三至八年級學生數學文字題的表徵轉換與等價能力

Analyzing Representation Transformation and Equivalence Abilities across Students in Grades Three to Eight in Mathematical Word Problems

代數思維是算術一般化的歷程，著重數量或變量間的關係，而掌握等價性為算術過渡到代數的關鍵之一。問題情境提供語意脈絡，有助於學生發展等價性。本研究為了瞭解橫跨國小到國中的等價性發展，針對2677位三至八年級學生設計數學文字題解題算式的多重是非題題組，採用認知診斷模型（cognitive diagnostic model [CDM]）界定出表徵轉換能力與等價能力兩種認知屬性，確認學生可分成困難者、單一表徵者、多元表徵者三種類型，並以試題反應理論（item response theory [IRT]）估計學生的能力值，以探究三至八年級學生在三種類型的百分比、能力值及其差異。研究結果顯示，隨著年級的增加，困難者百分比大致呈現下降，單一表徵者、多元表徵者百分比大致呈現上升的趨勢，而國小已有二至六成學生屬於多元表徵者，顯示國小階段即可培養初階代數能力，然而國中學生僅四成為多元表徵者。在能力值方面，困難者最低，單一表徵者居中，多元表徵者最高，且困難者文字題算式判斷能力成長遲滯，單一表徵者的能力隨年級增加明顯些，而多元表徵者的能力則隨年級有更顯著地增長，顯示等價能力會隨等式涉及的數學內涵而呈現多層次。此外，本研究示範了整合CDM與IRT以描繪長期發展之能力的研究方法，並檢討其未來優化的方向。

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.