Exploring the role of cognitive style and APOS-based scaffolding in developing students’ advanced mathematical thinking on function limits

INTRODUCTION

Function limits constitute a conceptual gateway to continuity, differentiation, integration, and mathematical analysis. However, students often approach limit tasks as routines for substitution and algebraic manipulation rather than as problems that require reasoning about covariation, approximation, and values that are approached but not necessarily attained. Research involving university students and prospective mathematics teachers continues to identify fragmented concept images, confusion between a function value and a limiting value, weak coordination across representations, and difficulty interpreting the indeterminate form 0/0 as a signal for further analysis rather than as a numerical result(Bansilal & Mkhwanazi, 2022; Borg & Fahlgren, 2026; Mirin, 2024; Tall & Vinner, 1981; Thompson & Harel, 2021). These difficulties have particular importance in mathematics teacher education. Prospective teachers must not only obtain a correct answer, but also identify the structure of an expression, justify each transformation, connect intermediate results, and communicate a mathematically defensible conclusion.

The central difficulty in learning limits is therefore not reducible to procedural accuracy. Understanding a limit requires students to coordinate processes, reorganize familiar algebraic relations, and treat the resulting limit as an object that can be examined and used in a broader argument(Cottrill et al., 1996). Advanced mathematical thinking (AMT) provides an appropriate framework for examining this transition from isolated operations to connected mathematical reasoning.Tall (1991)characterizes AMT through abstraction, conceptual reorganization, and formal reasoning. In the present study, abstraction refers to recognizing and reusing an underlying relation, such as a difference of squares or a conjugate structure. Conceptual reorganization refers to coordinating separate transformations and intermediate values into a coherent solution structure. Formal reasoning refers to constructing a logically connected justification for the transformations, limit evaluations, and final identity. These dimensions are treated as qualitative indicators of mathematical activity, not as independent psychometric scores.

Students may display these forms of reasoning through different patterns of information processing. The field dependence-independence framework concerns the extent to which learners can disembed relevant information from a surrounding context and reorganize it through an internally generated reference structure(Witkin et al., 1977). Field-independent (FI) students tend to identify and restructure relevant information more autonomously, whereas field-dependent (FD) students may rely more strongly on explicit organization, instructional cues, or interpersonal guidance. This distinction does not represent a hierarchy of mathematical ability and should not be interpreted as a basis for assigning a fixed teaching method to a cognitive-style category. Recent criticism of mechanical instructional matching further supports the use of cognitive style as an analytical lens for examining task entry and access to support, rather than as a deterministic learner label(Hattie & O’Leary, 2025).

APOS theory provides a complementary explanation of the mathematical constructions that students form while working with limits. In APOS theory, an Action is an externally prompted transformation; a Process emerges when the learner can mentally coordinate and reverse related actions; an Object is formed when a process can be examined, validated, and reused; and a Schema integrates relevant Actions, Processes, Objects, and their relations into a coherent conceptual structure(Arnon et al., 2014; Dubinsky, 2014). For function limits, these constructions may include direct substitution, factorization, conjugate rationalization, cancellation, evaluation of the resulting expression, and integration of separate limit values into a final proof. APOS-based scaffolding can support these constructions through targeted cues, comparison questions, sequencing prompts, requests for explanation, and questions that require students to validate and reuse completed results. Research indicates that APOS-informed activities can strengthen conceptual understanding in university mathematics, while the effectiveness of scaffolding depends on the learner’s available resources and on the specific mechanism targeted by the support(Borji et al., 2022; Chien et al., 2026; DiNapoli & Miller, 2022; Hofer & Reinhold, 2025; Martínez-Planell et al., 2024).

Despite these contributions, the relevant literature remains theoretically and empirically fragmented. Studies of function limits primarily document conceptual and representational difficulties. APOS studies identify the mental constructions required for advanced mathematical concepts. Scaffolding studies explain how contingent support can sustain strategy selection and reasoning. Cognitive-style research describes differences in how learners organize information and use external structure. Limited evidence connects these perspectives within the same instructional episode. In particular, little is known about whether FI and FD students differ mainly in selecting an initial Action, coordinating a Process, validating an Object, or integrating a Schema; how stage-specific APOS prompts change their subsequent reasoning; and how such prompt-response changes appear in abstraction, conceptual reorganization, and formal justification. Analyses based only on final correctness cannot reveal these mechanisms.

This study addresses that gap through an integrated diagnostic model. Cognitive style is used to examine how students initially identify, organize, and act on information in the same function-limit task. APOS theory is used to locate the construction that is already available, remains unstable, or emerges after support. AMT is used to describe the quality of the resulting mathematical reasoning. Within this model, APOS-based scaffolding does not replace students’ reasoning or prescribe identical support for all members of a cognitive-style group. Instead, it targets the next mathematical construction indicated by the student’s written work and interview response. The term developing in the title therefore refers to observable, short-term qualitative change in a student’s mathematical construction during scaffolded activity, such as recognizing a previously unnoticed relation, coordinating a valid sequence of transformations, validating a limit value, or integrating separate results into a coherent argument. It does not claim a statistically measured gain, a causal treatment effect,