P3, P5, and P2: ‘All the problems can be solved by applying the geometric mean’) and perceiving properties, which involves attention structured by seeing relationships as instantiations of properties. Finally, she used reasoning based on perceived properties, which is a structure of attention in which selected properties are attended to as the only basis for further reasoning, as Ronny did by applying the strategy of geometric mean to solve P2, P4, and P5.

4.3. Second Intervention Session: Solving Seven Problems by CCRSRC Actions

Description of the CCRSRC Flowchart

This time, Ronny looked for SCs between problems before attempting any solution.

Line 1. She first stated that all the problems were related to geometric progression, but initially noted this connection only between P3 and P5. However, she later recognized that ‘there is a connection between P2 and P4 that is also geometric progression’ because she identified similar data and concepts. She explained the source of connection as ‘Formulation’ (Reflection).

She chose to solve P7, stating that she knew how to solve problems for which she had built SCs (i.e., P2, P3, P4, and P5), but because P7 did not seem to have any SCs, its solution might help her in building more SCs. We can thus conjecture that Ronny viewed the act of building SCs as a reflective tool to suggest solutions and build her pool of resources. Finding the SCs was an independent act that helped her organize her thoughts about subjects and related algorithmic ideas. According to her perception, if she identified an algorithmic connection, she did not need to solve the problem because she knew she had the tools. However, if she did not see any SCs for a problem, she had to solve it to discover more SCs.

Explanation of Figure 3

Figure 3. CCRSRC Flowchart of Ronny’s Second Intervention Session. The shaded element signifies the P-S step that results from Building SCs. The red words describe the model’s actions.

Ronny solved P7 and then reflected, ‘Problem 7 and Problem 1 are similar because in both cases we must solve a quadratic equation’. This seems to be an algorithmic connection, but it is unclear whether she arrived at this conclusion after deep analysis or merely superficial observation of P1’s external structure.

Line 2. She repeated the process, observing the seven problems. She noted the SC between P7 and P1, and also the SC between P1 and P4, ‘… they both involve a1 and q’ (formulation connection). However, she chose to solve P6, reflecting that ‘I couldn’t create any SCs between this problem and others, so solving it may lead to new ideas regarding SCs’. Once she had solved P6 and applied independently the CCRSRC actions, she succeeded in finding an SC between P6 and P2 and then P6 and P4: ‘geometric progression’ problems.

Line 3. She repeated the process, choosing to solve P2. Although she properly sketched the circles, she did not know how to proceed. In other words, she could define the task but did not know how to solve it.

Line 4. At this point, Ronny chose to terminate the session.

Interpretation. Ronny exhibited a two-directional perception of determining SCs: SCs can aid in problem solving, and problem solving can lead to building SCs. She thus chose to solve problems for which she had not observed any SC, assuming that this would lead to other SCs.

Now, suppose we utilize the shift of attention theory. In this session, the connections were primarily formulation (the attention level seemed as holding wholes) and did not advance P-S development. However, successfully building some SCs (recognizing relationships) gave her the courage to choose unfamiliar problems and address their solution processes.

This session demonstrated the importance of constructing mathematical or heuristic mathematical SCs. Here, Ronny built only formulation SCs, which were insignificant and did not offer her any key to solve the problems. She failed to notice that she was not building any new SCs (in other words, she did not use the attention level reasoning based on perceived properties). According to Ormerod and MacGregor (2017), only significant ideas common to two problems will lead to a transfer of learning between them.

Here, Ronny advanced her perception of her P-S skills by realizing that if she could build SCs, she could improve her ability to solve different problems. Thus, she chose to solve problems for which she could not find any SCs. Ronny’s courage to solve problems for which no connections were built implies that she is trying to develop the level of attention that relates to ‘recognizing relationships’ because she realized that while solving them, she would find connections and relationships that hint at a solution process. Ronnie’s implementation of CCRSRC activities has led her to the insightful realization that connections serve as a catalyst for solutions, and solutions, in turn, drive further connections. This understanding has shifted her attention. Initially, she observed the problems at a general level (holding wholes) and identified objects. By solving problems, her Attention has shifted to emphasize the importance of connections, both in relationships and properties (recognizing relationships). In the current case, simply focusing attention was not enough to solve the problems because Ronny lacked mathematical knowledge and an insignificant connection.

4.4. Third Intervention Session: Solving Nine Problems by CCRSRC Actions

Description of the CCRSRC Flowchart

Explanation of Figure 4

Figure 4. CCRSRC Flowchart of Ronny’s Third Intervention Session. Shaded Elements Signify the P-S Steps that Result from Building Similarity connections. The red words describe the model’s actions.

Lines 1, 2, 3: Similarly to the second intervention, Ronny started by noting several SCs: P3, P4, P5—geometric mean; P5, P6—find q and then continue; P7, P8—one minus something [1-…]. The latter two seem to be heuristic.

Line 3 (cont.): She then chose to solve P9. However, she began by stating, ‘I need to find the probability that it did not rain on at least one of the above dates’ (heuristic). Upon reflection, she realized she was unsure whether she had summed up all the events and hesitantly stated, ‘It seems that I need to find one minus the answer to part B’. In other words, Ronny used the [1-…] method.

This is how she discovered the connection between P9 and P6 (‘one minus’), demonstrating a general (heuristic) connection. She then chose to forego solving P6 and chose P8 instead

Line 4: She repeats the cycle, finding SCs between P8 and P2: ‘In both, a good start to a solution is to draw a sketch’. Ronny initiates her attempt to solve P8 (which she previously claimed she could not) by sketching the problem and reflecting upon it to see how to proceed.

What is significant in this intervention is that Ronny’s SCs were mainly heuristic. She chose to solve P8, P9, and P6—problems she had been unsure about—based on heuristic SCs, evidenced by her mentioning general ideas for beginning the solution rather than specific (i.e., algorithmic) ways to find the solution. This reinforces the concept that a general key solution idea is a heuristic one.

Line 5: Repeating the process, she built SCs between P8 and P7, and then chose to solve P6. Then, Ronny reflected that she did not want to solve the other problems on the intervention problem sheet, stating, ‘I have to see what I know’.

Line 6: Thus, she chose to end the intervention and solve three random problems from a workbook. She succeeded by building SCs between these problems. Then, with a sense of success and satisfaction, she agreed to complete the problems on the problem sheet: ‘Now I feel that I truly know what I am doing … I will be able to build SCs to similar problems in the exam’.

Interpretation. In this session, Ronny comprehensively and independently solved three problems from the problem sheet and three from the workbook, declaring that she knew how to solve the ones she had chosen to forego. Throughout the session, she independently discovered significant SCs and, in their wake, considered planning solutions to problems she had previously solved, even if she had not built any SCs. She successfully solved problems she initially thought she could not in the order of her choosing and then tried to build SCs to the other problems on the problem sheet. This third session proved to her that she could solve problems that she previously thought she could not.

The ways of attention she used when she built her heuristic strategies were recognizing relationships, perceiving properties, and reasoning based on perceived properties; she then solved the problem by using connections. By verbalizing her thought processes and asking herself questions about the SCs between the problems, Ronny found the appropriate answers by building heuristic strategies. As far as Ronny was now concerned, any problem for which she could build a meaningful SC with another was one she could solve.

4.5. Ronny’s Intervention Summary

Ronny’s three-session CCRSRC actions intervention demonstrates how her evolving attentional structures, guided by J. Mason’s (2010) five levels of attention, enabled her to become a more strategic and autonomous problem solver. Her progress illustrates how increasing levels of attention correspond to cognitive growth in mathematical thinking, as reflected in the building of SCs and the solving of problems through them. Ronny’s problem-solving behavior can be identified and characterized during the implementation of intervention activities that include developing meaningful relationships and promoting attention through four phenomena:

Identifying Meaningful Mathematical Structures (Recognizing Relationships → Perceiving Properties): In Session 1, Ronny began moving beyond superficial observation (e.g., ‘exponential equations’) and started recognizing functional relationships, such as the use of the geometric mean across multiple problems. By Session 3, she had progressed to perceiving properties within problems (e.g., identifying shared heuristic patterns, such as the ‘1 minus something’ structure), which enabled her to discern deeper mathematical structures.

Constructing and Evaluating Strategies (Discerning Details → Recognizing Relationships): Early in the intervention, Ronny used trial strategies without evaluation. As she advanced, she reflected on and modified her strategies mid-process—e.g., adjusting her plan when solving exponential equations—indicating a shift toward consciously evaluating the relationships between strategies and problem structures.

Transferring Ideas Between Tasks (Recognizing Relationships → Reasoning Based on Perceived Properties): Ronny increasingly transferred solution strategies between problems. Notably, solving one problem would later trigger insights about another, demonstrating backward transfer. Her reasoning became grounded in shared mathematical properties, such as structure, strategy, or heuristic utility, indicating a high-level attentional structure.

Building Confidence and Autonomy (From Holding Wholes → Reasoning Based on Perceived Properties): Initially overwhelmed by the entirety of each problem, Ronny gradually learned to focus on specific details and relationships. By the final session, her independence and confidence were clear: she chose problems strategically, solved additional problems from a workbook, and stated she could transfer her approach to exam situations. Her attention structure had shifted toward fully autonomous reasoning, in particular, when she decided that if a problem “did not participate” in the building connections phase, she should solve it. This fact shows an attentional focus on autonomous solutions. These findings underscore that advancing through Mason’s levels of attention, from passive observation as “holding whole” to active reasoning as “reasoning based on perceived properties,” plays a crucial role in fostering meaningful mathematical problem-solving, strategic thinking, and learner agency.

4.6. Macro View: Participants’ Attention Levels in a One-on-One Intervention

Going beyond a single case study, what insights can be gained about analyzing the attention levels of other participants in a one-on-one intervention? Furthermore, how might analyzing attention levels teach us about their problem-solving process?

What can we learn from Table 2 (Section 3.6) about the development of learners’ attention and reflection? J. Mason’s (2010) theory of attentional shifts (discerning → relating → perceiving → reasoning) provided a foundational lens through which to interpret the learning processes of struggling students in mathematical problem-solving. He proposes that learners progress through structured stages of attention: from unconscious awareness, to noticing, discernment, recognizing relationships, and eventually to reasoned use. These attentional transitions reflect a learner’s movement from passive observation of mathematical content to active engagement with structure and meaning. Importantly, each new concept or representation in mathematics demands a corresponding reorganization in a learner’s structure of attention—what Mason calls a shift in the focal awareness of the learner. This view resonates with Gattegno’s (1970) notion of layered awareness (sensory, naming, reflective) and aligns with the idea that deep understanding is not about seeing more, but about seeing differently. The findings of the current study expand Mason’s theoretical ideas. In this study, it is observed that students display different levels of attention when solving problems. According to the table, it is evident that if students focus their attention on the levels of WH-holding wholes and DD-discerning details, they can generate initial ideas for solving a problem, but will not be able to solve it completely. On the other hand, if their attention focuses on recognizing relationships or perceiving properties, they may be able to solve a problem that requires applying algorithmic procedures fully. Only when their level of attention shifts to RPP-reasoning based on perceived properties, will they be able to activate heuristic ideas, understand concepts and procedures, and solve a problem completely.

The column showing the first session of the intervention indicates attention levels that still do not enable learners to analyze a problem by forming significant connections, and as a result, the solutions are also incomplete. In the column showing the second intervention, it is clear that students have connected to the model actions, and attention levels have become more meaningful. The connections between problems are also at a higher level, leading to more problems being solved. In the column showing the third session of the intervention, it is evident that participants’ attention levels are such that they absolutely help them analyze a problem and find a solution through the use of imagination connections. The intervention actions foster attention levels that are more meaningful for problem solving.

5. Discussion

Jitendra et al.’s meta-analysis (Jitendra et al., 2018) showed that struggling students do not receive sufficient instructional time that focuses on higher-level mathematics despite its importance. To close the gap, the current study was conducted in a learning environment that implemented the state curriculum, while also focusing on solving complex problems. This setting serves as an example of an applied model that promotes problem-solving skills of struggling students. Additionally, a theoretical contribution will be presented below.

Attention, reflection, and problem analysis. Attention, reflection, and problem analysis are essential for students who struggle with routine problem-solving. These students are unskilled in problem analysis processes, meaning they cannot connect new knowledge about the topic, concepts, or algorithms (Hord et al., 2020). During the intervention, they can improve by engaging in cognitive and metacognitive questioning that encourages them to recognize similarities and differences between set of problems. This type of action helps them focus on selecting the most effective connections as ideas to solve the problems. It is especially important because problem analysis and strategy selection are often ineffective or missing in students who struggle (e.g., Krawec, 2014). Another factor influencing individual differences in learning is cognitive flexibility, which is crucial for effective learning and has been shown to predict academic performance (e.g., Castro-Sánchez et al., 2019). The CCRSRC model, particularly the “connections” stage, encourages learners to build connections between problems. This approach encourages flexibility because it does not depend on a specific, fixed procedure. This method helps learners develop mental flexibility through connection-building, a crucial step in analyzing problems.

Attention, reflection, and responsibility. The format of the intervention, although structured, allowed students freedom regarding which problems they wished to solve or use to build SCs, and when to stop the intervention, i.e., they took responsibility for managing their process. To make decisions about choosing a problem to solve or deciding to end a process, they reflected on their ideas. They managed their attention so that it shifted from ideas for one problem to organizing ideas for another problem, or from one idea to a solution for another idea. This is unique because typical interventions for struggling students usually involve a teacher dictating specific procedures (Gersten et al., 2009) or targeted actions, and do not invite students’ autonomy (Jitendra et al., 2018).

Attention, reflection, and improved SCs quality. The varying levels of attention displayed by students during the intervention suggest that learners can enhance and direct their focus toward key problem-solving concepts and operations. These concepts and operations emerged from building connections, which then evolved into associations that support either algorithmic procedures or heuristic strategies, as outlined in the findings from the intervention sessions.

This attentional development aligns with prior research emphasizing the role of reflection and cognitive structuring in effective problem-solving. Fiorella and Mayer (2016) argue that general strategies are beneficial for developing problem-solving schemas. The SCs strategy—Solution-Conducive thinking—supports the formation of schemas that are specific to individual problems while also generating generalizable insights applicable to new tasks. Similarly, Hiebert and Grouws (2007), along with Holyoak et al. (2022, 2023), emphasize that associative thinking fosters analogical reasoning, enabling learners to transfer structural understanding across mathematical contexts. Together, these findings highlight the interdependence of attention, reflection, and schema construction in supporting meaningful and transferable mathematical thinking.

Attention, reflection, and promotion of long-term memory.

Attention and reflection play a central role in fostering long-term memory formation and cognitive flexibility in mathematical problem-solving. According to Sweller (2024), individual differences in learning outcomes are primarily determined by how knowledge is encoded and retrieved from long-term memory. He emphasizes the importance of solving problems that encourage learners to identify sources of relevant knowledge within contextual features, thereby facilitating the construction of meaningful cognitive schemas. Repeated engagement with such tasks as “building connections between problems” and reflecting on the action allows knowledge to be assimilated into long-term memory more effectively. In the current study, using the CCRSRC model, learners were required to interact with connections, reflect on their choices, and develop solutions. The model is interactive. According to Chen et al. (2023) Element interactivity effects both working and long-term memory.

This attentional-reflective process cultivated not only accurate problem-solving but also cognitive flexibility, allowing learners to adapt strategies across varying mathematical contexts.

Thus, the combination of focused attention, structured reflection, and repeated schema-building activities aligned with Sweller’s (2024) claim that meaningful practice enhances long-term learning by deepening the organization and accessibility of mathematical knowledge.

Attention, reflection, and struggling students’ problem solving:

Strohmaier et al. (2020), employing eye-tracking technology, found that students engaged in creative mathematical tasks exhibited more flexible and exploratory gaze patterns than those solving routine problems, highlighting the increased cognitive demands of attentional control in open-ended contexts. Similarly, Masson et al. (2024) reported that young learners demonstrate operation-specific attentional shifts during arithmetic problem-solving, suggesting that attentional dynamics are deeply embedded in even foundational mathematical activities. Collectively, these studies affirm that attentional strategies are both measurable through observable behaviors and responsive to instructional scaffolding. Building on these insights, the present study highlights the practical and theoretical value of structured one-on-one interventions for students struggling with routine problem-solving. The findings illustrate that a guided yet adaptive approach, grounded in the CCRSRC model, can effectively foster both attentional development and reflective thinking. Through focused interactions that promote shifts in attention, particularly toward discerning mathematical structure and reasoning based on perceived properties, students demonstrated marked improvements in their problem-solving performance. These results support the claim that targeted support can cultivate essential metacognitive resources and enable learners to become more autonomous and strategic mathematical thinkers.

Conclusions That Can Be Drawn

Together, the results show that struggling students can greatly benefit from interventions that combine attention (by building SCs), reflection, and iterative problem-solving cycles. The CCRSRC model provided a structure through which learners developed, paying close attention. Theoretically, this study expands understanding of attentional dynamics and reflective processes in mathematics education. Practically, it offers educators a clear and flexible framework for fostering meaningful, transferable problem-solving skills in students who typically find routine tasks challenging.