From Perplexity to Mathematical Certainty: Reflective Clarificative Thinking in Solving Systems of Linear Equations
DOI:
https://doi.org/10.51278/aj.v8i4.2953Keywords:
Reflective Clarificative Thinking, Perplexity, Systems of Linear Equations, Problem Solving, Pre-Service Mathematics TeachersAbstract
Solving systems of linear equations may generate doubt that requires scrutiny of information, procedures, and results. This study aims to describe and reconstruct pre-service mathematics teachers’ reflective clarificative thinking in addressing perplexity across Polya’s four problem-solving stages. A qualitative multiple-case study involved two purposively selected participants who solved the same traffic-flow conservation problem using substitution. Data were collected through a problem-solving test and in-depth task-based interviews, including a follow-up interview on S2’s solution checking, and examined through within-case and cross-case analyses. The findings show that S1 engaged in confirmatory clarification by verifying the equations to establish solution correctness. S2 engaged in corrective clarification by detecting an inconsistency, tracing a subtraction error, correcting the initial value and its dependent variables, and verifying the revised solution without changing the substitution method. Both forms of clarification supported mathematically justified certainty. This study contributes a distinction between confirmatory and corrective clarification as responses to doubt during problem solving. These findings highlight the importance of learning activities that encourage pre-service teachers to articulate uncertainty, examine sources of error, explain corrections, and verify solutions against all problem conditions, thereby strengthening their capacity to justify solutions and make mathematical judgments grounded in careful reasoning that can be substantiated.
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