<![CDATA[Linear algebra is an area of mathematics that is widely considered challenging. One of the subjects covered is the system of linear equations in two variables. During their first year of college, Pre-Service Teachers (PST) study this subject using increasingly sophisticated approaches and theories. The objective of this qualitative study is to classify and describe the Mathematical Thinking Trajectories (MTT) of PST in solving a system of linear equations in two variables problems. Thirty-three PST majoring in mathematics education who enrolled in the Errors and Misconceptions in Mathematics course participated in this study, which used written answers as research data. The Three Worlds of Mathematics theory framework is used in this study. Seven MTT classifications were identified by the research findings: (1) embodied, (2) symbolic, (3) embodied-symbolic, (4) symbolic-embodied, (5) formal, (6) embodied-symbolic-embodied, and (7) symbolic-embodied-symbolic. “Embodied” refers to how PST applies mathematical ideas that focus on actual objects, “symbolic” focuses on symbols, operations, and their properties, and “formal” focuses on definitions, theorems, and logical procedures. In learning practices, the results of this study can be applied to create learning that is meaningful and that fosters the growth of algebraic thinking abilities.]]>
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