Threshold Transitions and Expertise Reversal: The Impact of Procedural Mastery on Critical Thinking in Pre-Service Mathematics Teachers

Abstract

Background: Procedural mastery serves as a foundational component in the development of advanced critical thinking within educational ecosystems where teacher-training frameworks are continually developing. However, rigid instructional designs and automated cognitive patterns persist, heavily shaping pedagogical content knowledge and profoundly influencing the problem-solving capacity of future educators.

Objective: This study examines how procedural-conceptual mastery limits critical thinking performance in geometric sequences among pre-service mathematics teachers, while evaluating the structural boundaries of the expertise reversal effect and non-linear threshold transitions.

Methodology: The empirical analysis utilises a sequential mixed-methods explanatory design combining qualitative and quantitative frameworks. A sample of 29 pre-service mathematics teachers was selected from public university education programmes in Indonesia. Quantitative data were operationalised via a paper-based assessment that differentiated between pre-critical performance (procedural fluency and basic conceptual recall) and critical thinking capabilities (modelling, interpretation, justification, and strategic decision-making), with qualitative insights captured through comprehensive written solutions and protocol-driven think-aloud interviews.

Results: The findings indicate that a strong performance asymmetry exists within the sample, demonstrating that pre-critical thinking scores are significantly more stable (M = 35.99/50; CV = 22.3%) than higher-order critical thinking scores (M = 22.50/50; CV = 71.0%). In opposition to assumptions of linear learning models, pre-critical and critical thinking do not report strong positive linear associations (r = -0.34), whereas overall test scores remain strongly correlated with critical thinking performance (r = 0.84). Concurrently, score distributions suggest a distinct cognitive missing middle, with no participants falling within intermediate performance bands. Qualitative theme extraction reveals that over-automated formulas function as a procedural knowledge trap that actively weakens structural reasoning and obscures justificatory control during advanced geometric sequence tasks.

Conclusion: The study concludes that procedural fluency assumes control over complex reasoning processes and constrains critical thinking in advanced learning contexts. Structural focus on algorithmic automaticity yields suboptimal cognitive outcomes when pre-service teachers encounter non-linear conceptual thresholds that require flexible execution.

Unique Contribution: This research advances cognitive load theory and mathematics teacher education literature by providing an integrated empirical framework that links the expertise reversal effect directly to macro-level threshold transitions within an emerging economy context, explicitly illustrating how rigid algorithmic architectures disrupt the development of deeper critical mathematical thinking.

Key Recommendation: It is recommended that curriculum designers, mathematics educators, and university instruction boards in teacher training institutions design educational reforms that discourage rote algorithmic memorisation. Additionally, policymakers should implement targeted instructional strategies that emphasize procedural flexibility, de-automatisation paradigms, and explicit justification practices to safeguard conceptual transparency and enhance the critical thinking capabilities of future mathematics teachers.