教育學報27-1.indd

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1 * 摘要本研究目的是依據 Lamon(1993) 的比與比值語意結構的分類, 提出比與比值初始概念引入的替代性教學設計及試驗性教學初探採用文獻評析, 提出影響學生學習比值的重要因素及分析教科書的架構 ; 以內容分析法, 分析現行六年級教科書在比與比值單元的題目情境類型本文亦提出比與比值起始概念引入的關鍵概念 : 比的使用之適當性比感的建立及比值相對關係兩數量在題目中需符合固定倍數關係情境可類推性對應關係意義化三種條件, 使用比的符號, 才具意義性本研究的替代性教學設計包含教學目標及教學布題, 可作為教科書編輯及教師教學設計的參考責任編輯 : 楊龍立投稿日期 :2009 年 11 月 18 日,2010 年 5 月 17 日修改完畢,2010 年 6 月 15 日通過採用 * 林碧珍, 國立新竹教育大學數理教育研究所教授, linpj@mail.nhcue.edu.tw 127

題目中需符合固定倍數關係情境可類推性對應關係意義化三種條件, 使用比的符號, 才具意義性本研究的替代性教學設計包含教學目標及教學布題, 可作為教科書編輯及教師教學設計的參考責

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9 Jeong, Levine, & Huttenlocker, 2007; Steinthorsdottir, 2006 Lamon1993 ABCDXX ABCXABXD X XXBCD AXCD 3. ABCDABA CABC 1ABAC268a 2AB269a 3AC258a 4ABAC259a Karplus, Pulos, & Stage, 1983; Ruiz & Lupianez, 2009; Van Dooren, et al., 2006 AB 2 3ABCAB D

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30 a 2009b 本文所呈現的結果為國科會科教處資助的三年期研究計畫數學教學案例在教師專業發展的應用編號 NSC S MY3 之部分資料, 感謝國科會科教處對本研究計畫在經費上的支持感謝參與於本研究計畫現場教師在比與比值教學實務的貢獻, 讓本文得以融入了理論與實務的觀點 156

31 Hart, K. M. (1981). Fractions. In K. M. Hart, D. Kerslake, M. L. Brown, G. Ruddock, D. E. Kuchmann, & M. McCartney (Eds.), Children understanding of mathematics (pp ). Oxford, London: Northampton. Jeong, Y., Levine, S. C., & Huttenlocker, J. (2007). The development of proportional reasoning: Effect of continuous versus discrete quantities. Journal of Cognition and Development, 8(2), Karplus, R., Pulos, S., & Stage, E. (1983). Proportional reasoning of early adolescents. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes. New York: Academic Press. Lamon, S. (1993). Ration and proportion: Connecting content and children's thinking. Journal for Research in Mathematics Education, 24(1), Lamon, S. (1994). Ration and proportion: Cognitive foundations in utilizing and norming. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp ). Albany, NY: State University of New York Press. Lo, J., & Watanable, T. (1997). Developing ratio and proportion schemes: A story of fifth grader. Journal for Research in Mathematics Education, 28(2), Paige, D. D., Willcutt, R. E., & Wagenblast, J. M. (1968). Mathematics methods for junior high school teachers. Boston: Prindle. Ruiz, E. F., & Lupianez, J. L. (2009). Detecting psychological obstacles to teaching and learning the topics of ratio and proportion in sixth grade primary pupils. Electronic Journal of Research in Educational Psychology, 17(7), Steinthorsdottir, B. (2006). Proportional reasoning: Variable influencing the problem difficulty level and one's use of problem solving strategies. 157

32 Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, vol.5 (pp ). Prague: Charles University. Van Dooren, W., De Bock, D., Evers, M., & Verschaffel, L. (2006). Pupils' overuse of proportionality on missing vaule problems: How numbers may change solutions. Proceedings of the 30 th Conference of the International Group for the Psychology of Mathematics Education, vol.5 (pp ). Prague: Charles University. Verschaffel, L., De Corte, E., & Lasure, S. (1994). Realistic considerations in mathematical modeling of school arithmetic word problem. Learning and Instruction, 4,

33 The Exploration of Teaching Initial Concept of Ratio Pi-Jen Lin* Abstract The study was indented to explore an alternative approach of teaching initial concept of rate and ratio by taking the view of the nature of mathematics in which is based on Lamons four semantic structures of proportion problems. Main factors of influencing studentssolving proportion problems and the framework of analyzing current textbooks were addressed by literature reviewing. The semantic structures of the proportion problems were compared and contrast among three textbooks. The key initial concepts of ratio includes: the use of ratio, the sense of ratio, and the meaning of the relationship between two numbers. Particularly, the study claims that students learned the sense of ration by emphasizing the use of ratio in which ratio can only be used when the contexts of problem is inferred and the numbers are patterns. The inference of contexts being embedded into the problems are two essential elements of developing sense of ratio. The alternative instructional design can be as a reference for curriculum writing committees. Key words: rate, ratio, sense of ratio, semantic structure Section editor: Long-Lih Yang Received: November 18, 2009; Modified: May 17, 2010; Accepted: June 15, 2010 * Pi-Jen Lin, professor, Department of Graduate Institute of Mathematics and Science Education, National Hsinchu University of Education, linpj@mail.nhcue.edu.tw 159

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PROSPECT EXPLORATION 壹前言 20 90 066 第 9 卷第 2 期中華民國 100 年 2 月

PROSPECT EXPLORATION 壹前言 20 90 066 第 9 卷第 2 期中華民國 100 年 2 月專題研究法律與法制探析中國大陸現行土地估價法律制度體系探析中國大陸現行土地估價法律制度體系 A Study on the Legal Framework of Real Estate Appraisal in China 蔣國基 * (Chiang, Kuo-chi) 香港鴻裕公司法務長摘