• 作者： 林福來
• 作者服務機構： 國立台灣師範大學數學系
• 中文摘要： 調查國中生的比例推理能力時，曾建構了解單比例推理的四個了解層次。本文主要描述超越這四個了解層次的三個高等比例推理的了解層次。調查對象是具代表性的台灣高中一、二年級學生共1539名，選擇的概念領域包括複比與非線型比。調查方法以筆測為主，訪談為輔。 高等比例推理的三個了解層次中的學生特徵，包括各層次學生有能力處理的不同型態之比例問題，解題的策略，．解題時簡捷的想法，解題時易犯的錯誤，與較弱的能力等等，都是本文探討的重點。
• 英文摘要： A hierarchy of four levels of understanding in simple ratio were estalished by one of our pre-vious stuaies. The present study developed three more advanced understanding levels beyond thosefour levels. Among the various domains of advanced proportional problems, multiple ratio, i.e. aa bc, and the non-linear ratios, i.e. y a kx2∕kx3，were chosen as the concept domains for thisstudy. 1539 subjects who are representative of Taiwan senior high school students aged 16 and 17took part in this study. These three advanced hierarchical levels are linked nicely with the four levels of understand-ing simple ratios. They are characterized strongly by the different kinds of relationships betweenquantities involved in the tasks, such as inverse ratio, composition of two positive ratios, composi-tion of positive and inverse ratios, the quadratic relation and the cubic relation. The reasoningstrategies used by students and the meaning of the levels of understanding were analysed in thispaper.
• 中文關鍵字： ratio and proportion; advanced mathematics thinking; understanding levels
• 英文關鍵字： --