The study used a qualitative, bounded case study design to examine connections among cognitive style, APOS-oriented support, and students’ mathematical reasoning within one instructional context (Baxter & Jack, 2008; Yin, 2018). The case was bounded by the teaching of function limits in a Mathematics Education Program at an Islamic university in Sumatra, Indonesia, during the even semester of 2024/2025. The unit of in-depth analysis was each selected student’s developmental pathway from the initial written response to the explanation or revision produced after stage-specific scaffolding. The design was explanatory and comparative. It examined qualitative change within cases and contrast across cases without testing causal effects or statistical hypotheses.

The bounded classroom case involved 24 mathematics education students who had previously studied introductory differential calculus. Before the APOS sequence, cognitive style was classified using an Indonesian-language paper-and-pencil version of the Group Embedded Figures Test (GEFT), originally developed by Witkin et al. (1971) to examine field dependence-independence through the ability to locate a simple figure embedded in a complex visual configuration. The adapted booklet retained the original visual figures, item order, response procedure, and scoring structure, while the written instructions were presented in Indonesian. The GEFT contains three sections. Section I includes seven practice items and is not scored. Sections II and III contain nine scored items each. Each correctly traced figure receives one point, whereas an incorrect or unanswered item receives zero, producing a total score from 0 to 18. The complete classroom administration, including instructions and practice, lasted 20 minutes, consistent with the time allocation stated in the Indonesian booklet. Higher scores indicate a stronger field-independent tendency. For contrasting-case classification, scores of 0–11 were categorized as field-dependent and scores of 12–18 as field-independent, following an operational classification used in prior mathematics education research (Mirlanda & Pujiastuti, 2018). The cutoff was used for purposive grouping and case selection only; it was not treated as a measure of mathematical ability or as an absolute psychological diagnosis.

GEFT classified 6 students as FI and 18 as FD. Four information-rich cases were purposively selected for contrasting-case analysis: FI-1, FI-2, FD-1, and FD-2. Selection considered cognitive-style category, completeness of written work, and the potential to illuminate different solution pathways. The balanced four-case subsample supported within-group and cross-group comparison. It was not intended to represent the 24 students statistically or to claim thematic saturation (see Table 1).

APOS-based scaffolding was implemented across four Calculus I sessions on function limits. Each session lasted 135 minutes. The course lecturer delivered the sequence using a study-prepared guide, while the researcher observed enactment and collected student work. Action-oriented prompts focused attention on the source of 0/0 and on transformations such as factorization or multiplication by a conjugate. Process-oriented prompts asked students to explain, order, and predict transformation sequences. Object-oriented prompts required students to validate completed limit values and reuse them as mathematical entities. Schema-oriented prompts required integration of both limit results into a coherent justification. The same stage-specific logic guided the stimulated-recall interviews. Prompts began at the APOS construction indicated by the initial written response and were reduced when the student could continue independently. Scaffolds supplied cues and questions, not completed solutions (see Table 2).

An integrated, researcher-designed task served as the common elicitation context. Students determined and , and then verified a^² + 3b^² = 8. The task required two limit evaluations and one integrated justification. Students first produced an unaided written response. The lecturer then used the response during a stimulated-recall interview to identify the point of difficulty and provide the least intensive APOS-aligned prompt needed for continuation. The task elicited qualitative reasoning and short-term response to scaffolding; it did not generate a numerical AMT score.

Three data sources were used with an explicit hierarchy. The four students’ initial written solutions were the primary evidence because they displayed task entry, algebraic transformations, intermediate results, and final integration before interview support. Semi-structured stimulated-recall interviews then examined how students interpreted their own work and responded to stage-specific APOS prompts. For each case, the researchers documented the initial difficulty, the prompt delivered, the student’s subsequent explanation or revision, and the degree of independence shown. Representative excerpts were selected when they directly clarified strategy choice, the point of breakdown, the effect of a prompt, or the student’s ability to continue after support was reduced. The excerpts reported below are English translations of the Indonesian interview responses. Classroom observation was recorded on a structured sheet that logged the session, APOS focus, lecturer prompt, visible student response, degree of independence, and evidence of scaffold fading. The four observation records were coded OBS-S1 through OBS-S4. Reflection notes and learning materials provided supplementary evidence. Claims about initial mathematical construction were grounded in written work, claims about immediate development were grounded in the comparison between the initial response and the documented post-prompt response, and claims about implementation were checked against the observation record for the corresponding APOS session.

AMT was analyzed qualitatively through three dimensions derived from Tall’s framework: abstraction, conceptual reorganization, and formal reasoning. The open-ended task was an elicitation instrument, not a standardized AMT test. For each dimension, initial evidence was coded as not evident, emerging, or explicit using the observable criteria in Table 3. A developmental response was then recorded when stage-specific support enabled the student to identify a previously unrecognized relation, coordinate a previously unstable sequence, validate a result, or integrate completed values. Such changes were described as assisted emergence or consolidation, depending on the degree of independence. The codes were qualitative descriptors and were not converted into scores, summed, averaged, or compared statistically.

Data were examined through a framework-guided cross-case analysis. The researchers first coded each initial written solution for task entry, interpretation of 0/0, factorization, conjugate rationalization, cancellation, formation of limit values, and final integration. Interview records were then coded as a prompt-response sequence: initial APOS position, scaffold type, post-prompt action or explanation, revised APOS position, and change in AMT evidence. Observation records were coded by session and examined for five implementation indicators: correspondence between the planned APOS focus and the lecturer prompt, visibility of the intended mathematical action, student response to the prompt, reduction or continuation of support, and consistency with the collected written work. GEFT category, APOS construction, and the three AMT dimensions served as sensitizing concepts rather than variables for hypothesis testing. Case profiles were constructed for FI-1, FI-2, FD-1, and FD-2 and compared within and across cognitive-style categories through iterative coding, review, and theme refinement (Braun & Clarke, 2006).

Trustworthiness was supported through source and method triangulation, a prompt-response evidence matrix, and retention of disconfirming evidence (Nowell et al., 2017). The matrix linked each claim to the initial written step, the scaffold delivered, the observed response, the APOS interpretation, and the AMT code. Within-category variation was retained. FI-2, for example, remained a contrasting FI case because autonomous strategy selection did not initially produce an integrated Schema. Developmental claims were limited to changes explicitly documented during the scaffolded interaction. The analysis did not treat supported performance as evidence of independent mastery, retention, or transfer.

Ethical approval was obtained from the Institute for Research and Community Service (Lembaga Penelitian dan Pengabdian kepada Masyarakat, LP2M) under approval number STM/031/UI-AN/LP2M/III/2025. Participation was voluntary. Participants received information about the study, provided written informed consent, and were told that participation or withdrawal would not affect course grades. Codes FI-1, FI-2, FD-1, and FD-2 replaced personal identifiers in transcripts, analytic files, figures, and reporting. Digital and printed data were accessible only to the research team.

The cross-case analysis identified variation in initial task organization, APOS breakpoints, and responses to stage-specific scaffolding. The FI cases selected relevant transformations with less external structure, but differed in whether correct limit values were integrated. The FD cases required clearer cues, although the needed support began at different APOS constructions. The results distinguish the initial written pathway from the subsequent scaffolded response. Development refers to the observable reorganization produced during that interaction, not to long-term or statistically measured improvement. Short interview excerpts are presented with the written evidence to show how each student understood the initial difficulty and what changed after the APOS-oriented prompt.

The central finding is not that field-independent students possess stronger advanced mathematical thinking than field-dependent students. Rather, cognitive style shaped how students initially extracted, organized, and acted on information, while the location of the mathematical breakdown determined the support that became useful. Variation within each category is theoretically important. Autonomous selection of factorization or conjugate rationalization did not always produce a complete argument, and reliance on external structure did not imply an absence of relevant mathematical knowledge. These contrasting cases challenge a simple FI-success and FD-difficulty interpretation and support the use of field dependence-independence as an analytical lens rather than a hierarchy of mathematical ability (Hattie & O’Leary, 2025; Witkin et al., 1977).

This interpretation refines the role assigned to cognitive style in scaffolded calculus learning. Field independence can facilitate early disembedding of algebraic structures, whereas field dependence can increase the usefulness of visible organization and focused instructional cues. However, cognitive style alone does not identify what a student understands or which prompt should be delivered. The same learner category may contain students who differ in procedural availability, coordination, validation, and integration. This finding is consistent with evidence that scaffolding effects depend on the resources already available to the learner and on the specific mechanism targeted by the activity (Hofer & Reinhold, 2025). The relevant unit for instructional diagnosis is therefore the student’s current mathematical construction, not the cognitive-style label.

APOS theory clarifies the mechanism through which different prompts supported reasoning. The four forms of support performed distinct epistemic functions rather than representing increasing amounts of help. An Action-oriented cue made a viable transformation visible. A Process-oriented prompt coordinated the order and meaning of transformations. An Object-oriented question required the student to validate a completed limit value before reusing it. A Schema-oriented prompt connected separate limit results within one coherent argument. These constructions were not necessarily formed in a uniform or strictly linear sequence. A student could execute appropriate Actions without coordinating them, or could encapsulate an invalid result as an Object after an unstable Process. This diagnostic use of APOS extends its value beyond classifying solution steps and supports decisions about when scaffolding should be redirected, intensified, or faded (Arnon et al., 2014; Borji et al., 2022; Cottrill et al., 1996; Listiawati et al., 2025; Martínez-Planell et al., 2024).

The AMT framework explains why procedural correctness alone was insufficient evidence of advanced reasoning. In the limit task, abstraction became visible when students recognized and reused the difference-of-squares or conjugate structure. Conceptual reorganization appeared when separate symbolic transformations and intermediate values were coordinated into a valid solution. Formal reasoning required students to justify cancellation, distinguish an indeterminate form from a limit value, and connect the two limits to the final identity. This interpretation is consistent with accounts of limit understanding that emphasize coordinated processes, defensible concept structures, and meaningful relations rather than isolated routines (Cottrill et al., 1996; Santosa & Filiz, 2025; Tall, 1991; Tall & Vinner, 1981; Thompson & Harel, 2021). It also addresses persistent findings that prospective teachers may obtain results without constructing stable meanings for limits (Bansilal & Mkhwanazi, 2022; Mirin, 2024).

A further implication is that similar final answers can represent different developmental mechanisms. A student who already recognizes the relevant algebraic structure may need support only to make inferential links explicit or integrate completed values. Another student may know the required procedures but need help sequencing them. A third may require an initial structural cue before any viable transformation becomes available. Correctness should therefore be interpreted together with the origin of the strategy, the quality of the explanation, and the student’s ability to continue after support is reduced. This distinction prevents a correct answer produced through close guidance from being treated as equivalent to an independently constructed and justified solution.

The term developing is supported here through evidence of short-term qualitative reorganization during scaffolded activity. Development was inferred when a prompt enabled a student to notice a previously unused relation, coordinate an incomplete sequence, validate an intermediate result, or integrate separate results, and when the student subsequently continued with less assistance. By contrast, following a worked cue without being able to explain or reuse the transformation was interpreted as guided performance rather than a stable construction. This mechanism-based interpretation is compatible with conceptual scaffolding research (Kusmaryono et al., 2020; Setiyani et al., 2025), but it does not imply a statistically measured treatment effect or durable learning gain (Chien et al., 2026; DiNapoli & Miller, 2022; Hofer & Reinhold, 2025).

The findings suggest a practical diagnostic cycle for function-limit instruction. First, the lecturer examines where the student’s written reasoning stops or becomes invalid. Second, the lecturer selects the least directive prompt that targets the missing construction. A student who stops at 0/0 may need an Action cue that draws attention to factorization or a conjugate. A student who knows both procedures but applies them in an invalid order needs a Process prompt. Correct values accepted without checking call for Object-oriented validation, while separate results that are not connected require Schema-oriented integration. Third, the lecturer checks whether the student can explain and continue independently, then reduces support. This approach avoids both over-scaffolding students who can reorganize the task autonomously and under-supporting students whose initial Actions have not stabilized.

The study contributes a three-layer account of scaffolded AMT development. Cognitive style explains variation in initial task organization and access to external structure. APOS locates the construction that is available, unstable, or missing, while AMT describes the quality of abstraction, conceptual reorganization, and justification that emerges. Methodologically, triangulation of written work, interviews, and classroom observations keeps mathematical evidence central and retains disconfirming cases. The contribution is therefore not a typology prescribing one method for FI students and another for FD students, but a diagnostic model linking learner variation to the mathematical function of scaffolding.

Several limitations bound this interpretation. The study involved one institution, one integrated task, and four purposively selected cases, so it supports analytical transfer rather than statistical generalization. The observed changes were immediate and cannot establish retention, transfer, or independent long-term development. Because stimulated-recall interviews may extend reasoning beyond the initial written work, classroom learning and interview-elicited change must remain analytically distinct. GEFT categories were used for case selection, not as fixed diagnoses or measures of mathematical ability. Future studies should use repeated novel tasks, delayed assessments, systematic records of scaffold fading, and multi-site longitudinal designs.