從算式或文字閱讀數學科普文章：共變數分析與階層線性模式的比較

Reading Popular Mathematics from Equations or Words: Comparison of Analysis of Covariance and Hierarchical Linear Modeling

理解數學推導是困難的，而文獻上對文本中數學推導段落採用算式或文字表徵何者較具優勢的結論不一。本研究以299位來自四所高中職的高一學生為對象，比較科普文章以算式或文字表達數學推導對理解的影響，以及對高、低能力學生是否有不同的效果。由於高中生能力是巢套於學校，因此以共變數分析與階層線性模式來比較結果。研究材料為三篇幾何的科普文本及其理解測驗，算式與文字兩個版本的內容只在關鍵句（分別僅1、5、5句）用了不同表徵，其餘內容、附圖及測驗都相同。受試者被隨機分派到算式版或文字版，閱讀後的理解測驗包含是非、計算及說明題。此外，受試者也接受閱讀推理篩選測驗與數學先備知識測驗。共變數分析結果顯示能力和版本不具交互作用，且高能力學生顯著優於低能力學生，算式版多數優於文字版。而階層線性模式分析在排除學校平均數學能力的顯著影響後，凸顯了版本的效果，算式版低能力學生的閱讀表現與文字版高能力學生無顯著差異。本研究從算式和文字的語言結構討論算式優勢，也透過文本主題領域、認知負荷、評量重點及測驗題型，和受試者特性與文獻進行比較。

Understanding mathematical reasoning is challenging. No conclusive evidence exists on which external representation is more beneficial to comprehension: equation or words. This study involved 299 high school students and examined the effects of external representations and participants’ abilities on reading comprehension of popular mathematics. Because students were nested within schools, analysis of covariance (ANCOVA) and hierarchical linear modeling (HLM) were performed, and their results were compared. The materials included three popular mathematics and comprehension tests, all of which were in the domain of geometry. The main difference between the equation version and the verbal version was the method of representation used in key sentences (only one, five, and five sentences differed in each of the three passages, respectively). The other sentences, illustrations, and tests were the same in both versions. Students were randomly assigned into groups for each of the two versions and completed the reading comprehension tests, reading comprehension screening tests, and math prior knowledge tests. The ANCOVA results demonstrated that the equation readers outperformed the verbal readers and that the high-ability readers performed better than the low-ability ones, but the results did not indicate any interaction between version and ability. After the exclusion of the effects of school-average math ability, the HLM results found that low-ability equation readers demonstrated a nonsignificant difference in performance compared with high-ability verbal readers. The benefits of using equations were discussed by comparing the linguistic features of the two external representations. Furthermore, the results were compared with previous research in terms of passage domain, cognitive load, focus of measurement, and participant characteristics.