Supporting critical thinking in sequences and series through a problem-solving e-module with algebraic tasks
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Abstract
Examining sequences and series often leads students to rely on formulas, especially when assignments require algebraic modeling and generalization. This condition may hinder the advancement of critical thinking, especially in evaluation, inference, and reasoning. This study examined the extent to which a problem-solving e-module with algebraic challenges enhanced students' critical thinking and identified the elements that best facilitated critical thinking during the learning process. The study used a sequential explanatory mixed-methods design. The quantitative phase encompassed 89 senior high school students from three classes. Students completed a pretest, engaged with an e-module featuring guiding questions, scaffolding, feedback, and reflection, and then completed a posttest. The quantitative data were examined utilizing descriptive statistics, normalized gain, pretest-posttest comparisons, and effect size analysis. Data were gathered from students' e-module assignments, classroom observations, written reflections, and task-oriented interviews. The data were examined using theme coding and synthesized with the quantitative results. The results indicated an improvement in overall critical thinking and across all metrics; however, the improvements were inconsistent. Evaluation and interpretation improved more significantly than explanation. Feedback, directed inquiries, scaffolding, and reflection facilitated critical thinking; however, argument quality and algebraic reasoning varied among students.
INTRODUCTION
Students sometimes have difficulties with sequences and series when assignments require algebraic modeling(Kertil et al., 2019; Lee & Lee, 2021). A significant number of students can extend a pattern for the initial terms but find it challenging to accurately formulate the nth-term expression(Patterson & McGraw, 2018; Susanta et al., 2023; Wilkie, 2020). Students frequently confuse recursive rules with explicit forms, especially when assignments require conversion between the two(Sandefur & Manaster, 2022). In contextual concerns, students often misconstrue the given information, resulting in the creation of algebraic relationships that do not align with the actual scenario(Scheiner & Pinto, 2019). Errors in algebraic manipulation occur when students simplify expressions, formulate equations, or manage parameters(Soneira et al., 2018). Mathematical nomenclature, encompassing sigma notation and indices, often leads to inaccuracies(Yoon & Thompson, 2020). These problems undermine the quality of students' solutions and hinder their ability to provide logical justification.
Previous studies suggest that difficulties in sequences and series often arise from problems associated with generalization(Yao, 2022)and representation(Elia, 2020). Research on secondary school students shown heterogeneity in their levels of representation while articulating sequences, ranging from unstructured to fully ordered forms, and indicated that generalization strategies changed when tasks required extensive generalization(Yao, 2022). In close generalization, students mostly employed counting and addition operations, whereas in far generalization, strategies diversified, with correspondence methods emerging as the most common(Melhuish et al., 2020). Further research has highlighted pedagogical obstacles and the gap between knowledge obtained in secondary education and that necessary for higher education, leading students to depend on rote memorization of formulas and struggle with the transition between empirical and theoretical frameworks(Kop et al., 2020). A quasi-experimental study on numerical patterns in Singapore revealed that computational activity-based interventions did not consistently produce significant overall improvement; however, some students demonstrated substantial progress, suggesting that learning outcomes are greatly affected by the design of activities and the quality of instructional support(Vicente et al., 2022). These findings indicate that education should emphasize how students develop overarching principles, select strategies, and verify outcomes, rather than only rehearsing the application of formulas.
These issues have necessitated the implementation of teaching practices designed to improve critical thinking. Problem-solving instruction enables students to understand obstacles, formulate strategies, execute those methods, and assess their results(Voica et al., 2020). Problem-based(Xin et al., 2020)and inquiry-based learning(Pedersen & Haavold, 2023)require students to explain rationales, evaluate facts, and justify their strategic choices. In educational contexts, leading questions are often utilized to stimulate interpretation, analysis, and reasoning rather than just obtaining definitive answers(Foster et al., 2022). Scaffolding and metacognitive queries are employed to aid students in tracking their progress, recognizing faults, and enhancing their strategies(Çakmak Gürel, 2025). Delivering this support consistently, however, is difficult in conventional media. Printed worksheets fix a single solution path and cannot adapt prompts to a student's response; whole-class discussion distributes scaffolding unevenly and leaves little trace of individual reasoning; and video or static presentations convey content without eliciting or recording students' own justifications. An e-module addresses these constraints directly: it can sequence graduated tasks, embed guiding questions and scaffolding at the point of need, deliver staged feedback tied to reasoning rather than to final answers alone, and capture the process traces through which critical thinking becomes observable(Swidan et al., 2022). This last affordance is not merely convenient but methodologically necessary here, since analyzing which features elicit which critical thinking indicators during task performance (RQ2) requires a medium that records students' step-by-step interaction—data that conventional media do not provide. Such designs are expected to shift learning from mechanical formula application to analytical decision-making.
These strategies, however, share a limitation that constrains their capacity to promote critical thinking reliably: they attend to the products of student work rather than to the reasoning that generates them. Routine assessments reward procedural accuracy while leaving students' analysis and evaluation of strategies unexamined(Shaughnessy et al., 2019); where classroom discussion does occur, it is rarely anchored to explicit, measurable critical thinking criteria(Llinares, 2019); and e-modules are frequently reduced to digital worksheets that let students reproduce worked examples without constructing justifications(Amelia et al., 2026; Hidayat et al., 2025; Johar et al., 2025; Martin et al., 2021; Utami et al., 2018). The same product orientation recurs in how these approaches give feedback, which privileges the final answer over the quality of reasoning, the soundness of the method, and the plausibility of the result, and in how they are evaluated, typically stopping at pretest–posttest scores without documenting how students' cognitive processes change. This orientation also obscures an inconsistency in the evidence itself: even interventions that report positive effects do so unevenly, with substantial gains confined to only a subset of students(Vicente et al., 2022), yet without process-level data such divergent outcomes cannot be traced to the design features that drive them. Across otherwise different media and designs, then, the recurring gap is the absence of process-level evidence. This is why reported gains remain difficult to explain, and why the contribution of any specific design feature to those gains stays unidentified—the unresolved issue this study sets out to address.
This situation underscores the need for algebra-centric problem-solving activities that intentionally stimulate indicators of critical thinking. Assignments must require students to convert
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