Algebraic thinking involves generalizing arithmetic principles and focusing on the relationships between quantities or variables. Mastering the concept of equivalence is a key step in transitioning from arithmetic to algebra. Mathematical word problems provide a semantic context and aid in developing equivalence. To investigate the development of mathematical equivalence from elementary to junior high school, the study devised sets of multiple true or false questions based on mathematical word problems and recruited 2,677 students spanning grades three to eight. Based on the cognitive abilities of representation transformation and equivalence ability, students were divided into three groups. The cognitive diagnostic model (CDM) was used to examine the percentages of three ability types within grades three to eight: difficulties, single representations, and multiple representations. The item response theory (IRT) was used to estimate the students´ ability levels and discern variances across these three types of abilities for students in grades three to eight. The results showed that as students progressed through higher grade levels, the percentage of those difficult students decreased, while the percentages of students with single representation and multiple representation increased. It was observed that a considerable percentage of elementary students, ranging from 20% to 60%, exhibited proficiency in handling multiple representations, indicating that primary algebraic thinking could be nurtured at the primary school level. However, only about 40% of junior high school students demonstrated competency in multiple representations. In terms of ability levels, difficult students manifested the lowest values, followed by those with single representations; students with multiple representations displayed the highest values. Additionally, the growth of algebraic proficiency among difficult students lagged behind. Students with single representations showed noticeable improvement with increasing grade levels, while those with multiple representations exhibited more significant growth, indicating that equivalence abilities display multilevel characteristics depending on the mathematical content integrated into the equations. Furthermore, this study elucidated the research methodology that combines CDM and IRT to delineate long-term developmental abilities and discussed their potential optimization.
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Analyzing Representation Transformation and Equivalence Abilities across Students in Grades Three to Eight in Mathematical Word Problems
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