Mental Constructions of Male and Female Pre-Service Mathematics Teachers’ In Solving Systems of Linear Equations via APOS Theory

Abstract

Different solution strategies may produce the same correct answer while involving different mathematical constructions. This study addressed the need to examine these constructions and verification practices among male and female pre-service mathematics teachers by integrating Polya’s framework with APOS theory. A qualitative explanatory approach involved 38 students who had completed an Elementary Linear Algebra course. Six students were interviewed, and four two male and two female—were purposively selected for obtaining correct answers through distinct strategies. Written solutions and interviews were analyzed using Polya’s stages and APOS structures and mechanisms. All participants successfully understood the problem, devised a plan, and carried out their strategies. The male participants used Gauss–Jordan elimination and direct substitution, whereas the female participants incorporated arithmetic-sequence relationships through substitution or the arithmetic-series formula. Beyond these procedural differences, APOS analysis identified different conceptual connections and Schema organizations underlying the same correct solution. All demonstrated Action, Process, Object, and Schema structures through interiorization, coordination, and encapsulation. Within the study’s coding framework, reversal was evidenced through procedural rechecking in the male participants, but comparable evidence was not identified in the female participants. Nevertheless, no participant demonstrated complete final verification; evidence of de-encapsulation associated with verification was limited, and thematization was not clearly identified. These findings reveal individual construction pathways and distinguish procedural monitoring from final verification. Instruction should encourage strategy comparison, conceptual explanations, and explicit verification.