This study utilized an explanatory sequential mixed-methods design as its primary objective necessitated two complimentary outputs. The quantitative phase was undertaken initially to acquire a validated measure of enhancement, ensuring that the preliminary findings were not exclusively dependent on process-based interpretation. The qualitative phase was conducted subsequent to the initial quantitative analysis to inform case selection and exploration focus based on developing outcome patterns, including high, moderate, and low improvement groups. The study was conducted in standard classroom environments to evaluate the e-module intervention under genuine conditions, hence enhancing the practical significance and ecological validity of the results. The intervention focused exclusively on a singular unit concerning sequences and series throughout multiple meetings to ensure content alignment, implementation consistency, and the quantifiability of increases in students' proficiency. The synthesis of findings occurred during the interpretation phase by merging quantitative and qualitative results, as this method facilitates a cohesive elucidation of both "the degree of change" and "the elements that consistently elicited critical thinking indicators" within a unified scientific argument.

Because the study prioritizes explaining the mechanism through which particular e-module features elicit critical thinking indicators, rather than establishing comparative causal superiority over an alternative medium, it uses a single-group design without a control or comparison class. This design fits the second research question, which concerns how features activate indicators during task performance, something a between-group contrast would not reveal. The choice does, however, constrain causal inference: with no comparison group, the observed pretest-to-posttest gains cannot be attributed to the e-module alone, since maturation, repeated testing, and concurrent instruction may also contribute. The quantitative phase is therefore interpreted as evidence of change accompanying the intervention rather than proof of its exclusive effect, and the qualitative phase traces how that change arose at the feature and task levels.

This study involved 89 senior high school students drawn from three intact classes at the same grade level within a single school. The classes were selected through purposive sampling: [because they constituted the available classes at the target grade level in this school,] all sharing the same grade and institutional background, which held the students' educational context constant and allowed the e-module intervention to be examined under authentic classroom conditions rather than a reconstituted research group. Conducting the study across three classes of comparable background, rather than a single class, broadened the range of student work available for process analysis while keeping contextual variation controlled. The three classes comprised Class A (n = 28), Class B (n = 29), and Class C (n = 32); all participants were aged 15 to 17 years, consistent with the typical profile of students at this level. The sample was predominantly female (67 female, 22 male; 75.28% and 24.72% of the total), a distribution that was consistent across the three classes and is acknowledged as a limitation, since it constrains the extent to which the findings can be generalized across gender. Table 1 reports the composition of each class.

Prior to the intervention, all participants completed a pretest assessing critical thinking on sequences and series. Because the qualitative phase required cases that reflected a range of entry points, the pretest scores were used to stratify students purposively into high (n = 12), moderate (n = 53), and low (n = 24) initial-ability groups, from which participants for the process analysis were selected. This two-stage procedure—intact classes for the quantitative phase and purposive, stratified case selection for the qualitative phase—ensured that the process data captured diversity in initial ability and helped explain the patterns observed in the quantitative results.

This study utilized instruments designed to address the two research topics in accordance with the explanatory sequential mixed-methods framework. During the quantitative phase, the principal instrument utilized was a critical thinking assessment focused on sequences and series of algebraic problem-solving exercises. The assessment was conducted as both a pretest and a posttest utilizing the identical indication pattern to ensure that variations in students' abilities could be evaluated equitably and methodically. Assessment was performed utilizing an analytic rubric that delineated students' performance across critical thinking metrics, including interpretation, analysis, inference, assessment, and explanation. During the qualitative phase, the study supplemented the assessment with process-oriented tools that facilitated the identification of mechanisms that stimulated critical thinking while students engaged with tasks in the e-module. The process instruments were a worksheet for assessing students' work traces in the e-module, a task-based interview guide, and a feature engagement observation sheet to document evidence of leading questions, scaffolding, feedback, and reflection. Moreover, structured reflections or concise questionnaires, in conjunction with e-module activity logs, were employed as supplementary data to enhance triangulation and assess the consistency of feature engagement among students and between assignments. All instruments underwent expert assessment to guarantee content alignment with the measured constructs and were piloted on a limited scale to verify readability, implementation practicality, and scoring and coding consistency.

To measure students' critical thinking performance on sequences and series, the pretest and posttest were developed based on a blueprint that aligns each content focus with a corresponding algebraic task form, sample item, targeted critical thinking indicator, and evidence to be assessed. The blueprint spans six content areas (arithmetic sequences, geometric sequences, series, arithmetic series, geometric series, and mixed tasks) each constructed to elicit specific indicators such as interpretation, analysis, inference, evaluation, and explanation. For instance, the arithmetic sequence item requires students to generalize the formula for Un and justify their reasoning, whereas the geometric series item requires students to evaluate two competing solution strategies and identify the source of error in the incorrect one, thereby capturing evidence of evaluative and explanatory reasoning. The complete blueprint is presented in Table 2.

To address the second research question concerning the mechanisms that stimulate critical thinking as students engage with the e-module, the study employed a set of process-oriented instruments designed to capture qualitative evidence beyond test scores. These instruments (comprising a work-trace analysis worksheet, a task-based interview guide, a feature engagement observation sheet, a structured reflection or brief questionnaire, and an e-module activity log) differ in their unit of analysis and the type of data collected, ranging from solution-step segments and think-aloud transcripts to timestamped behavioral observations and activity logs. Each instrument was mapped to specific critical thinking indicators and accompanied by illustrative evidence, such as strategy revision following feedback or verification behavior prior to submission, allowing triangulation between cognitive outcomes and the underlying reasoning processes. Table 3 summarizes the purpose, unit of analysis, and indicators associated with each instrument.

To clarify how individual e-module features are theorized to elicit specific critical thinking indicators, the study operationally defined each feature and linked it to corresponding observation items, work-trace items, and interview questions. Four features were examined (guiding questions, step-by-step scaffolding, feedback, and reflection) each associated with distinct indicator profiles, such as interpretation, analysis, explanation, and evaluation for guiding questions, and evaluation and explanation for feedback and reflection. Concrete behavioral markers were specified for each feature, for example, a student opening the scaffolding and subsequently revising a solution step, or checking a result through substitution before submission, providing operational criteria for coding feature engagement across data sources. This feature-to-indicator mapping is detailed in Table 4.

The critical thinking test comprised six constructed-response items covering sequences and series. Its content validity was established through expert judgment by three experts in critical thinking scope research, yielding a Content Validity Index of 0.77. This value approaches, while falling marginally, the conventional threshold of 0.78 for a three-expert panel; the items were revised following the experts' feedback before administration. Internal-consistency reliability was acceptable, with Cronbach's alpha of 0.71, a value that is satisfactory for a six-item instrument given the known sensitivity of alpha to test length. For the qualitative instruments, two raters independently coded students' work traces and interview data against the critical thinking indicators. Inter-rater reliability reached 0.73, indicating substantial agreement. Because the second research question relies on this coding rather than on test scores alone, the substantial inter-rater agreement provides the primary basis for the reported feature-to-indicator patterns.

The data collecting phase commenced with procedural preparation to guarantee uniform execution across the three classes. The collection of quantitative data commenced with the administration of a pretest on critical thinking regarding sequences and series before to the implementation of the intervention. The pretest was conducted for all three courses within a tightly coordinated timeframe to mitigate the impact of variations in the academic calendar. The researcher standardized the duration, testing protocols, and instructions across all classes, while underscoring the significance of documenting both rationale and solution procedures. The pretest results were gathered, classified by class and student number, and archived as baseline data. Subsequent to the pretest, the instructional intervention employing the problem-solving e-module was executed throughout the three classrooms following a consistent learning sequence. Each meeting adhered to a consistent sequence of activities, commencing with problem orientation, succeeded by strategy exploration, methodical problem solving, and concluding with verification and reflection. The e-module featured algebraic problems necessitating generalization, modeling, justification, and result evaluation to facilitate the emergence of critical thinking indicators in a systematic manner. In specific portions, the e-module presented leading questions prompting students to elucidate their thinking, select a technique, or assess the validity of their results. When students faced challenges, the e-module offered incremental scaffolding that could be utilized as required, allowing for the tracking of support usage and subsequent strategic modifications. The e-module included feedback that encouraged students to amend their responses or rectify their solution processes, concluding each exercise with a structured reflection and self-assessment. During the implementation, the researcher ensured that the teacher employed identical procedures across all three classrooms to prevent discrepancies in outcomes due to instructional variation.

During each implementation meeting, the researcher methodically gathered process data as the primary source for RQ2. Learning artifacts were gathered from the e-module, encompassing students' responses to each task, answer changes, written rationales, and feedback responses. Students' reflection notes were gathered at the conclusion of each exercise as evidence of self-assessment and retrospective analysis of strategy utilization. When accessible, the researcher also obtained e-module activity logs, including the frequency of scaffolding access, the duration of feedback reading, and the number of answer revision attempts. Concurrently, observers executed feature engagement observations through event-based recording, namely by documenting students' behaviors in response to leading questions, scaffolding, feedback, and reflection. The observations concentrated on observable activities, like reviewing the problem, pausing to verify a finding, altering strategy, or providing justification. The researcher recorded field notes to describe the classroom atmosphere, including pertinent exchanges, technological challenges, and students' overall reactions. This sequence of data gathering meant that the study not only measured score changes but also elucidated the mechanisms that followed them.

Upon the completion of the instructional unit, the researcher conducted a critical thinking posttest on sequences and series, utilizing the identical indicator blueprint as the pretest. The posttest was conducted for all three classes using identical processes, encompassing the same duration, instructions, and testing regulations. The researcher reiterated the significance of documenting reasoning and solution processes to facilitate data analysis by indicator. The posttest results were gathered, encoded, and archived with the pretest data for comparative study. The researcher subsequently performed preliminary scoring to determine overall scores and indicator-level scores, along with the extent of progress for each student. The quantitative data provided the foundation for identifying patterns of enhancement at both the class and individual levels. This phase completed the quantitative data collecting and simultaneously initiated a more concentrated qualitative data collection phase.

The qualitative data collecting phase occurred subsequent to the completion of the preliminary quantitative analysis to ensure that participant selection was grounded in evidence of genuine progress. The researcher chose interview participants from each class by accounting for variations in improvement, categorizing them into high, moderate, and low-improvement groups, while also guaranteeing representation of original ability profiles. The chosen participants engaged in task-based interviews or think-aloud sessions utilizing various representative problems that mirrored the requirements of the critical thinking indicators and the primary e-module characteristics. The interviews addressed students' work artifacts and their experiences with certain aspects, allowing for the verification of their answers against their work traces. The researcher documented the interviews, made transcripts, and gathered supplementary materials, including revised solutions produced during the think-aloud sessions. To ensure consistency, the researcher employed a uniform interview guide for all participants, while permitting follow-up questions tailored to individual responses.

The quantitative analysis addressed RQ1 by comparing students' critical thinking scores before and after the intervention, at both the total-score and indicator levels, using the same analytic rubric. Descriptive statistics (mean, standard deviation, and median) characterized the score distributions. Because the pretest–posttest comparison relies on a parametric paired-samples test, the normality of the difference (gain) scores was examined using the Shapiro–Wilk test. Since the difference scores met the normality assumption, a paired-samples t-test was conducted to evaluate the change from pretest to posttest for the total score and for each of the five indicators. Practical significance was quantified using Cohen's dz for paired designs, complemented by the normalized gain computed at the individual level and aggregated to class and overall means. Results were reported at both class and aggregate levels and compared across indicators to assess the consistency of change. Given the single-group design, these results are interpreted as change accompanying the intervention rather than as evidence of its exclusive causal effect.

The qualitative analysis addressed RQ2 by examining the processes through which students engaged the e-module features. Data were drawn from three sources (students' work traces in the e-module, task-based interviews, and classroom observation notes) for 27 students selected from the high, moderate, and low improvement groups to capture variation in outcome patterns. The data were segmented into meaning units, each corresponding to a single solution step, a response to feedback, or a reflective remark, yielding [total number] units. Coding followed a hybrid approach. The five critical thinking indicators served as a predefined deductive scheme derived from the analytic rubric, while inductive codes were added as new patterns emerged from the data, including specific algebraic errors and strategic adjustments. Each unit was coded for the relevant critical thinking indicator and the specific e-module feature in use. Codes were then grouped into themes to identify the features that most consistently elicited critical thinking indicators. Coding was carried out using N.Vivo software / manual. Credibility was further supported by triangulating the three data sources against one another. The complete coding scheme, comprising the critical thinking indicators, e-module features, and inductive codes with their operational definitions, coded data examples, and data sources, is presented in Table 5.

This section cohesively presents the quantitative and qualitative findings to address the two research topics. The implementation of the problem-solving e-module improved critical thinking abilities about sequences and series. The improvement in overall scores and all metrics corresponds with the logic of problem-solving, requiring a sequence of cognitive tasks, from understanding the issue to assessing the final result. Polya described this process as a cyclical framework comprising understanding, strategizing, execution, and reflection, which automatically promotes reflective thinking in problem-solving activities (DiNapoli & Miller, 2022; Voica et al., 2020). The e-module consistently upheld this cycle due to the tasks being organized in a set sequence that could not be easily circumvented. Algebraic issues increased cognitive demand as students needed to develop generalizations and select the appropriate sequence structure, rather than merely calculate the next term (Blanton et al., 2019; Çekmez, 2020; Yao, 2022). This combination of process structure and work needs clarifies why the improvement was not merely incremental.

The overall improvement can be understood through a critical thinking paradigm that emphasizes key skills such as interpretation, analysis, inference, evaluation, and explanation. Facione’s paradigm views critical thinking as a conscious and self-regulatory judgment process, characterized by indications that emerge when students are engaged in processing information, forming connections, drawing conclusions, and evaluating the validity of their actions (Lehmann, 2023; Yasin et al., 2019). The problem-solving tasks in the e-module offer several opportunities to use these abilities specifically within the context of sequences and series (Scheibling-Sève et al., 2020; Xin et al., 2020). This topic requires students to transform numerical patterns, symbolic representations, and general formulas, often necessitating interpretative and analytical endeavors (El Mouhayar, 2018; Wilkie, 2021). Moreover, the imperative to conclude Un or Sn prompted inference, as students were compelled to transition from specific instances to generalizations. The repeated procedure within a single teaching unit reinforced these cognitive habits, hence the changes in scores can be viewed as the culmination of systematic practice (Hadar & Ruby, 2019).

The degree of improvement across indicators was inconsistent. The assessment and analysis enhanced considerably, as the topic of sequences and series provides an immediate chance for verification through the examination of starting terms or the validation of consistent common differences and ratios. Students assessed a formula in relation to U1, U2, or U3, participating in monitoring and correction, which are essential components of self-regulation according to Zimmerman (2002). Formative feedback improved this cycle by clarifying the difference between the current response and the proper one, so motivating students to adjust their methods, as emphasized in the formative feedback theories (Kontorovich, 2021). The expected increase in interpretation arose from the need for students to differentiate known knowledge from questions, understand notation, and categorize the sort of sequence before progressing. The instructional design increasingly requires the development of interpretation and assessment in observable and trainable activities.

Conversely, explanation advanced at a diminished pace as this metric requires explicit mathematical justification rather than merely selecting the suitable technique. Argumentation theory, as illustrated by Toulmin’s model, distinguishes between claims, data, and warrants; yet, many students stop at the claims stage, failing to furnish supporting evidence and conceptual underpinnings (Toulmin, 2003). In mathematics, justification refers to the criteria for proof and rationales that are subject to scrutiny, as detailed in the literature on mathematical justification and proof (Mata-Pereira & da Ponte, 2017). A predominantly procedural learning culture encourages the pursuit of definitive answers while undermining the expression of reasoning, making explanation the most difficult measure to reconcile with the others. This condition does not negate the observed progress; instead, it signifies that advancement in explanation requires more focused design support, such as prompts that urge students to illustrate the common difference or ratio and relate it to the fundamental notion. This result establishes a clear avenue for optimizing the design to attain a more equitable enhancement across indicators.

Mechanisms of Feature-Driven Critical Thinking

The components of the e-module did not stimulate critical thinking indicators to the same extent. Feedback (F-FB) emerged as the most dependable catalyst, particularly for evaluation (CT-EVA) and strategic adjustment, followed by guiding questions (F-GQ), which primarily stimulated analysis (CT-ANL) and elucidation (CT-EXP). Scaffolding (F-SC) enhanced inference (CT-INF), however its effectiveness was more vulnerable to difficulties in algebraic manipulation (IND-ALG). Reflection (F-RF) demonstrated the least consistency, as many responses were vague and lacked specific evaluative approaches. This trend suggests that the development of critical thinking is affected not just by the presence of traits but also by how those traits interact with monitoring, justification, and revision processes throughout task performance. In this study, these critical thinking indicators are understood as domain-specific manifestations of general cognitive processes: within sequences and series, evaluation, inference, and explanation take concrete form as validating an algebraic model, deriving an nth-term expression, and justifying a claim from the definitions of arithmetic and geometric sequences. A mathematical activity was treated as a critical thinking indicator only when it involved reasoning rather than procedural execution alone.

Feedback emerged as the most dependable trigger for assessment, as it activated a certain self-regulation mechanism inside the problem-solving process. Upon detecting a signal of inconsistency, students participated in a monitoring and regulation phase, which is essential to self-regulated learning (Kontorovich, 2021). This procedure entails evaluating results against predetermined standards, identifying the source of the error, and enacting modifications. In formative assessment frameworks, feedback provides insight into the gap between current performance and the desired goal, so encouraging corrective actions rather than simply classifying responses as correct or incorrect (Ayalon & Wilkie, 2020). The effect amplifies when feedback operates at both the task and process levels, allowing students to link the input to a specific step, then retracing their steps to initial words, analyzing frequent disparities or ratios, and modifying parameters. This pattern aligns that effective feedback addresses three questions: the learner's goal, the learner's current status, and the next steps to undertake. Thus, it is not surprising that CT-EVA became famous and was frequently followed by IND-SW, as correcting a mistake often required a strategic shift, such as moving from direct substitution to a term-table approach or doing iterative verification.

Guiding questions enabled analysis and clarification by requiring students to express their reasoning and link conclusions to evidence. From an argumentative perspective, questions such as “why” and “how do you know” require a reasoning structure that minimally includes a claim and supporting evidence, ideally supplemented by a warrant that links the evidence to the claim (Chorney et al., 2024; Lee et al., 2023). In sequences and series, this argumentative structure is realized mathematically: the supporting evidence is typically a constant difference or constant ratio, and the warrant derives from the definitions and properties of arithmetic or geometric sequences. Argument quality varied, as many students offered only claims without evidence or rationale, as in "it is arithmetic because it increases," yielding a superficial form of CT-EXP. More proficient students presented data producing a fuller representation of CT-ANL and CT-EXP. This variation reflects the standards of justification specific to mathematics, which are more rigorous than everyday explanation because reasoning must be scrutinized and grounded in formal concepts. Thus guiding questions prompted analysis and explanation, but the depth of argument depended on students' ability to marshal mathematical evidence and connect it to concepts.

Scaffolding demonstrated its value by providing temporary support during periods of impasse, promoting advancement within the zone of proximal development, as emphasized by Vygotsky and the scaffolding framework that requires gradual aid followed by ultimate removal (Kim et al., 2022; Rezat et al., 2022). In tasks requiring inference, such as deriving Un or converting a recursive form into an explicit form, scaffolding aided students in deconstructing the problem into manageable steps, including identifying the pattern of difference or ratio, formulating the general expression, and subsequently validating it. The effectiveness of scaffolding depended on algebraic problems, as mathematical reasoning requires accurate symbolic execution (Li et al., 2024). The principles of symbolic literacy and symbol sense are essential, as students may comprehend the general concept but encounter difficulties with manipulation, simplification, or substitution, resulting in an erroneous final conclusion. Scaffolding that provides only conceptual direction is insufficient when the main problem lies in the accuracy of symbolic operations. Meanwhile, reflection was frequently superficial, as many students regarded it only as a recounting of feelings or understanding, rather than as an evaluation of methodology and a demonstration of revision (DiNapoli & Miller, 2022). This suggests that metacognitive skills, especially accurate monitoring, were not sufficiently developed, leading to pupils' inability to recognize the stage at which an error occurred, the reasons behind it, and the strategies for rectification. Reflection would be enhanced if prompts included a distinct metacognitive structure, such as an error-cause-revision-verification format, to promote the consistent development of CT-EVA and CT-EXP.

The design implications of these findings are apparent when linked to the profile of indication augmentation. The most significant improvement was noted in indicators supported by mechanisms requiring visible revision, especially feedback, as it directly initiated criterion-based evaluation and encouraged the correction of solution phases. Thus, the e-module should prioritize feedback that is specific and process-oriented rather than merely score-based, thereby encouraging students to participate in focused verification and correction. Guiding questions should go beyond simple calls for justification, requiring explicit evidence so that arguments include claim, data, and warrant. Scaffolding should deliver targeted algebraic support at critical moments, such as precise symbolic manipulation or verification that preserves the integrity of the initial terms, preventing execution errors from degrading inference. Reflection should be structured as a systematic evaluation process rather than open exploration, enabling focused, verifiable assessment. These modifications strengthen the e-module's capacity to support explanation and to promote more balanced progress across indicators.