本研究檢證國小五年級學生在數學等值分數單元使用數學虛擬教具的學習成就、表徵流暢度、對數學學習態度、及對數學虛擬教具態度。研究採準實驗設計，實驗組為26位使用虛擬教具學習等值分數概念的學生，控制組為24位使用紙本示例學習等值分數概念的學生。以前測與後測量數檢證學生的學習成就與表徵流暢度；以問卷的數據結果歸納學生對數學學習態度與對數學虛擬教具的態度。研究發現，使用數學虛擬教具學習的學生之學習成就顯著高於使用紙本示例學習的學生。使用數學虛擬教具學習之學生在表徵流暢度上優於使用紙本示例學習的學生，尤其是文字轉文字表徵以及圖形轉文字表徵。使用虛擬教具學習的學生對數學學習態度與對虛擬教具使用上的表現都持正向看法。

This study examined the effects of mathematical virtual manipulative on fifth Graders’ performance and representational fluency in equivalent fraction. Studnets’ attitude toward mathematical learning and toward virtual manipulatives were also analyzed. Quasi-experiment design was implemented. A class with 26 students was assigned as the experimental group, and the other class with 24 students as the control group. The experimental group used virtual manipulatives, and the control group used teacher-made handouts. Data sources included a pretest and a posttest of students’ equivalent fraction concepts and a student attitude survey. The results indicated that students’ performance using virtual manipulatives were better than student performance using teacher-made handouts. In terms of representational fluency, students using virtual manipulatives were better than students using teacher-made handouts. Translations from symbolic to symbolic representation and translations from pictorial to symbolic representations were significant. Student attitude survey indicated the majority of students had positive attitudes toward using the fraction virtual manipulatives to learn fraction concepts.

Baker, M. (2008). Merging technology & mathematics instruction: The case of virtual manipulatives & geometric concepts. In G. Richards (Ed.), Proceedings of World Conference on E-Learning in Corporate, Government, Healthcare, and Higher Education (pp. 566-573). Chesapeake, VA: AACE.
Bruner, J.S. (1964). The course of cognitive growth. American Psychologist, 19, 1-15.
Cannon, L. O., Heal, E. R., & Wellman, R. (2000). Serendipity in interactive mathematics: Virtual (electronic) manipulatives for learning elementary mathematics. Proceedings of the Society for Information Technology and Teacher Education International Conference, USA, 2000, 1083-1088.
Goldin, G., & Shteingold, N. (2001). Systems of representations and the development of mathematical concepts. In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics (pp. 1-23). Reston, VA: National Council of Teachers of Mathematics.
Highfield, K., & Mulligan, J. T. (2007). The role of dynamic interactive technological tools in preschooler's mathematical patterning. In J. Watson & K. Beswick (Eds.), Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia, Hobart (Vol. 1, pp. 372-381). Adelaide: MERGA.
Janvier, C. (1987). Representation and understanding: The notion of function as an example. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 67-71). Hillsdale, NJ: Erlbaum.
Jigyel, K., & Afamasaga-Fuata’i, K. (2007). Students’ conceptions of models of fractions and equivalence. Aust. Math. Teach. 63(4), 17-25
Lesh, R. (1979), Mathematical learning disabilities: Considerations for identification, diagnosis, and remediation. In R. Lesh, D. Mierkiewicz & M. G. Kantowski (Eds.), Applied mathematical problem solving (pp.111-175), Columbus, OH: ERIC/SMEAC.
Post, T., Behr, M., & Lesh, R. (1982). Interpretations of Rational Number Concepts. In L. Silvey & J. Smart (Eds.), Mathematics for Grades 5-9, 1982 NCTM Yearbook (pp. 59-72). Reston, Virginia: National Council of Teachers of Mathematics.
Lesh, R., Landau, M., & Hamilton, E. (1983). Conceptual models in appliedmathematical problem solving research. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts & processes (pp. 263-343). New York: Academic Press.
Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier, (Ed.), Problems of Representations in the Teaching and Learning of Mathematics (pp. 33-40). Hillsdale, NJ: Lawrence Erlbaum.
Lindroth, L. K. (2005, Jan). How to... find online math manipulatives. Retrieved January 18, 2010, from http://www.teachingk-8.com/archives/how_to/how_tofind_online_math_manipulatives.html
Mamede, E., Nunes, T., & Bryant, P. (2005). The equivalence and ordering of fractions in part-whole and quotient situations. In Chick, H. L. & Vincent, J. L. (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (pp. 281-288). Melbourne: PME.
Moyer, P. S., Bolyard, J.J., & Spikell, M.A. (2002). What are virtual manipulatives? Teaching Children Mathematics, 8(6), 372-377.
Moyer, P. S., Niezgoda, D., & Stanley, J. (2005). Young children’s use of virtual manipulatives and other forms of mathematical representations. In W. J. Masalski & P.C. Elliott (Eds.), Technology-supported mathematics learning environments, 67th yearbook (pp. 17-34). Reston, VA: National Council of Teachers of Mathematics.
Moyer, P.S., & Bolyard, J.J. (2002). Exploring representation in the middle grades: Investigations in geometry with virtual manipulatives. The Australian Mathematics Teacher, 58(1), 19-25.
Moyer, P.S., Salkind, G., & Bolyard, J.J. (2008). Virtual manipulatives used by K-8 teachers for mathematics instruction: Considering mathematical, cognitive, and pedagogical fidelity. Contemporary Issues in Technology and Teacher Education, 8(3), 202-218.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics: An overview. Reston, VA: Author.
Reimer, K., & Moyer, P. S. (2005). Third graders learn about fractions using virtual manipulatives: A classroom study. Journal of Computers in Mathematics and Science Teaching, 24(1), 5-25.
Sayeski, K. L. (2008). Virtual manipulatives as an assistive technology support for students with high-incidence disabilities. Journal of Special Education Technology, 23(1), 47-53.
Scaptura, C., Suh, J. M., & McHaffey, G. (2007). Masterpieces to mathematics: Using art to teach fraction, decimal, and percent equivalents. Mathematics Teaching in the Middle School, 13(1), 24-28.
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.
Shari, A. B., & Vanessa , E. H. (2007). A virtual spin on the teaching of probability. Teaching Children Mathmatics, 13(9), 482-486
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. M. (2005). Third Graders’ Mathematics Achievement and Representation Preference Using Virtual and Physical Manipulatives for Adding Fractions and Balancing Equations. Unpublished doctoral dissertation, George Mason University, Fairfax, Virginia.
Suh, J. M. (2007). Tying it all together: Building mathematics proficiency for all students. Teaching Children Mathematics, 14(3), 163-169.
Suh, J. M., & Moyer, P. S. (2007). Developing students' representational fluency using virtual and physical algebra balances. Journal of Computers in Mathematics and Science Teaching, 26(2), 155-173.
Suh, J. M., Moyer, P. S., & Heo, H.-J. (2005). Examining technology uses in the classroom: Developing fraction sense using virtual manipulative concept tutorials. The Journal of Interactive Online Learning, 3(4), 1-22.
Thompson, D.R., & Chappell. M. F. (2007). Communication and representation as elements in mathematical literacy. Reading & Writing Quarterly, 23(2), 179-196.
Vance,J.H.（1992）.Understanding equivalence：A number by and other name. School Science and Mathematics, 92(5), 263-266.
Young, D. (2006, April 27). Virtual manipulatives in mathematics education. Retrieved January 18, 2010, from http://plaza.ufl.edu/youngdj/talks/vms_paper.doc
Yuan, Y. (2005). Design of virtual manipulatives for mathematical explorations using Flash ActionScript. Proceedings of the 10th Asia Technology Conference in Mathematics, 183-192.
Yuan, Y. (2007). A comparison study of polyominoes explorations in a physical and a virtual manipulative environment. Proceedings of the 4th East Asia Regional Conference on Mathematics Education, 186-190.
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，100，7-15。
王美仁 (2009)。電腦化九年一貫國小五年級學童等值分數能力診斷測驗編製之探討。國立屏東教育大學數理教育研究所碩士論文，未出版，屏東市。
王智弘 (2005)。多方塊虛擬教具的開發與教學研究。國立交通大學理學院網路學習學程碩士論文，未出版，新竹市。
呂玉琴（1991）。分數概念：文獻探討。國立臺北師範學院學報，4，573-606。
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(報告編號：NSC95-2520-S-033-003)，未出版。
袁媛、陳國龍、張世明(2007)。萬用揭示板(Magic Board)-國小特教老師的數學教學好幫手。特教論壇，3，1-13。
張漢宜(2002)。教導兒童學習數學的新工具-虛擬教具。國教輔導，42(1)，33-37。
教育部 (2008年5月23日)。97年國民中小學課程綱要。台北市：教育部。2010年1月18日，取自：http://teach.eje.edu.tw/9CC/context/97-03-3-7.html
陳霈頡、楊德清(2005)。數學表徵應用在教學上的探究。科學教育研究與發展季刊，40，48- 61。
曾振家 (2002)。國小五年級學生在動態多重表徵情境下建構分數加法概念之研究。國立臺南師範學院數學科教學碩士論文，碩士論文，未出版，台南市。
詹婉華、呂玉琴（2004）。國小高年級學童分數概念量表之設計研究。科學教育學刊，12(2)，241-263。
劉景聰 (2008)。虛擬教具融入國小六年級分數補救教學成效之研究。中原大學教育研究所碩士論文，未出版，桃園縣。
劉賢建 (2006)。虛擬計算機的開發與教學研究—以探索數字樣式一般化為例。國立交通大學理學院網路學習學程碩士論文，未出版，新竹市。
鄭千佑 (2008)。虛擬教具對國小學生等值分數彈性思考表現之影響。國立台灣師範大學資訊教育研究所碩士論文，未出版，台北市。
蕭登仲、謝哲仁、蔡玉花 (2004)。國小學生在動態多重表徵視窗環境下學習等值分數成效之研究。南師學報，38(1)，77-106。