本研究旨在探討範例融入方式及數學自我效能，對國小學生等值分數彈性思考表現和數學學習態度之影響。研究對象為台北市某國小五年級學生，共100位。研究設計採因子設計之準實驗研究法，自變項包含範例融入方式和數學自我效能，範例融入方式分為「連續型範例一般教學」、「連續型範例資訊技融入教學」、「整合型範例資訊科技融入教學」三種方式，數學自我效能依據數學自我效能量表總得分，將學習者分為高、低數學自我效能兩組，參與者分別於教學實驗前後接受測量。依變相為「等值分數彈性思考表現」及「數學學習態度」，等值分數彈性思考表現包括(1)基本彈性思考：基本補畫能力、分割能力；(2)進階彈性思考：進階補畫能力、組合能力、運作思考能力、單位形成能力，數學學習態度為學習興趣、學習動機、數學焦慮三個面向。

研究結果發現：(1)運用整合型範例資訊科技融入教學，能促進學習者等值分數彈性思考；(2)運用虛擬教具於數學教學中，可提升學習者的學習興趣；(3)高數學自我效能學習者透過不同範例融入方式學習等值分數，均有較好的等值分數彈性思考表現及學習態度。本研究結果與建議可供國小數學科教學與未來相關研究參考。

The purpose of this study was to investigate the effect of type of example and mathematics self-efficacy on the five graders’ flexible thinking ability and mathematical attitudes. The participants in this study were 100 fifth graders from three classes of an elementary school in Taipei, Taiwan. The quasi-experimental design with factorial design was employed in the study. The independent variables were methods of example integrated into instruction and self-efficacy toward mathematics. The methods of example integrated into instruction include general instruction of continuous example, continuous example integrated into instruction with technology, and mixed example integrated into instruction with technology. Learners were categorized into two groups, which were high self-efficacy toward mathematics and low self-efficacy toward mathematics based on the scores measured by a self-efficacy scale on mathematics. The dependent variables were flexible thinking ability on equivalent fractions and mathematical attitudes. The flexible thinking ability include (a) basic flexible thinking: basic drawing ability and division ability, (b) advanced flexible thinking: advanced drawing ability, assembly ability, operative thinking ability and unitize ability.
The results showed that (a) the application of mixed example integrated into instruction with technology promoted the flexible thinking ability on equivalent fractions, (b) the application of virtual manipulative on the mathematics teaching promoted the students’ learning interest, and (c) the higher mathematical self-efficacy, the better flexible thinking ability on equivalent fractions and mathematical attitudes.

中文部份
Booth, L. R. (1987)。分數的學習困難(Booth 專題演講，林麗惠整理)。科學教育月刊，100，8-16。
中小學數學科教材教法 (張英傑、周菊美譯) (2005)。台北市：五南。(原著出版年﹕2004年)
王全世 (2000)。對資訊科技融入各科教學之資訊情境的評估標準。資訊與教育，77，36-47。
王春生 (2004)。教師專業成長資訊融入教學提昇學生學習。北縣教育，50，55-58。
何榮桂 (2002)。台灣資訊教育的現況與發展－兼論資訊科技融入教學。資訊與教育，87，22-48。
余民寧 (2005)。心理與教育統計學。台北市：三民。
吳知賢 (1996)。歸因再訓練在國小兒童自我效能及學業成就之影響研究。國民教育研究集刊，2，1-36。
吳麗玲 (2005)。台灣、美國與新加坡國小五、六年級分數教材內容之分析比較。國立嘉義大學數學教育研究所，未出版，嘉義市。
呂玉琴 (1991a)。分數概念：文獻探討。國立台北師範學院學報，4，573-606。
呂玉琴 (1991b)。影響分數二分之一概念的因素：個案分析。國民教育，31(11,12)，16-21。
呂玉琴 (1994)。國小教師分數教學之相關知識研究。國立台灣師範大學科學教育研究所博士論文，未出版，台北市。
呂玉琴 (1998a)。國小教師分數教學之相關知識研究。臺北師院學報，11，393-438。
呂玉琴 (1998b)。分數的四則運算與等值分數的設計。國民小學數學科新課程概說【高年級】(114-131頁)。台北市：台灣省國民學校教師研習會。
林碧珍 (1990)。從圖形表徵與符號表徵之間的轉換探討國小學生的分數概念。新竹師院學報，4，295-347。
林福來、黃敏晃、呂玉琴 (1996)。分數啟蒙的學習與教學之發展性研究。科學教育學刊，4(2)，161-196。
邱瓊慧 (2002)。中小學資訊科技融入教學之實踐。資訊與教育雜誌，88，3-10。
洪素敏、楊德清 (2002)。創意教學：分數的補救教學。科學教育研究與發展季刊，29，33-52。
徐新逸、吳佩謹 (2002)。資訊融入教學的現代意義與具體作為。教學科技與媒體，3，63-73。
袁媛、吳東光、林素微 (2007)。國中小數學虛擬教具的研發與教學研究。行政院國家科學委員會專題研究計畫成果報告(編號：NSC95-2520-S-033-003)。執行單位：中原大學。
張國恩 (2002)。從學習科技的發展看資訊融入教學的內涵。北縣教育，41，16-25。
教育部 (2004)。國民中小學九年一貫課程綱要。台北：教育部。
曹萬春 (2004)。應用鷹架理論輔助國小分數迷思概念課程效益之探究。國立台中師範學院數學教育學系碩士論文，未出版，台中市。
陳玉玲 (1995)。目標設定、目標投入與自我效能對國小學生數學作業表現的影響。國立高雄師範大學教育學系碩士論文，未出版，高雄市。
陳淑貞 (2004)。國小教師對於資訊科技融入教學的迷思與省思。師說，180，11-13。
陳雅芬 (2003)。國小學童等值分數概念的試題編製與分析之研究。國立台中師範學院數學教育學系碩士論文，未出版，台中市。
陳靜姿 (1997)。國小四年級兒童等值分數瞭解之初探。國立台中師範學院初等教育研究所碩士論文，未出版，台中市。
彭海燕 (1998)。國小學童等值分數概念了解之研究。國立台北師範學院國民教育研究所碩士論文，未出版，台北市。
游自達 (1993)。美國國小兒童對分數大小之理解及思考策略。國立台中師範學院初等教育研究所初等教育研究集刊，1，121-145。
湯錦雲 (2002)。國小五年級學童分數概念與運算錯誤類型之研究。國立屏東師範學院數理教育研究所碩士論文，未出版，屏東市。
黃敏晃、周筱亭 (2001)。國小數學教材分析-分數的數概念與運算。台北縣：教育部台灣省國民學校教師研習會。
黃馨緯 (1995)。國小高年級學童分數數線表示法了解之研究。國立台中師範學院初等教育研究所碩士論文，未出版，台中市。
甯自強 (1997)。量的子分割(二)：真分數的引入。教師之友，38(4)，33-39。
楊壬孝 (1988)。國中小學生分數概念的發展。行政院國家科學委員會專題研究計劃成果報告(編號：NSC-77-0111-S-003-09A)。執行單位：國立台灣師範大學數學系。
楊瑞智 (2000)。探究師院生之分數基本概念及分數概念的課室教學。台北市立師範學院學報，31，357-382。
萬志祥 (2004)。資訊融入教學的省思與推動。北縣教育，50，42-45。
詹志禹 (1997)。全方位對話。教育研究雙資訊月刊，17，6-7。
劉世雄 (2002)。探討資訊科技融入教學之課程設計。生活科技教育，35(6)，24-31。
劉信雄 (1992)。國小學生認知風格、學習策略、自我效能與學業成就之研究。國立政治大學教育研究所博士論文，未出版，台北市。
劉秋木 (2002)。國小數學科教學研究。台北：五南。
數學學習心理學 (林義雄、陳澤民譯) (1995)。台北市：九章。(原著出版年：1987年)
蔡麗蓉 (2003)。國小數學科審定本教科書分數教材之內容分析。國立台中師範學院國民教育研究所，未出版，台中市。
賴阿福 (2004)。數位化教學與學習環境之變革。國教新知，51(1)，19-32。
魏麗敏 (1996)。影響國小兒童數學成就之自我調節學習與情感因素分析及其策略訓練效果之研究。國立台灣師範大學教育心理與輔導研究所博士論文，未出版，台北市。

英文部份
Ainsa, T. (1999). Success of using technology and manipulatives to introduce numerical problem solving skills in monolingual/bilingual early childhood classrooms. Journal of Computers in Mathematics and Science, 18(4), 361-369.
Anderson, J. R., Farrell, R., & Sauers, R. (1984). Learning to program in LISP. Cognitive Science, 8(2), 87-129.
Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. (2000). Learning form examples: Instructional principles from the worked examples research. Review of Educational Research, 70(2), 181-214.
Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. Englewood Cliffs, NJ: Prentice Hall.
Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In R. Lesh, & M. Landau, (Eds.), Acquisition of Mathematics Concepts and Processes (pp.91-126). New York: Academic Press.
Behr, M. & Post, T. (1992). Teaching rational number and decimal concepts. In T. Post (Ed.), Teaching mathematics in grades K-8: Research-based methods (pp. 201-248). Boston: Alley & Bacon.
Behr, M. J., Wachsmuth, I., & Post, T. R. (1985). Construct a sum: A measure of children’s understanding of fraction size. Journal for Research in Mathematics Education, 16(2), 120-131.
Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15(5), 323-341.
Bergeron, M. J. & Herscovics, H. (1987). Unit fractions of a continuous whole. Psychology of Mathematics Education, 11(1), 357-365.
Bishop, A. J. (1989). Review of research on visualization in mathematics education. Focus on Learning Problems in Mathematics, 11(1-2), 7-16.
Bouffard-Bouchard, T. (1989). Influence of self-efficacy of performance in a cognitive task. Journal of Social Psychology, 130(3), 353-363.
Bright, G. W., Behr, M. J., Lesh, R., & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education, 19(3), 215-232.
Brown, S. D. (1989). Effects of self-efficacy: Aptitude incogruence on career behavior. The American psychological Association. (ERIC Document Reproduction Service No. ED302747)
Clement, D. H. (1999). Concrete manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45-60.
Clements, D. H., Nastasi, B. K., & Swaminathan, S. (1993). Young children and computers: Crossroads and directions from research. Young Children, 48(2), 56-64.
Columba, H. L. (1989). Equivalent fraction concepts: A teaching experiment. Unpublished doctoral dissertation, University of Louisville, Louisville, KY.
Dias, L. B. (1999). Integrating technology: Some things you should know. Learning & Leading with Technology, 27(3), 10-13.
Drickey, N. A. (2000). A comparison of virtual and physical manipulatives in teaching virtualization and spatial reasoning to middle school mathematics students. Unpublished doctoral dissertation, Utah State University.
Duarte, V., Young, M., & DeFranco, T. (2000). What experts say and do regarding the use of technology in the mathematics classroom. Journal of Research and Development in Education, 33(4), 223-231.
Enderson, M. (1997). Old problems, new questions: Using technology to enhance math education. Learning and Leading with Technology, 25(2), 28-32.
Godding, P. R. & Glasgow, R. E. (1985). Self-efficacy and outcome expectancy as predictors of controlled smoking status. Cognitive Therapy and Research, 9(1), 23-31.
Hackett, G. & Betz, N. E. (1989). An exploration of the mathematics self-efficacy/ mathematics performance correspondence. Journal for Research in Mathematics Education, 20(3), 261-273.
Hart, K. (1981). Children's understanding of mathematics. London: John Murray Ltd.
Hunting, R. P. (1983). A case study of knowledge of units and performance with fractions. Journal for Reacher in Mathematics Education, 14(3), 182-197.
Izydorczak, A. (2003). A study of virtual manipulatives for elementary mathematics. Unpublished doctoral dissertation, State University of New York-Buffalo.
Johansen, D. H. (2000). Computer as mindtools for schools. NJ: Prentice Hall, Inc.
Jonassen, D. H., Peck, K.C., & Wilson, B. G.. (1999). Learning with technology: A constructivist perspective. Upper Saddle River, NJ: Merrill/ Prentice Hall.
Kamii, C. & Clark, F. B. (1995). Equivalent fractions: Their difficulty and educational implications. Journal of Mathematical Behavior, 14(4), 365-378.
Kerslake, D. (1986). Fraction: Children’s strategies and errors. A report of the strategies and errors in secondary mathematics project. Windsor, Berkshire, England: NFER-NELSON.
Kong, S. C. & Kwok, L. F. (2005). A cognitive tool for teaching the addition/ subtraction of common fractions: A model of affordances. Computers & Education, 45(2), 245-265.
LeFevre, J. A. & Dixon, P. (1986). Do written instructions need examples? Cognition and Instruction, 3(1), 1-30.
Lent, R. W. Brown, S. D., & Larkin, K. C. (1984). Relation of self-efficacy expectations to academic achievement and persistence. Journal of Counseling Psychology, 31(3), 356-362.
Moyer, P., Bolyard, J., & Spikell, M. (2002). What are virtual manipulatives? Teaching Children Mathematics, 8(6), 372-377.
Moyer, P., Niezgoda, D., & Stanley, M. (2005). Young children’s use of virtual manipulatives and other forms of mathematical representation. In W. Masalski & P. Elliott (Eds.), Technology-supported mathematics learning environments (pp. 17-34). Reston, VA: NCTM.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Olkun, S. (2003). Comparing computer versus concrete manipulatives in learning 2D geometry. The Journal of Computers in Mathematics and Science Teaching, 22(1), 43-56.
Olkun, S., Altun, A., & Smith, G.. (2005). Computers and 2D geometric learning of Turkish fourth and fifth graders. British Journal of Educational Technology, 36(2), 317-326.
Panagos, R. J. & DuBois, D. L. (1999). Career self-efficacy development and students with learning disabilities. Learning Disabilities Research and Practice, 14(1), 25-34.
Parrat-Dayan, S. & Voneche, J. (1992). Conservation and the notion of “half”. In J. Bideaud, C. Meljac, & J. P. Fischer (Eds.), Pathways to number. Hillsdale, NJ: Erlbaum.
Piaget, J. (1987). Possibility and necessity. Minneapolis: University of Minnesota Press. (Original work published 1983)
Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of geometry. New York: Basic Book.
Post, T. R. (1988). Some notes on the nature of mathematics learning. In T. R. Post (Eds.), Teaching mathematics in grades K-8 (pp. 1-19). Boston, MA: Allyn and Bacon.
Randhawa, B. S., Beamer, J. E., & Lundberg, I. (1993). Role of mathematics self-efficacy in the structure model of mathematics achievement. Journal of Educational Psychology, 85(1), 41-48.
Reimer, K. & Moyer, P. S. (2005). Third-graders learn about fractions using virtual manipulatives: a classroom study. The Journal of Computers in Mathematics and Science Teaching, 24(1), 5-25.
Recker, M. & Pirolli, P. (1995). Modeling individual differences in learning strategies. Journal of the Learning Sciences, 4(1), 1-38.
Roth, W. G. (1985). Treatment implications derived from self-efficacy research with children (Report No. CG019361). Doctor of Psychology Research Paper. (ERIC Document Reproduction Service No. ED273897)
Saenz-Ludlow, A. (1994). Michael’s Fraction Scheme. Journal of Research in Mathematics Education, 25(1), 50-85.
Saenz-Ludlow, A. (1995). Ann’s fraction schemes. Educationl Studies in Mathematics Education, 28(2), 101-132.
Schunk, D. H. (2000). Learning theories: An educational perspective. (3rd ed.). Upper Saddle River, NJ: Prentice Hall.
Sherer, M. & Maddux J. (1982). The self-efficacy scale: Construction and validation. Psychological Reports, 51(2), 663-671.
Shade, D. & Watson, J. A. (1990). Computers in early education: Issues put to rest, theoretical links to sound practice, and the potential contribution of microworlds. Journal of Educational Computing Research, 6(4), 375-392.
Siegle, D. & McCoach, D. B. (2007). Increasing student mathematics self-efficacy through teacher training. Journal of Advanced Academics, 18(2), 278-312.
Skinner, E. A. Wellborn, J. G., & Connell, J. P. (1990). What it take to do well in school and whether I’ve got it: A process model of perceived control and children’s engagement and achievement in school. Journal of Educational Psychology, 82(1), 22-32.
Steen, K., Brooks, D., & Lyon, T. (2006). The impact of virtual manipulatives on first grade geometry instruction and learning. Journal of Computers in Mathematics and Science Teaching, 25(4), 373-391.
Suh, J., Moyer, P. S., & Heo, H. (2005). Examining technology uses in the classroom: students developing fraction sense by using virtual manipulative concept tutorials. Journal of Interactive Online Learning, 3(4), 1-21.
Thomas, J. W., Iventosch, L., & Rohwer, W. D. Jr. (1987). Relationships among student characteristics, study activities, and achievement as a function of course characteristics. Contemporary Educational Psychology, 12(4), 344-364.
Vance, J. H. (1992). Understanding equivalence: A number by any other name. School Science and Mathematics, 92(5), 263-266.
Woolfolk, A. E. & Hony, W. K. (1990). Prospective teachers’ sense of efficacy and beliefs about control. Journal of Educational Psychology, 82(1), 81-91.
Yuan, Y. (2005). Design of virtual manipulatives for mathematical explorations using Flash ActionScript. Asia Technology Conference in Mathematics, 182-193.
Zimmerman, B. J. (2000). Attaining self-regulation: A social cognitive perspective. In M. Boekaerts, P. R. Pintrich, & M. Zeidner (Eds.), Handbook of self-regulation (pp. 13-39). New York: Academic Press.