Designing for Deep Thinking: A Theory-Driven Inquiry into Problem-Chain Teaching in Primary Mathematics

Designing for Deep Thinking: A Theory-Driven Inquiry into Problem-Chain Teaching in Primary Mathematics | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Designing for Deep Thinking: A Theory-Driven Inquiry into Problem-Chain Teaching in Primary Mathematics Siliang Yu, Qiaoqiao Xue This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6782865/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Background Developing higher-order thinking skills (HOTS) is critical for fostering adaptive and flexible cognitive abilities in children. However, conventional mathematics instruction often neglects the cultivation of complex reasoning and creative problem-solving. This study aimed to design and evaluate a theory-driven instructional model—problem-chain teaching—targeting the enhancement of HOTS in fourth-grade students, grounded in cognitive and educational psychology frameworks. Methods A quasi-experimental design was implemented with 50 fourth-grade students randomly assigned to experimental (problem-chain teaching) and control (conventional teaching) groups. The intervention was informed by inquiry-based learning theory, cognitive load theory, and HOTS developmental models. Pre- and post-intervention assessments measured problem-solving, critical thinking, and creative thinking using validated psychometric instruments. Statistical analyses included paired-sample t-tests and ANCOVA, with effect sizes calculated using Cohen’s d to assess the magnitude of intervention effects. Results Students receiving problem-chain instruction demonstrated significant improvements in all HOTS domains compared to the control group (p 3.0). Notably, creative thinking exhibited the greatest gains. ANCOVA results indicated significant between-group differences favoring the experimental group, with partial η² values exceeding 0.65, suggesting a substantial impact of the intervention. Conclusions The problem-chain teaching model effectively enhanced higher-order cognitive processes in primary school students. These findings highlight the value of integrating cognitive psychology principles into instructional design to promote complex reasoning skills. The study provides a replicable framework for educators seeking to foster critical, creative, and problem-solving abilities in early mathematical learning, contributing to the broader psychological understanding of cognitive skill development in children. Cognitive development Cognitive load theory Educational intervention Higher-order thinking skills Inquiry-based learning Problem-chain teaching Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction In an era increasingly shaped by artificial intelligence and digital transformation, the ability to think critically, solve complex problems, and apply knowledge creatively has become a defining competency for 21st-century learners (González-Pérez & Ramírez-Montoya, 2022). As automation rapidly replaces routine cognitive tasks, education systems must respond by cultivating students’ higher-order thinking skills (HOTS) that support deep reasoning and adaptive learning (González-Salamanca et al., 2020; Ramírez-Montoya, 2022; Tang et al., 2024). Primary education, particularly mathematics, offers a foundational context for nurturing such competencies. Mathematics is not only a discipline of logic and abstraction but also a fertile ground for structured inquiry, problem-solving, and conceptual transfer (Jablonka, 2020; Peng & Kievit, 2020). However, traditional instructional methods—often centered on procedural fluency and rote practice—tend to underutilize the potential of early mathematics learning as a platform for higher-order thinking development. To address this gap, problem-chain teaching has emerged as a promising pedagogical approach that scaffolds student thinking through logically sequenced, interrelated problems. Grounded in constructivist principles and aligned with Vygotsky’s (1978) zone of proximal development, this approach guides learners from basic understanding toward more complex cognitive operations such as analysis, synthesis, and innovation (Tang et al., 2020; Ding et al., 2021). Building on this foundation, the present study incorporates three complementary theoretical frameworks to inform the design of a structured instructional pathway. De Jong’s (2006) inquiry-based learning framework supports the progression from teacher-guided orientation to student-led exploration and reflection, ensuring that knowledge is actively constructed rather than passively received. Sweller’s (2011) cognitive load theory offers principles for managing mental effort during instruction—particularly through techniques such as worked examples and visual scaffolds—to prevent overload and facilitate schema acquisition. Meanwhile, King et al.’s (1998) three-tier HOTS development model provides a developmental scaffold that sequences learning from foundational knowledge acquisition, to cognitive bridging, and ultimately to advanced application and transfer. Together, these theories contribute not only to the structure of the teaching pathway but also to its cognitive coherence—ensuring that each phase of instruction aligns with targeted thinking outcomes. Despite the conceptual potential of problem-chain teaching, the literature remains fragmented—either focused on macro-level frameworks or on technology-enhanced learning environments—leaving a critical need for theoretically grounded, classroom-tested models specifically designed to foster HOTS in primary mathematics (Çakıroğlu et al., 2024; Hu et al., 2025). This study responds to that need by designing and empirically validating a structured problem-chain teaching pathway that supports the systematic cultivation of higher-order thinking. Drawing upon De Jong’s (2006) inquiry-based learning framework and King et al.’s (1998) three-tier model of HOTS development, the study aims to construct a replicable instructional model that bridges theory and classroom practice. To address the gap between theoretical promise and classroom applicability, this study aims to design and empirically validate a structured, theory-informed problem-chain teaching pathway for primary mathematics education. Specifically, the research is guided by three questions: What is the current profile of fourth-grade students’ higher-order thinking skills (HOTS) in mathematics? How can a problem-chain teaching pathway be systematically constructed to foster HOTS based on established cognitive and instructional theories? To what extent does the proposed pathway significantly outperform conventional instruction in enhancing students’ HOTS? This inquiry contributes to the growing body of design-based research that seeks to align pedagogical innovation with cognitive development goals in elementary education. Literature Review Higher-Order Thinking Skills: Conceptualizations and Developmental Models Higher-order thinking skills (HOTS) are widely regarded as essential competencies for 21st-century learners, enabling students to engage in complex reasoning, knowledge transfer, and adaptive problem-solving. Traditionally, Bloom’s ( 1956 ) taxonomy identified analysis, synthesis, and evaluation as markers of higher-order cognition. Later, Anderson and Krathwohl’s ( 2001 ) revision elevated “creation” as the apex of cognitive processing, thereby emphasizing the role of innovation in contemporary education. Contemporary frameworks have further refined HOTS into multidimensional constructs. Alkhatib (2019, 2022), for example, proposed a tripartite model comprising problem-solving, critical thinking, and creative thinking—dimensions particularly aligned with the goals of mathematics education and interdisciplinary integration. These dimensions reflect not only cognitive complexity but also the ability to apply knowledge flexibly and reflectively across novel contexts. Building on these conceptualizations, King et al. ( 1998 ) introduced a three-tier model of HOTS development, distinguishing between foundational knowledge acquisition, bridging through scaffolding, and advanced application. This developmental progression provides a scaffolded framework for instructional design that gradually transitions students from lower-order cognition to higher-order performance. Instructional Pathways for HOTS: From Cognitive Theory to Design Principles Translating HOTS into effective classroom practice requires alignment with cognitive and instructional theories. Sweller’s ( 2011 ) cognitive load theory (CLT) highlights the importance of managing mental effort during learning, particularly through strategies such as worked examples that reduce extraneous load and facilitate schema acquisition. These principles are especially relevant for novice learners encountering complex problem-solving tasks (Sweller et al., 2019 ). In parallel, inquiry-based learning frameworks (De Jong, 2006 ) advocate for student-driven exploration, emphasizing the active construction of knowledge through question posing, investigation, and reflection. Although rooted in different assumptions—CLT focusing on efficiency, inquiry learning emphasizing autonomy—these theories can be integrated through instructional sequencing that transitions from structured modeling to exploratory engagement. Furthermore, dialogic and collaborative learning paradigms (Lu, 2021 ) underscore the role of peer interaction and discourse in deepening cognitive engagement and meta-cognitive regulation. These perspectives collectively inform the design of instructional models that aim not only to develop HOTS, but to sustain them through reflective and socially mediated processes. Problem-Chain Teaching: A Promising Yet Under-Specified Approach Within mathematics education, problem-chain teaching has gained traction as a method for fostering cognitive progression through a series of logically sequenced, interdependent problems. Grounded in constructivist theory and Vygotsky’s ( 1978 ) concept of the zone of proximal development, this approach enables learners to build upon prior knowledge and gradually reach higher levels of abstraction and transfer (Tang et al., 2020 ; Ding et al., 2021 ). Recent empirical studies have offered valuable typologies for problem-chain design. Hu et al. ( 2024 , 2025 ) identified three core types—contextual, trap-based, and summarizing questions—each serving distinct pedagogical functions. Similarly, Kinnear et al. ( 2024 ) demonstrated that structured questioning sequences can enhance engagement and deepen conceptual understanding. Despite these advances, two key limitations persist. First, most research remains at the conceptual or theoretical level, with limited translation into replicable classroom models tailored to primary education (Feng et al., 2023 ; Qing-li et al., 2024 ). Second, studies often emphasize digital tool integration or teacher training, while neglecting the core instructional architecture of problem sequences themselves. This study addresses these gaps by synthesizing insights from inquiry learning, cognitive load theory, and HOTS developmental models into a classroom-tested, theoretically grounded pathway for problem-chain teaching. Unlike prior approaches that treat questioning as discrete or intuitive, the present model offers a staged, cognitively aligned instructional framework specifically designed to promote HOTS in primary mathematics. Methodology Research Design This study adopted a theory-driven mixed-methods research design, consisting of three sequential phases: (1) diagnostic assessment of students’ higher-order thinking skills (HOTS), (2) development of a problem-chain teaching pathway grounded in cognitive and instructional theory, and (3) empirical validation of the pathway’s effectiveness through a quasi-experimental intervention. The quasi-experimental component followed a pre-test/post-test control group design. To ensure methodological rigor under naturalistic classroom conditions, random assignment was conducted at the student level within one school. The independent variable was the instructional method (problem-chain teaching vs. conventional teaching), and the dependent variable was students’ HOTS performance, measured across three dimensions: problem-solving, critical thinking, and creative thinking. Each research phase was directly aligned with a guiding research question and supported by specific theoretical frameworks. The diagnostic phase (RQ1) drew upon King et al.’s ( 1998 ) three-level HOTS model to map students’ cognitive profiles. The design phase (RQ2) integrated De Jong’s ( 2006 ) inquiry-based learning and Sweller’s ( 2011 ) cognitive load theory to develop a staged instructional sequence. The empirical phase (RQ3) validated the effectiveness of this design through quantitative outcome measures. In addition, the instructional design process was informed by classroom observations and teacher consultations. Each stage of the problem-chain pathway was mapped to a cognitive objective and piloted in a non-sample class to evaluate clarity, progression, and alignment with HOTS dimensions. Revisions were made iteratively to optimize both pedagogical coherence and cognitive scaffolding. Participants and Sampling A total of 316 fourth-grade students from three public primary schools in District A of Chengdu, China, participated in the initial diagnostic phase. These schools implement the 2022 National Mathematics Curriculum (Ministry of Education of the People's Republic of China, 2022 ), which emphasizes problem-solving and mathematical reasoning. Students had received minimal prior exposure to structured inquiry or question-based pedagogies. For the quasi-experimental phase, one school (School B) was purposively selected based on prior cooperation and logistical feasibility. From 115 eligible fourth graders, 50 students were randomly selected and evenly assigned to the experimental and control groups (n = 25 per group). Group equivalence in age and gender was confirmed via independent samples t-tests prior to intervention. All participants and their guardians provided informed consent, and pseudonymity was ensured throughout the study. Table 1 summarizes the demographic characteristics of the experimental and control groups. Both groups were comparable in terms of gender distribution and age, confirming baseline equivalence prior to the intervention. Table 1 Participants Demographics Variable Experimental Group (n = 25) Control Group (n = 25) Total (n = 50) Male 11 12 23 Female 14 13 27 Mean Age (SD)* 9.8(0.4) 9.7(0.5) 9.75(0.45) Instruments Two complementary instruments were employed to assess students’ HOTS from both subjective and objective perspectives: Higher-Order Thinking Questionnaire Adapted from Yu ( 2021 ) and refined through expert consultation with mathematics educators, this 21-item scale measures students’ self-perceived abilities in problem-solving, critical thinking, and creative thinking. Responses were recorded on a six-point Likert scale (1 = strongly disagree to 6 = strongly agree). The questionnaire demonstrated strong internal consistency (Cronbach’s α = 0.897), and construct validity was supported by a KMO value of 0.886 and a significant Bartlett’s test (p < 0.001). Higher-Order Thinking Performance Test This instrument was developed in accordance with the 2022 National Mathematics Curriculum Standards (Ministry of Education of the People's Republic of China, 2022 ) and drew on Bloom’s taxonomy and Alkhatib’s (2019) three-dimensional HOTS framework. The test integrated real-life, open-ended mathematical scenarios and was structured around three domains: problem-solving, critical thinking, and creative thinking. Scoring rubrics were pre-defined and applied using a double-blind grading protocol. Psychometric properties were satisfactory: test-retest reliability = 0.85, split-half reliability = 0.82, and CVI = 0.92. Construct validity was supported (KMO = 0.83), and item discrimination was within acceptable range. Appendix 1 and Appendix 2 present the full questionnaire and performance test respectively, including sample items and scoring criteria. Procedures The intervention lasted eight weeks, with three 40-minute mathematics sessions per week. Both groups covered the same mathematical content prescribed by the national curriculum. The experimental group received instruction through the seven-stage problem-chain teaching pathway: Preparation, Orientation, Conceptualization, Investigation, Conclusion, Discussion, and Reflection. Each lesson phase was deliberately aligned with specific HOTS dimensions and instructional goals. For example, the Conceptualization stage incorporated worked examples to reduce cognitive load, while the Discussion and Reflection stages emphasized dialogic learning and peer evaluation based on Lu’s ( 2021 ) model. To ensure fidelity of implementation, the instructor participated in two preparatory workshops on problem-chain pedagogy and received detailed lesson scripts. Two independent observers used a structured checklist to assess adherence to the teaching pathway in each session. Weekly debriefings with the instructor supported consistency and reflection. The control group followed conventional whole-class instruction based on the same curriculum but without the structured problem-chain sequence or explicit HOTS scaffolding. Pre- and post-tests were administered to both groups using the validated instruments. Scoring was conducted by assessors blinded to group assignment to minimize bias. Data Analysis All data were analyzed using SPSS 26.0. Descriptive statistics summarized demographic variables and pre-test HOTS levels. Group equivalence at baseline was examined using independent samples t-tests. To evaluate intervention effects, paired-sample t-tests were conducted for within-group comparisons, while ANCOVA was used to compare post-test performance between groups, controlling for pre-test scores. Cohen’s d was calculated to estimate effect sizes (0.2 = small, 0.5 = medium, 0.8 = large). All statistical tests adopted a significance threshold of p < 0.05. Results Diagnostic Survey of Students’ Higher-Order Thinking Skills Baseline diagnostic data from 316 fourth-grade students revealed a differentiated profile of higher-order thinking skills (HOTS). As shown in Table 2 , students reported the highest proficiency in problem-solving (M = 3.67), followed by critical thinking (M = 3.58), with creative thinking scoring the lowest (M = 3.44) on a 6-point Likert scale. Item-level analysis provided further insight into internal variance within each dimension. In particular, Item 18, which assessed the ability to connect mathematical concepts with other disciplines, recorded the lowest average score (M = 2.99), as shown in Fig. 1 . Table 2 Descriptive Statistics of HOTS Diagnostic Survey (N = 316) Dimension Min Max Mean Interpretation Problem-Solving 1 6 3.67 Moderate level, room for improvement Critical Thinking 1 6 3.58 Moderate to low, needs enhancement Creative Thinking 1 6 3.44 Relatively weak, priority for intervention Note: Responses were recorded on a 6-point Likert scale (1 = strongly disagree, 6 = strongly agree). These findings suggest that while students possess reasonable procedural fluency, they struggle with interdisciplinary reasoning and divergent thinking. This weakness informed the instructional design focus of the intervention, particularly through creative inquiry and peer-based reflection. Instructional Design of the Problem-Chain Teaching Pathway Design Rationale and Theoretical Alignment The problem-chain teaching pathway was designed as a structured, cognitively aligned instructional model to foster students’ higher-order thinking skills (HOTS) in primary mathematics. Drawing upon De Jong’s ( 2006 ) inquiry-based learning framework, Sweller’s ( 2011 ) cognitive load theory (CLT), and King et al.’s ( 1998 ) three-level HOTS model, the pathway scaffolds learners' progression from conceptual understanding to reflective reasoning. Specifically: De Jong’s model informed the inquiry-driven progression from orientation to conclusion; CLT guided the use of worked examples and visual scaffolds to manage intrinsic and extraneous cognitive load; King’s model framed the developmental logic of HOTS across three levels: acquisition, bridging, and application. This design also responded to findings from the diagnostic phase, which revealed weaknesses in students’ critical and creative thinking. These gaps were addressed through dialogic questioning, trap-based problems, and reflective tasks embedded throughout the pathway. To visualize how these theoretical perspectives converge into a coherent instructional design, the study developed a structured model that maps each theory onto specific teaching phases and targeted cognitive outcomes. As shown in Fig. 2 , the pathway aligns De Jong’s inquiry cycle, Sweller’s cognitive load principles, and King’s developmental model of HOTS with a seven-stage instructional process. This process scaffolds learners’ thinking from schema activation to self-regulation, progressively cultivating problem-solving, critical thinking, and creative thinking. (This figure illustrates a multi-layered instructional framework integrating De Jong’s inquiry learning, Sweller’s cognitive load theory (CLT), and King’s HOTS model. Seven teaching stages are aligned with specific mediating mechanisms—knowledge construction, reasoning, and meta-cognitive monitoring—that lead to the development of problem-solving, critical thinking, and creative thinking) The Seven-Stage Problem-Chain Pathway The teaching pathway consists of seven sequential stages, each mapped to a specific instructional function and cognitive target. This structure forms a “thinking arc” that supports progressive engagement from prior knowledge activation to meta-cognitive regulation. For clarity, the seven stages and their corresponding instructional objectives are shown in Table 3 . Each phase was guided by scaffolding materials including teacher prompts, sentence stems, and cognitive supports. The complete cycle was typically implemented over one or two 40-minute lessons depending on task complexity. Table 3 The Seven-Stage Problem-Chain Teaching Pathway No. Stage Instructional Focus Cognitive Target (HOTS) Theoretical Support 1 Preparation Activate prior knowledge; introduce big idea Conceptual readiness CLT (schema activation) 2 Orientation Pose initial open question; create cognitive tension Engagement & motivation De Jong (problem posing) 3 Conceptualization Provide worked examples & visual representations Concept understanding CLT (example-based learning) 4 Investigation Engage in student-led problem exploration Problem-solving, creativity Inquiry learning 5 Conclusion Synthesize findings and generalize strategies Abstraction, transfer Knowledge construction 6 Discussion Justify, critique, compare solutions Critical thinking, argumentation Dialogic learning 7 Reflection Reflect on strategies and thinking process Meta-cognition, self-regulation King’s HOTS model (level 3) Sample Implementation Snapshot Mini-Case: Trap-Based Reasoning in a Division Context In a Grade 4 unit on division with remainders, the following trap-based problem was used during the Investigation stage: “ A school has 37 students and 6 buses. If each bus carries 6 students, how many buses are needed? ” Most students initially divided 37 ÷ 6 = 6 remainder 1 and answered “6 buses,” overlooking that the remainder represents one more student still needing transportation. This common misconception was deliberately embedded to trigger cognitive conflict. In the Conclusion stage, students discussed why 7 buses were needed, revisiting the division logic and visualizing the “leftover.” The Discussion stage focused on evaluating reasoning patterns, using guiding questions like: “ What made this problem tricky? ” “ How did the remainder affect your solution? ” In Reflection, students completed self-assessment prompts such as: “ Today I learned that in real-life problems, the remainder can change the answer. I need to ask what the numbers mean, not just calculate .” This task exemplifies how the pathway scaffolds both problem-solving persistence and critical reinterpretation of intuitive answers, two components identified as weak in the diagnostic phase. Alignment with HOTS Dimensions To ensure that the instructional model systematically supports students’ higher-order thinking development, the seven-stage problem-chain teaching pathway was deliberately aligned with the three core HOTS dimensions proposed by Alkhatib (2019): problem-solving, critical thinking, and creative thinking. Rather than treating these dimensions as parallel or isolated, the design aimed to embed them in a progressive, mutually reinforcing manner across the instructional sequence. Problem-solving is primarily addressed through the stages of Preparation, Conceptualization, Investigation, and Conclusion. These phases scaffold the cognitive process from schema activation to abstraction and generalization. For instance, students begin by activating prior knowledge (Preparation), analyzing worked examples (Conceptualization), exploring novel problems (Investigation), and then synthesizing their findings (Conclusion). This sequence aligns with Sweller’s ( 2011 ) cognitive load theory and supports efficient cognitive progression for novice learners. Critical thinking is emphasized in the Orientation and Discussion stages. Here, students are introduced to open-ended or trap-based questions that create cognitive dissonance and are later encouraged to critique reasoning and compare alternative solutions in peer dialogue (Lu, 2021 ). These stages promote argumentation, justification, and reflective reasoning—key components of critical mathematical engagement. Creative thinking is fostered through the Investigation, Conclusion, and Reflection stages. During these phases, students are prompted to generate original strategies, apply knowledge across contexts, and reflect on their problem-solving approaches. The Reflection phase, in particular, supports metacognitive development and self-regulation in line with King et al.’s ( 1998 ) third level of HOTS progression. Importantly, the alignment is not rigidly compartmentalized. Students often engage in multiple forms of higher-order thinking simultaneously—for example, solving a novel problem (problem-solving), evaluating a peer’s reasoning (critical thinking), and proposing an alternative method (creative thinking). This interplay reflects the integrated nature of HOTS and reinforces the cognitive coherence of the pathway. Table 4 below summarizes the alignment between the instructional stages and each HOTS dimension, illustrating how the pathway systematically maps cognitive goals to specific pedagogical actions. Table 4 Instructional Alignment Between Teaching Stages and HOTS Dimensions HOTS Dimension Supported Through Stages Problem-Solving Stages 1, 3, 4, 5: Activate schema, example analysis, student exploration, generalization Critical Thinking Stages 2, 6: Pose cognitively dissonant tasks, critique reasoning, compare solutions Creative Thinking Stages 4, 5, 7: Generate alternative strategies, transfer across contexts, reflect and self-direct Intervention Outcomes Cognitive Gains in the Experimental Group Following the implementation of the problem-chain teaching pathway, students in the experimental group (n = 25) exhibited statistically significant pre-to-post gains in all three HOTS dimensions. Paired-sample t-tests showed large effect sizes: Problem-Solving: Mₚ r ₑ = 23.0 → Mₚₒₛₜ = 26.3, t(24) = -16.14, p < .001, d = 3.23 Critical Thinking: Mₚ r ₑ = 23.1 → Mₚₒₛₜ = 26.0, t(24) = -15.45, p < .001, d = 3.09 Creative Thinking: Mₚ r ₑ = 35.8 → Mₚₒₛₜ = 38.8, t(24) = -16.77, p < .001, d = 3.35 Notably, creative thinking showed the most substantial improvement, suggesting that the intervention effectively stimulated students’ capacity for generating novel ideas and applying mathematical reasoning in flexible ways. These gains were likely supported by the open-ended tasks, trap-based inquiry, and reflective dialogue embedded in the Investigation, Discussion, and Reflection stages of the pathway. These quantitative results are detailed in Table 5 , which presents the pre- and post-test means, standard deviations, t-statistics, and effect sizes for both experimental and control groups. The experimental group exhibited large and statistically significant gains in all three HOTS dimensions, with Cohen’s d values exceeding 3.0—indicating not only statistical significance but also strong instructional impact. By contrast, the control group showed minimal or no improvement across dimensions. Table 5 Pre- and Post-Test Mean Scores, t-Values, and Effect Sizes by Group (N = 50) Group Dimension Pre-test M (SD) Post-test M(SD) t p Cohen's d Experimental Problem-Solving 23.0 (1.0) 26.3 (0.9) -16.14 < .001 3.23 Experimental Critical Thinking 23.1 (1.0) 26.0 (0.9) -15.45 < .001 3.09 Experimental Creative Thinking 35.8 (1.1) 38.8 (0.8) -16.77 < .001 3.35 Control Problem-Solving 23.1 (1.2) 23.5 (1.3) 2.09 .047 — Control Critical Thinking 23.0 (1.1) 23.1 (1.2) 0.53 .600 — Control Creative Thinking 35.7 (1.3) 35.8 (1.4) 0.39 .703 — Note. All p-values 3.0 indicates large effect sizes. Figure 3 visualizes these changes, demonstrating clear upward trajectories across all dimensions for the experimental group, contrasted with flat or negligible gains in the control group. Stability in the Control Group In contrast, the control group (n = 25), which followed a conventional curriculum, exhibited minimal change. A slight improvement was observed in problem-solving (p = .047), but critical thinking and creative thinking remained statistically unchanged: Critical Thinking: Mₚ r ₑ = 23.0 → Mₚₒₛₜ = 23.1, t(24) = 0.53, p = .600 Creative Thinking: Mₚ r ₑ = 35.7 → Mₚₒₛₜ = 35.8, t(24) = 0.39, p = .703 This stagnation reinforces the claim that without targeted, cognitively rich instruction, students are unlikely to develop higher-order thinking skills beyond surface-level procedural comfort. Between-Group Comparison via ANCOVA To assess the differential effects of instruction, ANCOVA was conducted using pre-test scores as covariates. Results showed that the experimental group significantly outperformed the control group on all three HOTS dimensions at post-test (p .65). Figure 4 highlights the post-test means between groups. Across all dimensions, the experimental group held a clear advantage, with the largest margin in creative thinking. These results suggest that the design-based intervention was effective not only in fostering general improvement, but in specifically addressing cognitive areas previously identified as weak. Hypothesis Evaluation and Summary of Findings The findings provide robust support for both research hypotheses: H1: Students who received problem-chain teaching performed significantly better on HOTS than their peers in the control group → Supported H2: The experimental group demonstrated significant cognitive gains from pre- to post-test → Supported The convergence of quantitative trends and theoretical alignment affirms the design validity and pedagogical effectiveness of the problem-chain teaching pathway. The most notable impact in creative thinking suggests that carefully structured inquiry tasks and student-driven discourse can meaningfully shift students’ cognitive trajectories in mathematics learning. Discussion The present study provides robust empirical evidence that a structured problem-chain teaching pathway, grounded in cognitive and instructional theory, can significantly enhance primary students’ higher-order thinking skills (HOTS) in mathematics. This section discusses the findings in relation to previous research, theoretical underpinnings, and practical implications. It also critically reflects on the study’s limitations and proposes directions for future inquiry. Integrating Theory and Practice: Advancing HOTS through Structured Design The results confirm that elementary students are capable of engaging in complex reasoning tasks when scaffolded through well-sequenced instruction. The intervention’s most pronounced impact was observed in creative thinking (Cohen’s d = 3.35), followed closely by critical thinking and problem-solving. As shown in Figure 3, the experimental group demonstrated steep gains across all three HOTS domains, particularly in creative thinking. Furthermore, Figure 4 illustrates the post-test advantage of the experimental group over the control group in each dimension, with the most notable gap in creative thinking—an outcome that aligns with King et al.’s (1998) emphasis on meta-cognitive regulation and strategic transfer. These findings echo prior research suggesting that HOTS are not innate cognitive traits but developable competencies when aligned with appropriate pedagogical supports (Anderson & Krathwohl, 2001; Alkhatib, 2022; Dilekçi & Karatay, 2023). Unlike traditional instruction, which often isolates cognitive skills from instructional delivery, the problem-chain model explicitly integrated HOTS development into each stage of the lesson. The alignment between De Jong’s (2006) inquiry cycle, Sweller’s (2011) cognitive load principles, and King et al.’s (1998) HOTS model created a coherent cognitive architecture that moved students from schema activation to metacognitive reflection. This staged approach resonates with Wu et al. (2024), who demonstrated that systematically embedded thinking routines in instructional design produce measurable cognitive growth. Furthermore, the seven-stage structure fostered recursive engagement, with early stages (e.g., Orientation and Conceptualization) introducing cognitive tension and schema refinement, while later stages (e.g., Discussion and Reflection) promoted critical evaluation and creative transfer. These findings support the contention that instructionally embedded reflection is essential to consolidate and extend cognitive gains (Zimmerman, 2002; Lu, 2021). Creative Thinking as a Differentiator: From Procedures to Possibilitie s The marked improvement in creative thinking challenges the widespread assumption that creativity is peripheral or too abstract for primary mathematics (Yayuk & As’ari, 2020). Instead, the open-ended and trap-based (cognitively dissonant) tasks embedded in the Investigation and Reflection stages appeared to activate divergent thinking and encourage flexible strategy generation. These design features mirror those in recent HOTS-enhancement studies that highlight the power of ambiguity, error analysis, and analogical reasoning in stimulating creative insight (Szabo et al., 2020; Dilekçi & Karatay, 2023). Such results suggest that HOTS dimensions are not parallel domains but dynamically interrelated. For instance, students often engaged in problem-solving not as a procedural exercise but as an exploratory process that required reframing the task—a hallmark of creative cognition (Peng & Kievit, 2020). Similarly, critical thinking was frequently mobilized in service of creativity, as students compared multiple representations and reflected on solution plausibility. This reconceptualization aligns with Alkhatib’s (2019) tripartite HOTS framework, which advocates for integration rather than compartmentalization of cognitive skills. The problem-chain pathway operationalized this integration through instructional prompts that moved students beyond “correctness” to “appropriateness,” thus reinforcing the generative dimensions of mathematical reasoning. Instructional Design as Cognitive Architecture The intervention’s effectiveness can also be attributed to its careful attention to cognitive load. By sequencing activities from worked examples to independent inquiry, the pathway mitigated extraneous load while supporting germane processing—a core proposition of cognitive load theory (Sweller et al., 2019). In this way, the model supports novice learners’ progression toward expert-like reasoning, particularly in tasks that demand abstraction and transfer. These insights extend previous research on instructional coherence. Wu et al. (2024) emphasized that teaching interventions grounded in computational thinking (CT) principles promote critical literacy by reducing unnecessary complexity and increasing task transparency. Similarly, the present study shows that when tasks are designed with clear cognitive targets and sequencing logic, students are more likely to exhibit deep engagement and transfer. Additionally, the use of trap-based problems (e.g., the division-with-remainders scenario) introduced productive struggle, a concept supported by recent design-based research emphasizing the cognitive value of encountering and resolving contradictions (Kinnear et al., 2024; Nahar, 2022). Such cognitive dissonance may play a central role in triggering meta-cognitive monitoring, which King et al. (1998) identify as a key indicator of HOTS maturity. Empirical Contributions and Educational Significance The between-group ANCOVA results revealed significant post-test differences across all HOTS dimensions, with partial η² exceeding 0.65. These large effect sizes suggest not only statistical significance but practical utility, particularly in light of the modest sample size and real-classroom implementation. The design’s effectiveness under typical instructional constraints demonstrates its scalability and potential for adoption in other curricular contexts. Beyond statistical outcomes, the study contributes to the growing body of evidence that supports theory-driven pedagogical innovation in primary education. While many studies explore inquiry learning or cognitive load principles in isolation, this research demonstrates how their integration—when aligned with HOTS developmental theory—can yield compounding cognitive benefits. This triangulated design approach aligns with the findings of Preckel et al. (2020) and Greiff & Borgonovi (2022), who argue that cross-domain instructional synthesis is key to 21st-century competence development. Furthermore, the study addresses persistent concerns in mathematics education regarding the marginalization of critical and creative thinking in favor of procedural fluency (Jablonka, 2020; Yayuk & As' ari, 2020). By embedding HOTS development into the very structure of problem sequencing, the model repositions mathematical instruction as a venue for deep thinking rather than rote application. Pedagogical and Policy Implications The findings of this study underscore the feasibility of integrating higher-order thinking skills into early mathematics instruction without sacrificing curricular coherence or instructional clarity. When supported by deliberate scaffolding and cognitively sequenced design, even young learners can meaningfully engage with abstraction, critical evaluation, and creative synthesis. This challenges conventional assumptions that such capacities emerge only in later stages of schooling and supports growing advocacy for embedding cognitive development objectives across all grade levels (González-Salamanca et al., 2020; Ramírez-Montoya, 2022). From a professional development perspective, the problem-chain teaching pathway offers a replicable framework for cultivating teacher capacity in HOTS-oriented instruction. Rather than relying on intuitive or fragmented approaches, teachers can draw upon a theoretically grounded sequence of stages that guide the orchestration of inquiry, cognitive challenge, and reflective discourse. This approach aligns with Qing-li et al.'s (2024) conception of teachers as “curriculum makers” who design for deep thinking through purposeful, cross-disciplinary integration. At the policy level, the study illustrates how micro-level instructional decisions—such as task sequencing, questioning strategy, and reflection prompts—can serve as levers for macro-level educational transformation. Grounding these decisions in cognitive science and classroom data allows for scalable curriculum reform that transcends textbook-driven routines. In doing so, the model contributes to the broader agenda of fostering 21st-century competencies through evidence-based pedagogy. Limitations and Future Research Despite its strengths, the study has several limitations. The relatively small and geographically concentrated sample limits generalizability. Moreover, the study focused primarily on cognitive outcomes; affective, motivational, and social dimensions were not systematically examined. While fidelity of implementation was supported through observation and teacher debriefing, teacher beliefs and agency—key mediators of instructional innovation—warrant further exploration. Future research could expand the model to other subject areas or cultural contexts to examine its cross-domain adaptability. Longitudinal studies could also assess the sustainability of HOTS gains over time. Additionally, integrating digital tools such as AI tutors or adaptive platforms might enhance personalization and expand the design’s reach—an avenue explored in recent work on GPT-supported instructional planning (Hu et al., 2024; Wang et al., 2025). Conclusion This study set out to design, implement, and evaluate a problem-chain teaching pathway aimed at enhancing fourth-grade students’ higher-order thinking skills (HOTS) in mathematics. Grounded in established theoretical frameworks—including inquiry-based learning, cognitive load theory, and HOTS development models—the pathway offered a structured yet flexible sequence of instructional stages that embedded cognitive challenge into everyday classroom practice. The results of the quasi-experimental study provide compelling evidence that the intervention significantly improved students’ problem-solving, critical thinking, and creative thinking, with particularly large gains in the latter two dimensions. These findings support the view that elementary students are capable of engaging in deep mathematical reasoning when supported by intentionally designed, cognitively coherent instruction. They also demonstrate that such gains are not incidental, but the result of pedagogical design that aligns tasks, discourse, and reflection with cognitive development principles. In addition to affirming the effectiveness of the intervention, the study contributes to the growing literature on design-based educational research. It highlights the potential of using diagnostic data to inform instructional design, of treating lesson planning as a form of cognitive architecture, and of empowering teachers to orchestrate learning environments that support not only procedural fluency, but adaptive reasoning and strategic innovation. While the study is not without limitations—including sample scope, short-term measurement, and limited focus on teacher implementation—it offers a compelling proof of concept for advancing HOTS in primary mathematics through theory-driven pedagogy. The problem-chain teaching pathway, with its clear structure and empirical grounding, holds promise as a model for future curriculum design, teacher professional development, and educational policy aiming to promote deeper learning in the early years. Declarations Ethics approval and consent to participate This study was submitted to and approved by the Ethics Committee of the Faculty of Education, Sichuan Normal University. The study complied with the ethical standards outlined in the Declaration of Helsinki. All procedures involving human participants were conducted in accordance with the institutional guidelines of Sichuan Normal University and the collaborating primary schools. Written informed consent was obtained from the legal guardians of all participating students, and verbal assent was obtained from the students. Participation was voluntary, and participants could withdraw at any time without any negative consequences. Data confidentiality and anonymity were strictly maintained throughout the study. Consent for publication Consent for publication was not required as no identifying personal information or images of participants are included in this manuscript. All data are anonymized and reported in aggregate form to ensure confidentiality. Competing interests The author declares that there are no conflicts of interests regarding the publication of this paper. Funding This study received no specific financial support. Authors' contributions SY conceptualized and designed the study, developed the instructional model, performed the statistical analysis, and drafted the manuscript. QX contributed to data collection, preliminary data analysis, and critical revision of the manuscript. All authors read and approved the final manuscript. Availability of Data and Materials The datasets used and analysed in the current study are available from the first author on reasonable request. Acknowledgements We are grateful to the participating schools, teachers, and students for their generous cooperation and engagement in this study. Special thanks are extended to colleagues at Sichuan Normal University for their valuable feedback and support. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References Alkhatib OJ. (2019, March). A framework for implementing higher-order thinking skills (problem-solving, critical thinking, creative thinking, and decision-making) in engineering & humanities. In 2019 Advances in science and engineering technology international conferences (ASET) (pp. 1–8). IEEE. https://doi.org/10.1109/ICASET.2019.8714232 Alkhatib OJ. (2022, February). An effective assessment method of higher-order thinking skills (problem-solving, critical thinking, creative thinking, and decision-making) in engineering and humanities. In 2022 Advances in Science and Engineering Technology International Conferences (ASET) (pp. 1–6). IEEE. https://doi.org/10.1109/ASET53988.2022.9734856 Anderson LW, Krathwohl DR. A taxonomy for learning, teaching, and assessing: A revision of Bloom’s taxonomy of educational objectives. Allyn & Bacon; 2001. Bloom B, editor. Taxonomy of educational objectives: Book I, cognitive domain. New York: Longman; 1956. Çakıroğlu Ü, Güler M, Dündar M, Coşkun F. Virtual reality in realistic mathematics education to develop mathematical literacy skills. Int J Human–Computer Interact. 2024;40(17):4661–73. https://doi.org/10.1080/10447318.2023.2219960 . De Jong T. Technological advances in inquiry learning. Science. 2006;312(5773):532–3. https://doi.org/10.1126/science.1127750 . Dilekçi A, Karatay H. The effects of the 21st century skills curriculum on the development of students’ creative thinking skills. Think skills creativity. 2023;47:101229. https://doi.org/10.1016/j.tsc.2022.101229 . Ding F, Zhang W, Tang H. Problem chain teaching design for mathematics key competencies. Educational Sci Res. 2021;9:62–6. [In Chinese]. Feng YY, Su GH, Zhang DQ. Dao FA Shu Qi in Mathematics Problem Chain Teaching. J Math Educ. 2023;32(5):42–6. [In Chinese]. González-Pérez LI, Ramírez-Montoya MS. Components of Education 4.0 in 21st century skills frameworks: systematic review. Sustainability. 2022;14(3):1493. https://doi.org/10.3390/su14031493 . González-Salamanca JC, Agudelo OL, Salinas J. Key competences, education for sustainable development and strategies for the development of 21st century skills. A systematic literature review. Sustainability. 2020;12(24):10366. https://doi.org/10.3390/su122410366 . Greiff S, Borgonovi F. Teaching of 21st century skills needs to be informed by psychological research. Nat Reviews Psychol. 2022;1(6):314–5. https://doi.org/10.1038/s44159-022-00064-w . Hu B, Zheng L, Zhu J, Ding L, Wang Y, Gu X. Teaching plan generation and evaluation with GPT-4: Unleashing the potential of LLM in instructional design. IEEE Trans Learn Technol. 2024. https://doi.org/10.1109/tlt.2024.3384765 . Hu B, Zhu J, Pei Y, Gu X. Exploring the potential of LLM to enhance teaching plans through teaching simulation. npj Sci Learn. 2025;10(1):7. https://doi.org/10.1038/s41539-025-00300-x . Jablonka E. (2020). Critical thinking in mathematics education (pp. 159–163). In S. Lerman, editor, Encyclopedia of mathematics education (2nd ed.). Springer. https://doi.org/10.1007/978-3-030-15789-0_35 King FJ, Goodson L, Rohani F. (1998). Higher order thinking skills: Definition, teaching strategies, assessment. Goodson, F. Rohani. Tallahasee, FL: Center for the Advancement of Learning and Assessment Florida State University.[Electronic resource].– URL : http://www.cala.fsu.edu/files/higher_order_ thinking_skills . pdf (date of treatment: 07.09. 2021). Kinnear G, Hood G, Lardet E, Sheard C, Foster C. Lecturers' use of questions in undergraduate mathematics lectures. J Math Behav. 2024;76:101190. https://doi.org/10.1016/j.jmathb.2024.101190 . Lu S. Collaborative inquiry in primary mathematics: Enhancing critical thinking through peer dialogue. J Math Educ. 2021;14(3):212–29. https://doi.org/10.1080/14794802.2021.1937059 . Ministry of Education of the People's Republic of China. Compulsory education mathematics curriculum standards (2022 edition). Beijing: People's Education; 2022. Nahar S. Improving Students' Collaboration Thinking Skill under the Implementation of the Quantum Teaching Model. Int J Instruction. 2022;15(3):451–64. https://doi.org/10.29333/iji.2022.15325a . Peng P, Kievit RA. The development of academic achievement and cognitive abilities: A bidirectional perspective. Child Dev Perspect. 2020;14(1):15–20. https://doi.org/10.1111/cdep.12352 . Preckel F, Golle J, Grabner R, Jarvin L, Kozbelt A, Müllensiefen D, Worrell FC. Talent development in achievement domains: A psychological framework for within-and cross-domain research. Perspect Psychol Sci. 2020;15(3):691–722. https://doi.org/10.1177/1745691619895030 . Qing-li H, Meiqi F, Yu T, Xiaoxiao T, An-duo L. Evolving into curriculum makers: the pivotal role of geography teachers as boundary teachers. Int Res Geographical Environ Educ. 2024;1–21. https://doi.org/10.1080/10382046.2024.2426398 . Ramírez-Montoya MS, McGreal R, Agbu JFO. Horizontes digitales complejos en el futuro de la educación 4.0: luces desde las recomendaciones de UNESCO. RIED. Revista Iberoamericana de Educación Distancia. 2022;25(2):09–21. https://doi.org/10.5944/ried.25.2.33843 . Sweller J. Cognitive load theory. Psychol Learn Motivation. 2011;55:37–76. https://doi.org/10.1016/B978-0-12-387691-1.00002-8 . Sweller J, van Merriënboer JJG, Paas F. Cognitive architecture and instructional design: 20 years later. Educational Psychol Rev. 2019;31(2):261–92. https://doi.org/10.1007/s10648-019-09465-5 . Szabo ZK, Körtesi P, Guncaga J, Szabo D, Neag R. Examples of problem-solving strategies in mathematics education supporting the sustainability of 21st-century skills. Sustainability. 2020;12(23):10113. https://doi.org/10.3390/su122310113 . Tang F. Understanding the role of digital immersive technology in educating the students of english language: does it promote critical thinking and self-directed learning for achieving sustainability in education with the help of teamwork? BMC Psychol. 2024;12(1):144. https://doi.org/10.1186/s40359-024-01636-6 . Tang HJ, Zhang WZ, Chen BF. Question-Chain Teaching for Deep Understanding. Educational Dev Res. 2020;40(4):53–7. https://doi.org/10.14121/j.cnki.1008-3855.2020.04.011 . [In Chinese]. Vygotsky LS. Mind in society: The development of higher psychological processes. Volume 86. Harvard University Press; 1978. Wang WS, Lin CJ, Lee HY, Huang YM, Wu TT. Enhancing self-regulated learning and higher-order thinking skills in virtual reality: the impact of ChatGPT-integrated feedback aids. Educ Inform Technol. 2025;1–27. https://doi.org/10.1007/s10639-025-13557-x . Wu TT, Silitonga LM, Murti AT. Enhancing English writing and higher-order thinking skills through computational thinking. Comput Educ. 2024;213:105012. https://doi.org/10.1016/j.compedu.2024.105012 . Yayuk E. & As' ari, A. R. (2020). Primary School Students' Creative Thinking Skills in Mathematics Problem Solving. European Journal of Educational Research , 9 (3), 1281–1295. https://doi.org/10.12973/eu-jer.9.3.1281 Yu JJ. (2021). Research on Senior School Mathematics Teaching Strategies Based on the Cultivation of Higher-Order Thinking Ability [Master's thesis, Shandong Normal University].[In Chinese]. https://link.cnki.net/doi/10.27280/d.cnki.gsdsu.2021.001318 Zimmerman BJ. Becoming a self-regulated learner: An overview. Theory Into Pract. 2002;41(2):64–70. https://doi.org/10.1207/s15430421tip4102_2 . Additional Declarations No competing interests reported. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6782865","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":483237165,"identity":"5a569ea5-d12c-4d25-9d7c-3ac994c3dcac","order_by":0,"name":"Siliang 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Item\u003c/em\u003e\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-6782865/v1/ad233025bf190eff13dcaf96.png"},{"id":86665865,"identity":"6c06596a-a4ea-42e8-b813-48eea6641150","added_by":"auto","created_at":"2025-07-14 11:07:35","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":71274,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eTheoretical Instructional Pathway for HOTS via Problem-Chain Design\u003c/em\u003e\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6782865/v1/5bf3d3eef98a39002c38c63a.png"},{"id":86667370,"identity":"9ed59a32-0f5a-4a45-b80f-0f1a481392c8","added_by":"auto","created_at":"2025-07-14 11:15:35","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":47681,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eComparison of Pre- and Post-Test Mean Scores by Group and Dimension\u003c/em\u003e\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-6782865/v1/737c3e44bff1546d232560b6.png"},{"id":86667373,"identity":"95e009e8-970b-4a6d-95da-2eb6e150fca6","added_by":"auto","created_at":"2025-07-14 11:15:35","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":73045,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003ePost-Test HOTS Profile: Experimental vs. Control Group\u003c/em\u003e\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-6782865/v1/cebd0afdf5c809821e408856.png"},{"id":103310531,"identity":"463108fe-f24d-42d0-9e6d-1e44822b6a58","added_by":"auto","created_at":"2026-02-24 09:58:20","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1527453,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6782865/v1/9799d28a-ab7f-4c95-ae19-502634ff38cb.pdf"},{"id":86665868,"identity":"2d466563-970c-44b0-b725-eb80aa7dd804","added_by":"auto","created_at":"2025-07-14 11:07:35","extension":"docx","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":29080,"visible":true,"origin":"","legend":"","description":"","filename":"Appendix.docx","url":"https://assets-eu.researchsquare.com/files/rs-6782865/v1/220f260fca2e4dc7a47667d9.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Designing for Deep Thinking: A Theory-Driven Inquiry into Problem-Chain Teaching in Primary Mathematics","fulltext":[{"header":"Introduction","content":"\u003cp\u003eIn an era increasingly shaped by artificial intelligence and digital transformation, the ability to think critically, solve complex problems, and apply knowledge creatively has become a defining competency for 21st-century learners (Gonz\u0026aacute;lez-P\u0026eacute;rez \u0026amp; Ram\u0026iacute;rez-Montoya, 2022). As automation rapidly replaces routine cognitive tasks, education systems must respond by cultivating students\u0026rsquo; higher-order thinking skills (HOTS) that support deep reasoning and adaptive learning (Gonz\u0026aacute;lez-Salamanca et al., 2020; Ram\u0026iacute;rez-Montoya, 2022; Tang et al., 2024).\u003c/p\u003e\n\u003cp\u003ePrimary education, particularly mathematics, offers a foundational context for nurturing such competencies. Mathematics is not only a discipline of logic and abstraction but also a fertile ground for structured inquiry, problem-solving, and conceptual transfer (Jablonka, 2020; Peng \u0026amp; Kievit, 2020). However, traditional instructional methods\u0026mdash;often centered on procedural fluency and rote practice\u0026mdash;tend to underutilize the potential of early mathematics learning as a platform for higher-order thinking development.\u003c/p\u003e\n\u003cp\u003eTo address this gap, problem-chain teaching has emerged as a promising pedagogical approach that scaffolds student thinking through logically sequenced, interrelated problems. Grounded in constructivist principles and aligned with Vygotsky\u0026rsquo;s (1978) zone of proximal development, this approach guides learners from basic understanding toward more complex cognitive operations such as analysis, synthesis, and innovation (Tang et al., 2020; Ding et al., 2021).\u003c/p\u003e\n\u003cp\u003eBuilding on this foundation, the present study incorporates three complementary theoretical frameworks to inform the design of a structured instructional pathway. De Jong\u0026rsquo;s (2006) inquiry-based learning framework supports the progression from teacher-guided orientation to student-led exploration and reflection, ensuring that knowledge is actively constructed rather than passively received. Sweller\u0026rsquo;s (2011) cognitive load theory offers principles for managing mental effort during instruction\u0026mdash;particularly through techniques such as worked examples and visual scaffolds\u0026mdash;to prevent overload and facilitate schema acquisition. Meanwhile, King et al.\u0026rsquo;s (1998) three-tier HOTS development model provides a developmental scaffold that sequences learning from foundational knowledge acquisition, to cognitive bridging, and ultimately to advanced application and transfer. Together, these theories contribute not only to the structure of the teaching pathway but also to its cognitive coherence\u0026mdash;ensuring that each phase of instruction aligns with targeted thinking outcomes.\u003c/p\u003e\n\u003cp\u003eDespite the conceptual potential of problem-chain teaching, the literature remains fragmented\u0026mdash;either focused on macro-level frameworks or on technology-enhanced learning environments\u0026mdash;leaving a critical need for theoretically grounded, classroom-tested models specifically designed to foster HOTS in primary mathematics (\u0026Ccedil;akıroğlu et al., 2024; Hu et al., 2025).\u003c/p\u003e\n\u003cp\u003eThis study responds to that need by designing and empirically validating a structured problem-chain teaching pathway that supports the systematic cultivation of higher-order thinking. Drawing upon De Jong\u0026rsquo;s (2006) inquiry-based learning framework and King et al.\u0026rsquo;s (1998) three-tier model of HOTS development, the study aims to construct a replicable instructional model that bridges theory and classroom practice.\u003c/p\u003e\n\u003cp\u003eTo address the gap between theoretical promise and classroom applicability, this study aims to design and empirically validate a structured, theory-informed problem-chain teaching pathway for primary mathematics education. Specifically, the research is guided by three questions:\u003c/p\u003e\n\u003col\u003e\n \u003cli\u003eWhat is the current profile of fourth-grade students\u0026rsquo; higher-order thinking skills (HOTS) in mathematics?\u003c/li\u003e\n \u003cli\u003eHow can a problem-chain teaching pathway be systematically constructed to foster HOTS based on established cognitive and instructional theories?\u003c/li\u003e\n \u003cli\u003eTo what extent does the proposed pathway significantly outperform conventional instruction in enhancing students\u0026rsquo; HOTS?\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThis inquiry contributes to the growing body of design-based research that seeks to align pedagogical innovation with cognitive development goals in elementary education.\u003c/p\u003e"},{"header":"Literature Review","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003eHigher-Order Thinking Skills: Conceptualizations and Developmental Models\u003c/h2\u003e\u003cp\u003eHigher-order thinking skills (HOTS) are widely regarded as essential competencies for 21st-century learners, enabling students to engage in complex reasoning, knowledge transfer, and adaptive problem-solving. Traditionally, Bloom\u0026rsquo;s (\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1956\u003c/span\u003e) taxonomy identified analysis, synthesis, and evaluation as markers of higher-order cognition. Later, Anderson and Krathwohl\u0026rsquo;s (\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2001\u003c/span\u003e) revision elevated \u0026ldquo;creation\u0026rdquo; as the apex of cognitive processing, thereby emphasizing the role of innovation in contemporary education.\u003c/p\u003e\u003cp\u003eContemporary frameworks have further refined HOTS into multidimensional constructs. Alkhatib (2019, 2022), for example, proposed a tripartite model comprising problem-solving, critical thinking, and creative thinking\u0026mdash;dimensions particularly aligned with the goals of mathematics education and interdisciplinary integration. These dimensions reflect not only cognitive complexity but also the ability to apply knowledge flexibly and reflectively across novel contexts.\u003c/p\u003e\u003cp\u003eBuilding on these conceptualizations, King et al. (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e1998\u003c/span\u003e) introduced a three-tier model of HOTS development, distinguishing between foundational knowledge acquisition, bridging through scaffolding, and advanced application. This developmental progression provides a scaffolded framework for instructional design that gradually transitions students from lower-order cognition to higher-order performance.\u003c/p\u003e\u003c/div\u003e\n\u003ch3\u003eInstructional Pathways for HOTS: From Cognitive Theory to Design Principles\u003c/h3\u003e\n\u003cp\u003eTranslating HOTS into effective classroom practice requires alignment with cognitive and instructional theories. Sweller\u0026rsquo;s (\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2011\u003c/span\u003e) cognitive load theory (CLT) highlights the importance of managing mental effort during learning, particularly through strategies such as worked examples that reduce extraneous load and facilitate schema acquisition. These principles are especially relevant for novice learners encountering complex problem-solving tasks (Sweller et al., \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2019\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eIn parallel, inquiry-based learning frameworks (De Jong, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2006\u003c/span\u003e) advocate for student-driven exploration, emphasizing the active construction of knowledge through question posing, investigation, and reflection. Although rooted in different assumptions\u0026mdash;CLT focusing on efficiency, inquiry learning emphasizing autonomy\u0026mdash;these theories can be integrated through instructional sequencing that transitions from structured modeling to exploratory engagement.\u003c/p\u003e\u003cp\u003eFurthermore, dialogic and collaborative learning paradigms (Lu, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) underscore the role of peer interaction and discourse in deepening cognitive engagement and meta-cognitive regulation. These perspectives collectively inform the design of instructional models that aim not only to develop HOTS, but to sustain them through reflective and socially mediated processes.\u003c/p\u003e\n\u003ch3\u003eProblem-Chain Teaching: A Promising Yet Under-Specified Approach\u003c/h3\u003e\n\u003cp\u003eWithin mathematics education, problem-chain teaching has gained traction as a method for fostering cognitive progression through a series of logically sequenced, interdependent problems. Grounded in constructivist theory and Vygotsky\u0026rsquo;s (\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e1978\u003c/span\u003e) concept of the zone of proximal development, this approach enables learners to build upon prior knowledge and gradually reach higher levels of abstraction and transfer (Tang et al., \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Ding et al., \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eRecent empirical studies have offered valuable typologies for problem-chain design. Hu et al. (\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2024\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2025\u003c/span\u003e) identified three core types\u0026mdash;contextual, trap-based, and summarizing questions\u0026mdash;each serving distinct pedagogical functions. Similarly, Kinnear et al. (\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2024\u003c/span\u003e) demonstrated that structured questioning sequences can enhance engagement and deepen conceptual understanding.\u003c/p\u003e\u003cp\u003eDespite these advances, two key limitations persist. First, most research remains at the conceptual or theoretical level, with limited translation into replicable classroom models tailored to primary education (Feng et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Qing-li et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Second, studies often emphasize digital tool integration or teacher training, while neglecting the core instructional architecture of problem sequences themselves.\u003c/p\u003e\u003cp\u003eThis study addresses these gaps by synthesizing insights from inquiry learning, cognitive load theory, and HOTS developmental models into a classroom-tested, theoretically grounded pathway for problem-chain teaching. Unlike prior approaches that treat questioning as discrete or intuitive, the present model offers a staged, cognitively aligned instructional framework specifically designed to promote HOTS in primary mathematics.\u003c/p\u003e"},{"header":"Methodology","content":"\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\u003ch2\u003eResearch Design\u003c/h2\u003e\u003cp\u003eThis study adopted a theory-driven mixed-methods research design, consisting of three sequential phases: (1) diagnostic assessment of students’ higher-order thinking skills (HOTS), (2) development of a problem-chain teaching pathway grounded in cognitive and instructional theory, and (3) empirical validation of the pathway’s effectiveness through a quasi-experimental intervention.\u003c/p\u003e\u003cp\u003eThe quasi-experimental component followed a pre-test/post-test control group design. To ensure methodological rigor under naturalistic classroom conditions, random assignment was conducted at the student level within one school. The independent variable was the instructional method (problem-chain teaching vs. conventional teaching), and the dependent variable was students’ HOTS performance, measured across three dimensions: problem-solving, critical thinking, and creative thinking.\u003c/p\u003e\u003cp\u003eEach research phase was directly aligned with a guiding research question and supported by specific theoretical frameworks. The diagnostic phase (RQ1) drew upon King et al.’s (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e1998\u003c/span\u003e) three-level HOTS model to map students’ cognitive profiles. The design phase (RQ2) integrated De Jong’s (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2006\u003c/span\u003e) inquiry-based learning and Sweller’s (\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2011\u003c/span\u003e) cognitive load theory to develop a staged instructional sequence. The empirical phase (RQ3) validated the effectiveness of this design through quantitative outcome measures.\u003c/p\u003e\u003cp\u003eIn addition, the instructional design process was informed by classroom observations and teacher consultations. Each stage of the problem-chain pathway was mapped to a cognitive objective and piloted in a non-sample class to evaluate clarity, progression, and alignment with HOTS dimensions. Revisions were made iteratively to optimize both pedagogical coherence and cognitive scaffolding.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003eParticipants and Sampling\u003c/h2\u003e\u003cp\u003eA total of 316 fourth-grade students from three public primary schools in District A of Chengdu, China, participated in the initial diagnostic phase. These schools implement the 2022 National Mathematics Curriculum (Ministry of Education of the People's Republic of China, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), which emphasizes problem-solving and mathematical reasoning. Students had received minimal prior exposure to structured inquiry or question-based pedagogies.\u003c/p\u003e\u003cp\u003eFor the quasi-experimental phase, one school (School B) was purposively selected based on prior cooperation and logistical feasibility. From 115 eligible fourth graders, 50 students were randomly selected and evenly assigned to the experimental and control groups (n = 25 per group). Group equivalence in age and gender was confirmed via independent samples t-tests prior to intervention.\u003c/p\u003e\u003cp\u003e All participants and their guardians provided informed consent, and pseudonymity was ensured throughout the study. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e summarizes the demographic characteristics of the experimental and control groups. Both groups were comparable in terms of gender distribution and age, confirming baseline equivalence prior to the intervention.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eParticipants Demographics\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eVariable\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eExperimental Group (n = 25)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eControl Group (n = 25)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eTotal (n = 50)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eMale\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e11\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e12\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e23\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eFemale\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e14\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e13\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e27\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eMean Age (SD)*\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e9.8(0.4)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e9.7(0.5)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e9.75(0.45)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\n\u003ch3\u003eInstruments\u003c/h3\u003e\n\u003cp\u003eTwo complementary instruments were employed to assess students’ HOTS from both subjective and objective perspectives:\u003c/p\u003e\n\u003ch3\u003eHigher-Order Thinking Questionnaire\u003c/h3\u003e\n\u003cp\u003eAdapted from Yu (\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) and refined through expert consultation with mathematics educators, this 21-item scale measures students’ self-perceived abilities in problem-solving, critical thinking, and creative thinking. Responses were recorded on a six-point Likert scale (1 = strongly disagree to 6 = strongly agree). The questionnaire demonstrated strong internal consistency (Cronbach’s α = 0.897), and construct validity was supported by a KMO value of 0.886 and a significant Bartlett’s test (p \u0026lt; 0.001).\u003c/p\u003e\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\u003ch2\u003eHigher-Order Thinking Performance Test\u003c/h2\u003e\u003cp\u003eThis instrument was developed in accordance with the 2022 National Mathematics Curriculum Standards (Ministry of Education of the People's Republic of China, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) and drew on Bloom’s taxonomy and Alkhatib’s (2019) three-dimensional HOTS framework. The test integrated real-life, open-ended mathematical scenarios and was structured around three domains: problem-solving, critical thinking, and creative thinking. Scoring rubrics were pre-defined and applied using a double-blind grading protocol. Psychometric properties were satisfactory: test-retest reliability = 0.85, split-half reliability = 0.82, and CVI = 0.92. Construct validity was supported (KMO = 0.83), and item discrimination was within acceptable range. Appendix 1 and Appendix 2 present the full questionnaire and performance test respectively, including sample items and scoring criteria.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\u003ch2\u003eProcedures\u003c/h2\u003e\u003cp\u003eThe intervention lasted eight weeks, with three 40-minute mathematics sessions per week. Both groups covered the same mathematical content prescribed by the national curriculum.\u003c/p\u003e\u003cp\u003eThe experimental group received instruction through the seven-stage problem-chain teaching pathway: Preparation, Orientation, Conceptualization, Investigation, Conclusion, Discussion, and Reflection. Each lesson phase was deliberately aligned with specific HOTS dimensions and instructional goals. For example, the Conceptualization stage incorporated worked examples to reduce cognitive load, while the Discussion and Reflection stages emphasized dialogic learning and peer evaluation based on Lu’s (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) model.\u003c/p\u003e\u003cp\u003eTo ensure fidelity of implementation, the instructor participated in two preparatory workshops on problem-chain pedagogy and received detailed lesson scripts. Two independent observers used a structured checklist to assess adherence to the teaching pathway in each session. Weekly debriefings with the instructor supported consistency and reflection.\u003c/p\u003e\u003cp\u003eThe control group followed conventional whole-class instruction based on the same curriculum but without the structured problem-chain sequence or explicit HOTS scaffolding.\u003c/p\u003e\u003cp\u003ePre- and post-tests were administered to both groups using the validated instruments. Scoring was conducted by assessors blinded to group assignment to minimize bias.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\u003ch2\u003eData Analysis\u003c/h2\u003e\u003cp\u003eAll data were analyzed using SPSS 26.0. Descriptive statistics summarized demographic variables and pre-test HOTS levels. Group equivalence at baseline was examined using independent samples t-tests.\u003c/p\u003e\u003cp\u003eTo evaluate intervention effects, paired-sample t-tests were conducted for within-group comparisons, while ANCOVA was used to compare post-test performance between groups, controlling for pre-test scores. Cohen’s d was calculated to estimate effect sizes (0.2 = small, 0.5 = medium, 0.8 = large). All statistical tests adopted a significance threshold of p \u0026lt; 0.05.\u003c/p\u003e\u003c/div\u003e"},{"header":"Results","content":"\u003ch2\u003eDiagnostic Survey of Students\u0026rsquo; Higher-Order Thinking Skills\u003c/h2\u003e\n\u003cp\u003eBaseline diagnostic data from 316 fourth-grade students revealed a differentiated profile of higher-order thinking skills (HOTS). As shown in Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, students reported the highest proficiency in problem-solving (M\u0026thinsp;=\u0026thinsp;3.67), followed by critical thinking (M\u0026thinsp;=\u0026thinsp;3.58), with creative thinking scoring the lowest (M\u0026thinsp;=\u0026thinsp;3.44) on a 6-point Likert scale.\u003c/p\u003e\n\u003cp\u003eItem-level analysis provided further insight into internal variance within each dimension. In particular, Item 18, which assessed the ability to connect mathematical concepts with other disciplines, recorded the lowest average score (M\u0026thinsp;=\u0026thinsp;2.99), as shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003e\u003cem\u003eDescriptive Statistics of HOTS Diagnostic Survey (N\u0026thinsp;=\u0026thinsp;316)\u003c/em\u003e\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eDimension\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMin\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMax\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMean\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eInterpretation\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eProblem-Solving\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3.67\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eModerate level, room for improvement\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCritical Thinking\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3.58\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eModerate to low, needs enhancement\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCreative Thinking\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3.44\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eRelatively weak, priority for intervention\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003ctfoot\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"5\"\u003eNote: Responses were recorded on a 6-point Likert scale (1\u0026thinsp;=\u0026thinsp;strongly disagree, 6\u0026thinsp;=\u0026thinsp;strongly agree).\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tfoot\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eThese findings suggest that while students possess reasonable procedural fluency, they struggle with interdisciplinary reasoning and divergent thinking. This weakness informed the instructional design focus of the intervention, particularly through creative inquiry and peer-based reflection.\u003c/p\u003e\n\u003ch2\u003eInstructional Design of the Problem-Chain Teaching Pathway\u003c/h2\u003e\n\u003ch2\u003eDesign Rationale and Theoretical Alignment\u003c/h2\u003e\n\u003cp\u003eThe problem-chain teaching pathway was designed as a structured, cognitively aligned instructional model to foster students\u0026rsquo; higher-order thinking skills (HOTS) in primary mathematics. Drawing upon De Jong\u0026rsquo;s (\u003cspan class=\"CitationRef\"\u003e2006\u003c/span\u003e) inquiry-based learning framework, Sweller\u0026rsquo;s (\u003cspan class=\"CitationRef\"\u003e2011\u003c/span\u003e) cognitive load theory (CLT), and King et al.\u0026rsquo;s (\u003cspan class=\"CitationRef\"\u003e1998\u003c/span\u003e) three-level HOTS model, the pathway scaffolds learners\u0026apos; progression from conceptual understanding to reflective reasoning. Specifically:\u003c/p\u003e\n\u003cp\u003eDe Jong\u0026rsquo;s model informed the inquiry-driven progression from orientation to conclusion;\u003c/p\u003e\n\u003cp\u003eCLT guided the use of worked examples and visual scaffolds to manage intrinsic and extraneous cognitive load;\u003c/p\u003e\n\u003cp\u003eKing\u0026rsquo;s model framed the developmental logic of HOTS across three levels: acquisition, bridging, and application.\u003c/p\u003e\n\u003cp\u003eThis design also responded to findings from the diagnostic phase, which revealed weaknesses in students\u0026rsquo; critical and creative thinking. These gaps were addressed through dialogic questioning, trap-based problems, and reflective tasks embedded throughout the pathway.\u003c/p\u003e\n\u003cp\u003eTo visualize how these theoretical perspectives converge into a coherent instructional design, the study developed a structured model that maps each theory onto specific teaching phases and targeted cognitive outcomes. As shown in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, the pathway aligns De Jong\u0026rsquo;s inquiry cycle, Sweller\u0026rsquo;s cognitive load principles, and King\u0026rsquo;s developmental model of HOTS with a seven-stage instructional process. This process scaffolds learners\u0026rsquo; thinking from schema activation to self-regulation, progressively cultivating problem-solving, critical thinking, and creative thinking.\u003c/p\u003e\n\u003cp\u003e(This figure illustrates a multi-layered instructional framework integrating De Jong\u0026rsquo;s inquiry learning, Sweller\u0026rsquo;s cognitive load theory (CLT), and King\u0026rsquo;s HOTS model. Seven teaching stages are aligned with specific mediating mechanisms\u0026mdash;knowledge construction, reasoning, and meta-cognitive monitoring\u0026mdash;that lead to the development of problem-solving, critical thinking, and creative thinking)\u003c/p\u003e\n\u003ch2\u003eThe Seven-Stage Problem-Chain Pathway\u003c/h2\u003e\n\u003cp\u003eThe teaching pathway consists of seven sequential stages, each mapped to a specific instructional function and cognitive target. This structure forms a \u0026ldquo;thinking arc\u0026rdquo; that supports progressive engagement from prior knowledge activation to meta-cognitive regulation. For clarity, the seven stages and their corresponding instructional objectives are shown in Table \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eEach phase was guided by scaffolding materials including teacher prompts, sentence stems, and cognitive supports. The complete cycle was typically implemented over one or two 40-minute lessons depending on task complexity.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003ctable id=\"Tab3\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003e\u003cem\u003eThe Seven-Stage Problem-Chain Teaching Pathway\u003c/em\u003e\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eNo.\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eStage\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eInstructional Focus\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eCognitive Target (HOTS)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eTheoretical Support\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePreparation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eActivate prior knowledge; introduce big idea\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eConceptual readiness\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCLT (schema activation)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eOrientation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePose initial open question; create cognitive tension\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eEngagement \u0026amp; motivation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDe Jong (problem posing)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eConceptualization\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eProvide worked examples \u0026amp; visual representations\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eConcept understanding\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCLT (example-based learning)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eInvestigation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eEngage in student-led problem exploration\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eProblem-solving, creativity\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eInquiry learning\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eConclusion\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSynthesize findings and generalize strategies\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAbstraction, transfer\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eKnowledge construction\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDiscussion\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eJustify, critique, compare solutions\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCritical thinking, argumentation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDialogic learning\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eReflection\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eReflect on strategies and thinking process\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMeta-cognition, self-regulation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eKing\u0026rsquo;s HOTS model (level 3)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch2\u003eSample Implementation Snapshot\u003c/h2\u003e\n\u003cp\u003eMini-Case: Trap-Based Reasoning in a Division Context\u003c/p\u003e\n\u003cp\u003eIn a Grade 4 unit on division with remainders, the following trap-based problem was used during the Investigation stage:\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;\u003cem\u003eA school has 37 students and 6 buses. If each bus carries 6 students, how many buses are needed?\u003c/em\u003e\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eMost students initially divided 37\u0026thinsp;\u0026divide;\u0026thinsp;6\u0026thinsp;=\u0026thinsp;6 remainder 1 and answered \u0026ldquo;6 buses,\u0026rdquo; overlooking that the remainder represents one more student still needing transportation. This common misconception was deliberately embedded to trigger cognitive conflict.\u003c/p\u003e\n\u003cp\u003eIn the Conclusion stage, students discussed why 7 buses were needed, revisiting the division logic and visualizing the \u0026ldquo;leftover.\u0026rdquo; The Discussion stage focused on evaluating reasoning patterns, using guiding questions like:\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;\u003cem\u003eWhat made this problem tricky?\u003c/em\u003e\u0026rdquo;\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;\u003cem\u003eHow did the remainder affect your solution?\u003c/em\u003e\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eIn Reflection, students completed self-assessment prompts such as:\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;\u003cem\u003eToday I learned that in real-life problems, the remainder can change the answer. I need to ask what the numbers mean, not just calculate\u003c/em\u003e.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eThis task exemplifies how the pathway scaffolds both problem-solving persistence and critical reinterpretation of intuitive answers, two components identified as weak in the diagnostic phase.\u003c/p\u003e\n\u003ch2\u003eAlignment with HOTS Dimensions\u003c/h2\u003e\n\u003cp\u003eTo ensure that the instructional model systematically supports students\u0026rsquo; higher-order thinking development, the seven-stage problem-chain teaching pathway was deliberately aligned with the three core HOTS dimensions proposed by Alkhatib (2019): problem-solving, critical thinking, and creative thinking. Rather than treating these dimensions as parallel or isolated, the design aimed to embed them in a progressive, mutually reinforcing manner across the instructional sequence.\u003c/p\u003e\n\u003cp\u003eProblem-solving is primarily addressed through the stages of Preparation, Conceptualization, Investigation, and Conclusion. These phases scaffold the cognitive process from schema activation to abstraction and generalization. For instance, students begin by activating prior knowledge (Preparation), analyzing worked examples (Conceptualization), exploring novel problems (Investigation), and then synthesizing their findings (Conclusion). This sequence aligns with Sweller\u0026rsquo;s (\u003cspan class=\"CitationRef\"\u003e2011\u003c/span\u003e) cognitive load theory and supports efficient cognitive progression for novice learners.\u003c/p\u003e\n\u003cp\u003eCritical thinking is emphasized in the Orientation and Discussion stages. Here, students are introduced to open-ended or trap-based questions that create cognitive dissonance and are later encouraged to critique reasoning and compare alternative solutions in peer dialogue (Lu, \u003cspan class=\"CitationRef\"\u003e2021\u003c/span\u003e). These stages promote argumentation, justification, and reflective reasoning\u0026mdash;key components of critical mathematical engagement.\u003c/p\u003e\n\u003cp\u003eCreative thinking is fostered through the Investigation, Conclusion, and Reflection stages. During these phases, students are prompted to generate original strategies, apply knowledge across contexts, and reflect on their problem-solving approaches. The Reflection phase, in particular, supports metacognitive development and self-regulation in line with King et al.\u0026rsquo;s (\u003cspan class=\"CitationRef\"\u003e1998\u003c/span\u003e) third level of HOTS progression.\u003c/p\u003e\n\u003cp\u003eImportantly, the alignment is not rigidly compartmentalized. Students often engage in multiple forms of higher-order thinking simultaneously\u0026mdash;for example, solving a novel problem (problem-solving), evaluating a peer\u0026rsquo;s reasoning (critical thinking), and proposing an alternative method (creative thinking). This interplay reflects the integrated nature of HOTS and reinforces the cognitive coherence of the pathway. Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e below summarizes the alignment between the instructional stages and each HOTS dimension, illustrating how the pathway systematically maps cognitive goals to specific pedagogical actions.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003ctable id=\"Tab4\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003e\u003cem\u003eInstructional Alignment Between Teaching Stages and HOTS Dimensions\u003c/em\u003e\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eHOTS Dimension\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSupported Through Stages\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eProblem-Solving\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eStages 1, 3, 4, 5: Activate schema, example analysis, student exploration, generalization\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCritical Thinking\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eStages 2, 6: Pose cognitively dissonant tasks, critique reasoning, compare solutions\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCreative Thinking\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eStages 4, 5, 7: Generate alternative strategies, transfer across contexts, reflect and self-direct\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch2\u003eIntervention Outcomes\u003c/h2\u003e\n\u003ch2\u003eCognitive Gains in the Experimental Group\u003c/h2\u003e\n\u003cp\u003eFollowing the implementation of the problem-chain teaching pathway, students in the experimental group (n\u0026thinsp;=\u0026thinsp;25) exhibited statistically significant pre-to-post gains in all three HOTS dimensions. Paired-sample t-tests showed large effect sizes:\u003c/p\u003e\n\u003cp\u003eProblem-Solving: Mₚ\u003csub\u003er\u003c/sub\u003eₑ = 23.0 \u0026rarr; Mₚₒₛₜ = 26.3, t(24) = -16.14, p\u0026thinsp;\u0026lt;\u0026thinsp;.001, d\u0026thinsp;=\u0026thinsp;3.23\u003c/p\u003e\n\u003cp\u003eCritical Thinking: Mₚ\u003csub\u003er\u003c/sub\u003eₑ = 23.1 \u0026rarr; Mₚₒₛₜ = 26.0, t(24) = -15.45, p\u0026thinsp;\u0026lt;\u0026thinsp;.001, d\u0026thinsp;=\u0026thinsp;3.09\u003c/p\u003e\n\u003cp\u003eCreative Thinking: Mₚ\u003csub\u003er\u003c/sub\u003eₑ = 35.8 \u0026rarr; Mₚₒₛₜ = 38.8, t(24) = -16.77, p\u0026thinsp;\u0026lt;\u0026thinsp;.001, d\u0026thinsp;=\u0026thinsp;3.35\u003c/p\u003e\n\u003cp\u003eNotably, creative thinking showed the most substantial improvement, suggesting that the intervention effectively stimulated students\u0026rsquo; capacity for generating novel ideas and applying mathematical reasoning in flexible ways. These gains were likely supported by the open-ended tasks, trap-based inquiry, and reflective dialogue embedded in the Investigation, Discussion, and Reflection stages of the pathway.\u003c/p\u003e\n\u003cp\u003eThese quantitative results are detailed in Table \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e, which presents the pre- and post-test means, standard deviations, t-statistics, and effect sizes for both experimental and control groups. The experimental group exhibited large and statistically significant gains in all three HOTS dimensions, with Cohen\u0026rsquo;s d values exceeding 3.0\u0026mdash;indicating not only statistical significance but also strong instructional impact. By contrast, the control group showed minimal or no improvement across dimensions.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003ctable id=\"Tab5\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003e\u003cem\u003ePre- and Post-Test Mean Scores, t-Values, and Effect Sizes by Group (N\u0026thinsp;=\u0026thinsp;50)\u003c/em\u003e\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eGroup\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eDimension\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePre-test M (SD)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePost-test M(SD)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003et\u003c/em\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003ep\u003c/em\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eCohen\u0026apos;s d\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eExperimental\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eProblem-Solving\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e23.0 (1.0)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e26.3 (0.9)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-16.14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.23\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eExperimental\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCritical Thinking\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e23.1 (1.0)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e26.0 (0.9)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-15.45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.09\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eExperimental\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCreative Thinking\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35.8 (1.1)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e38.8 (0.8)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-16.77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.35\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eControl\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eProblem-Solving\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e23.1 (1.2)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e23.5 (1.3)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2.09\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e.047\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026mdash;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eControl\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCritical Thinking\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e23.0 (1.1)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e23.1 (1.2)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.53\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e.600\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026mdash;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eControl\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCreative Thinking\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35.7 (1.3)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35.8 (1.4)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e.703\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026mdash;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003ctfoot\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"7\"\u003eNote. All p-values\u0026thinsp;\u0026lt;\u0026thinsp;.001 for experimental group results; Cohen\u0026rsquo;s d\u0026thinsp;\u0026gt;\u0026thinsp;3.0 indicates large effect sizes.\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tfoot\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e visualizes these changes, demonstrating clear upward trajectories across all dimensions for the experimental group, contrasted with flat or negligible gains in the control group.\u003c/p\u003e\n\u003ch2\u003eStability in the Control Group\u003c/h2\u003e\n\u003cp\u003eIn contrast, the control group (n\u0026thinsp;=\u0026thinsp;25), which followed a conventional curriculum, exhibited minimal change. A slight improvement was observed in problem-solving (p\u0026thinsp;=\u0026thinsp;.047), but critical thinking and creative thinking remained statistically unchanged:\u003c/p\u003e\n\u003cp\u003eCritical Thinking: Mₚ\u003csub\u003er\u003c/sub\u003eₑ = 23.0 \u0026rarr; Mₚₒₛₜ = 23.1, t(24)\u0026thinsp;=\u0026thinsp;0.53, p\u0026thinsp;=\u0026thinsp;.600\u003c/p\u003e\n\u003cp\u003eCreative Thinking: Mₚ\u003csub\u003er\u003c/sub\u003eₑ = 35.7 \u0026rarr; Mₚₒₛₜ = 35.8, t(24)\u0026thinsp;=\u0026thinsp;0.39, p\u0026thinsp;=\u0026thinsp;.703\u003c/p\u003e\n\u003cp\u003eThis stagnation reinforces the claim that without targeted, cognitively rich instruction, students are unlikely to develop higher-order thinking skills beyond surface-level procedural comfort.\u003c/p\u003e\n\u003ch2\u003eBetween-Group Comparison via ANCOVA\u003c/h2\u003e\n\u003cp\u003eTo assess the differential effects of instruction, ANCOVA was conducted using pre-test scores as covariates. Results showed that the experimental group significantly outperformed the control group on all three HOTS dimensions at post-test (p\u0026thinsp;\u0026lt;\u0026thinsp;.001), with large effect sizes (partial \u0026eta;\u0026sup2; \u0026gt;.65). Figure \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e highlights the post-test means between groups. Across all dimensions, the experimental group held a clear advantage, with the largest margin in creative thinking.\u003c/p\u003e\n\u003cp\u003eThese results suggest that the design-based intervention was effective not only in fostering general improvement, but in specifically addressing cognitive areas previously identified as weak.\u003c/p\u003e\n\u003ch2\u003eHypothesis Evaluation and Summary of Findings\u003c/h2\u003e\n\u003cp\u003e\u003cstrong\u003eThe findings provide robust support for both research hypotheses:\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eH1: Students who received problem-chain teaching performed significantly better on HOTS than their peers in the control group \u0026rarr; Supported\u003c/p\u003e\n\u003cp\u003eH2: The experimental group demonstrated significant cognitive gains from pre- to post-test \u0026rarr; Supported\u003c/p\u003e\n\u003cp\u003eThe convergence of quantitative trends and theoretical alignment affirms the design validity and pedagogical effectiveness of the problem-chain teaching pathway. The most notable impact in creative thinking suggests that carefully structured inquiry tasks and student-driven discourse can meaningfully shift students\u0026rsquo; cognitive trajectories in mathematics learning.\u003c/p\u003e"},{"header":"Discussion","content":"\u003cp\u003eThe present study provides robust empirical evidence that a structured problem-chain teaching pathway, grounded in cognitive and instructional theory, can significantly enhance primary students’ higher-order thinking skills (HOTS) in mathematics. This section discusses the findings in relation to previous research, theoretical underpinnings, and practical implications. It also critically reflects on the study’s limitations and proposes directions for future inquiry.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eIntegrating Theory and Practice: Advancing HOTS through Structured Design\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe results confirm that elementary students are capable of engaging in complex reasoning tasks when scaffolded through well-sequenced instruction. The intervention’s most pronounced impact was observed in creative thinking (Cohen’s d = 3.35), followed closely by critical thinking and problem-solving. As shown in Figure 3, the experimental group demonstrated steep gains across all three HOTS domains, particularly in creative thinking. Furthermore, Figure 4 illustrates the post-test advantage of the experimental group over the control group in each dimension, with the most notable gap in creative thinking—an outcome that aligns with King et al.’s (1998) emphasis on meta-cognitive regulation and strategic transfer. These findings echo prior research suggesting that HOTS are not innate cognitive traits but developable competencies when aligned with appropriate pedagogical supports (Anderson \u0026amp; Krathwohl, 2001; Alkhatib, 2022; Dilekçi \u0026amp; Karatay, 2023).\u003c/p\u003e\n\u003cp\u003eUnlike traditional instruction, which often isolates cognitive skills from instructional delivery, the problem-chain model explicitly integrated HOTS development into each stage of the lesson. The alignment between De Jong’s (2006) inquiry cycle, Sweller’s (2011) cognitive load principles, and King et al.’s (1998) HOTS model created a coherent cognitive architecture that moved students from schema activation to metacognitive reflection. This staged approach resonates with Wu et al. (2024), who demonstrated that systematically embedded thinking routines in instructional design produce measurable cognitive growth.\u003c/p\u003e\n\u003cp\u003eFurthermore, the seven-stage structure fostered recursive engagement, with early stages (e.g., Orientation and Conceptualization) introducing cognitive tension and schema refinement, while later stages (e.g., Discussion and Reflection) promoted critical evaluation and creative transfer. These findings support the contention that instructionally embedded reflection is essential to consolidate and extend cognitive gains (Zimmerman, 2002; Lu, 2021).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eCreative Thinking as a Differentiator: From Procedures to Possibilitie\u003c/em\u003e\u003c/strong\u003es\u003c/p\u003e\n\u003cp\u003eThe marked improvement in creative thinking challenges the widespread assumption that creativity is peripheral or too abstract for primary mathematics (Yayuk \u0026amp; As’ari, 2020). Instead, the open-ended and trap-based (cognitively dissonant) tasks embedded in the Investigation and Reflection stages appeared to activate divergent thinking and encourage flexible strategy generation. These design features mirror those in recent HOTS-enhancement studies that highlight the power of ambiguity, error analysis, and analogical reasoning in stimulating creative insight (Szabo et al., 2020; Dilekçi \u0026amp; Karatay, 2023).\u003c/p\u003e\n\u003cp\u003eSuch results suggest that HOTS dimensions are not parallel domains but dynamically interrelated. For instance, students often engaged in problem-solving not as a procedural exercise but as an exploratory process that required reframing the task—a hallmark of creative cognition (Peng \u0026amp; Kievit, 2020). Similarly, critical thinking was frequently mobilized in service of creativity, as students compared multiple representations and reflected on solution plausibility.\u003c/p\u003e\n\u003cp\u003eThis reconceptualization aligns with Alkhatib’s (2019) tripartite HOTS framework, which advocates for integration rather than compartmentalization of cognitive skills. The problem-chain pathway operationalized this integration through instructional prompts that moved students beyond “correctness” to “appropriateness,” thus reinforcing the generative dimensions of mathematical reasoning.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eInstructional Design as Cognitive Architecture\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe intervention’s effectiveness can also be attributed to its careful attention to cognitive load. By sequencing activities from worked examples to independent inquiry, the pathway mitigated extraneous load while supporting germane processing—a core proposition of cognitive load theory (Sweller et al., 2019). In this way, the model supports novice learners’ progression toward expert-like reasoning, particularly in tasks that demand abstraction and transfer.\u003c/p\u003e\n\u003cp\u003eThese insights extend previous research on instructional coherence. Wu et al. (2024) emphasized that teaching interventions grounded in computational thinking (CT) principles promote critical literacy by reducing unnecessary complexity and increasing task transparency. Similarly, the present study shows that when tasks are designed with clear cognitive targets and sequencing logic, students are more likely to exhibit deep engagement and transfer.\u003c/p\u003e\n\u003cp\u003eAdditionally, the use of trap-based problems (e.g., the division-with-remainders scenario) introduced productive struggle, a concept supported by recent design-based research emphasizing the cognitive value of encountering and resolving contradictions (Kinnear et al., 2024; Nahar, 2022). Such cognitive dissonance may play a central role in triggering meta-cognitive monitoring, which King et al. (1998) identify as a key indicator of HOTS maturity.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eEmpirical Contributions and Educational Significance\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe between-group ANCOVA results revealed significant post-test differences across all HOTS dimensions, with partial η² exceeding 0.65. These large effect sizes suggest not only statistical significance but practical utility, particularly in light of the modest sample size and real-classroom implementation. The design’s effectiveness under typical instructional constraints demonstrates its scalability and potential for adoption in other curricular contexts.\u003c/p\u003e\n\u003cp\u003eBeyond statistical outcomes, the study contributes to the growing body of evidence that supports theory-driven pedagogical innovation in primary education. While many studies explore inquiry learning or cognitive load principles in isolation, this research demonstrates how their integration—when aligned with HOTS developmental theory—can yield compounding cognitive benefits. This triangulated design approach aligns with the findings of Preckel et al. (2020) and Greiff \u0026amp; Borgonovi (2022), who argue that cross-domain instructional synthesis is key to 21st-century competence development.\u003c/p\u003e\n\u003cp\u003eFurthermore, the study addresses persistent concerns in mathematics education regarding the marginalization of critical and creative thinking in favor of procedural fluency (Jablonka, 2020; Yayuk \u0026amp; As' ari, 2020). By embedding HOTS development into the very structure of problem sequencing, the model repositions mathematical instruction as a venue for deep thinking rather than rote application.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003ePedagogical and Policy Implications\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe findings of this study underscore the feasibility of integrating higher-order thinking skills into early mathematics instruction without sacrificing curricular coherence or instructional clarity. When supported by deliberate scaffolding and cognitively sequenced design, even young learners can meaningfully engage with abstraction, critical evaluation, and creative synthesis. This challenges conventional assumptions that such capacities emerge only in later stages of schooling and supports growing advocacy for embedding cognitive development objectives across all grade levels (González-Salamanca et al., 2020; Ramírez-Montoya, 2022).\u003c/p\u003e\n\u003cp\u003eFrom a professional development perspective, the problem-chain teaching pathway offers a replicable framework for cultivating teacher capacity in HOTS-oriented instruction. Rather than relying on intuitive or fragmented approaches, teachers can draw upon a theoretically grounded sequence of stages that guide the orchestration of inquiry, cognitive challenge, and reflective discourse. This approach aligns with Qing-li et al.'s (2024) conception of teachers as “curriculum makers” who design for deep thinking through purposeful, cross-disciplinary integration.\u003c/p\u003e\n\u003cp\u003eAt the policy level, the study illustrates how micro-level instructional decisions—such as task sequencing, questioning strategy, and reflection prompts—can serve as levers for macro-level educational transformation. Grounding these decisions in cognitive science and classroom data allows for scalable curriculum reform that transcends textbook-driven routines. In doing so, the model contributes to the broader agenda of fostering 21st-century competencies through evidence-based pedagogy.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eLimitations and Future Research\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eDespite its strengths, the study has several limitations. The relatively small and geographically concentrated sample limits generalizability. Moreover, the study focused primarily on cognitive outcomes; affective, motivational, and social dimensions were not systematically examined. While fidelity of implementation was supported through observation and teacher debriefing, teacher beliefs and agency—key mediators of instructional innovation—warrant further exploration.\u003c/p\u003e\n\u003cp\u003eFuture research could expand the model to other subject areas or cultural contexts to examine its cross-domain adaptability. Longitudinal studies could also assess the sustainability of HOTS gains over time. Additionally, integrating digital tools such as AI tutors or adaptive platforms might enhance personalization and expand the design’s reach—an avenue explored in recent work on GPT-supported instructional planning (Hu et al., 2024; Wang et al., 2025).\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eThis study set out to design, implement, and evaluate a problem-chain teaching pathway aimed at enhancing fourth-grade students\u0026rsquo; higher-order thinking skills (HOTS) in mathematics. Grounded in established theoretical frameworks\u0026mdash;including inquiry-based learning, cognitive load theory, and HOTS development models\u0026mdash;the pathway offered a structured yet flexible sequence of instructional stages that embedded cognitive challenge into everyday classroom practice.\u003c/p\u003e\n\u003cp\u003eThe results of the quasi-experimental study provide compelling evidence that the intervention significantly improved students\u0026rsquo; problem-solving, critical thinking, and creative thinking, with particularly large gains in the latter two dimensions. These findings support the view that elementary students are capable of engaging in deep mathematical reasoning when supported by intentionally designed, cognitively coherent instruction. They also demonstrate that such gains are not incidental, but the result of pedagogical design that aligns tasks, discourse, and reflection with cognitive development principles.\u003c/p\u003e\n\u003cp\u003eIn addition to affirming the effectiveness of the intervention, the study contributes to the growing literature on design-based educational research. It highlights the potential of using diagnostic data to inform instructional design, of treating lesson planning as a form of cognitive architecture, and of empowering teachers to orchestrate learning environments that support not only procedural fluency, but adaptive reasoning and strategic innovation.\u003c/p\u003e\n\u003cp\u003eWhile the study is not without limitations\u0026mdash;including sample scope, short-term measurement, and limited focus on teacher implementation\u0026mdash;it offers a compelling proof of concept for advancing HOTS in primary mathematics through theory-driven pedagogy. The problem-chain teaching pathway, with its clear structure and empirical grounding, holds promise as a model for future curriculum design, teacher professional development, and educational policy aiming to promote deeper learning in the early years.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003e\u003cem\u003eEthics approval and consent to participate\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis study was submitted to and approved by the Ethics Committee of the Faculty of Education, Sichuan Normal University. The study complied with the ethical standards outlined in the Declaration of Helsinki. All procedures involving human participants were conducted in accordance with the institutional guidelines of Sichuan Normal University and the collaborating primary schools. Written informed consent was obtained from the legal guardians of all participating students, and verbal assent was obtained from the students. Participation was voluntary, and participants could withdraw at any time without any negative consequences. Data confidentiality and anonymity were strictly maintained throughout the study.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eConsent for publication\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eConsent for publication was not required as no identifying personal information or images of participants are included in this manuscript. All data are anonymized and reported in aggregate form to ensure confidentiality.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eCompeting interests\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author declares that there are no conflicts of interests regarding the publication of this paper.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eFunding\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis study received no specific financial support.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eAuthors' contributions\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eSY conceptualized and designed the study, developed the instructional model, performed the statistical analysis, and drafted the manuscript. QX contributed to data collection, preliminary data analysis, and critical revision of the manuscript. All authors read and approved the final manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eAvailability of Data and Materials\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe datasets used and analysed in the current study are available from the first author on reasonable request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eAcknowledgements\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe are grateful to the participating schools, teachers, and students for their generous cooperation and engagement in this study. Special thanks are extended to colleagues at Sichuan Normal University for their valuable feedback and support. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAlkhatib OJ. (2019, March). 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Theory Into Pract. 2002;41(2):64\u0026ndash;70. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1207/s15430421tip4102_2\u003c/span\u003e\u003cspan address=\"10.1207/s15430421tip4102_2\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Cognitive development, Cognitive load theory, Educational intervention, Higher-order thinking skills, Inquiry-based learning, Problem-chain teaching","lastPublishedDoi":"10.21203/rs.3.rs-6782865/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6782865/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003ch2\u003eBackground\u003c/h2\u003e\u003cp\u003eDeveloping higher-order thinking skills (HOTS) is critical for fostering adaptive and flexible cognitive abilities in children. However, conventional mathematics instruction often neglects the cultivation of complex reasoning and creative problem-solving. This study aimed to design and evaluate a theory-driven instructional model\u0026mdash;problem-chain teaching\u0026mdash;targeting the enhancement of HOTS in fourth-grade students, grounded in cognitive and educational psychology frameworks.\u003c/p\u003e\u003ch2\u003eMethods\u003c/h2\u003e\u003cp\u003eA quasi-experimental design was implemented with 50 fourth-grade students randomly assigned to experimental (problem-chain teaching) and control (conventional teaching) groups. The intervention was informed by inquiry-based learning theory, cognitive load theory, and HOTS developmental models. Pre- and post-intervention assessments measured problem-solving, critical thinking, and creative thinking using validated psychometric instruments. Statistical analyses included paired-sample t-tests and ANCOVA, with effect sizes calculated using Cohen\u0026rsquo;s d to assess the magnitude of intervention effects.\u003c/p\u003e\u003ch2\u003eResults\u003c/h2\u003e\u003cp\u003eStudents receiving problem-chain instruction demonstrated significant improvements in all HOTS domains compared to the control group (p\u0026thinsp;\u0026lt;\u0026thinsp;.001), with large effect sizes (Cohen\u0026rsquo;s d\u0026thinsp;\u0026gt;\u0026thinsp;3.0). Notably, creative thinking exhibited the greatest gains. ANCOVA results indicated significant between-group differences favoring the experimental group, with partial η\u0026sup2; values exceeding 0.65, suggesting a substantial impact of the intervention.\u003c/p\u003e\u003ch2\u003eConclusions\u003c/h2\u003e\u003cp\u003eThe problem-chain teaching model effectively enhanced higher-order cognitive processes in primary school students. These findings highlight the value of integrating cognitive psychology principles into instructional design to promote complex reasoning skills. The study provides a replicable framework for educators seeking to foster critical, creative, and problem-solving abilities in early mathematical learning, contributing to the broader psychological understanding of cognitive skill development in children.\u003c/p\u003e","manuscriptTitle":"Designing for Deep Thinking: A Theory-Driven Inquiry into Problem-Chain Teaching in Primary Mathematics","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-14 11:07:30","doi":"10.21203/rs.3.rs-6782865/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"5ade43fb-dc44-456b-b2c8-d16599df33fa","owner":[],"postedDate":"July 14th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-02-24T09:57:18+00:00","versionOfRecord":[],"versionCreatedAt":"2025-07-14 11:07:30","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6782865","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6782865","identity":"rs-6782865","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}