五、 結論與建議
5.2 建議
簡報。AMA系統是一種新的繪圖法，能解決設計教材時定位不易的問題，與PowerPoint
同時使用兩種教具比單獨使用任何一種好（Terry, 1995）。Suh and Moyer（2007）
研究也顯示使用虛擬或是實體的代數天平來建立學生表徵，成就都一樣顯著，且在轉換
設計適合常態學生程度的課程，探索主要的規則鑲嵌、半規則鑲嵌、次規則鑲嵌簡介、
學校的國二學生，各選取一班為實驗組，一班為控制組，有樣本數過小的研究限制。雖 然接受教學的學生就讀之學校採常態編班，入學編班依據入學時的測驗成績做 S 型編 排，但仍不能強推到其他地區。因此，日後的研究者可以增加樣本數量，以取得更有代 表性的研究結果。
2.增加教學研究的時間
考量學校教學進度安排無法任意更動，本研究為延伸課程，旨在將數學融入日常生 活中，並非正式課堂必教課程，為了避免壓縮正式課程的教學時間，僅安排三堂課程，
有教學研究時間過短的研究限制，因此本實驗教學對於學習成效的結果論定可能造成影 響。因此，日後研究者可以考慮增加教學研究時間，進行更深入的探究。
3.AMA 簡報系統與不同教學素材結合的開發研究
AMA 簡報系統以數學概念所發展出來的教學軟體，能簡便地繪圖且能同時製作教學 簡報，成為一個數位內容設計及展演、繪本寫作及創意的平台。利用激發式動態呈現的 功能，能協助教學者適時的呈現數位內容，降低學習者的認知負荷。因此，未來研究者 若能善用 AMA 簡報系統的這些性質，就能輕鬆地將資訊融入數學教學，也可看到 AMA 簡 報系統與不同的教學素材結合的可能契機，例如：尺規作圖、直角座標、函數圖形等。
4.虛擬教具在於不同教學環境的成效研究
一些研究報告顯示虛擬教具可以運用在特殊教育、學前教育與補救教學的學習環境 中，但是目前在教學現場虛擬教具的使用並不普及，且對於是否能真的取代傳統實體教 具，學者間也有不同的看法。而國內對於虛擬教具教學成效的正式研究，數量其實很少
（Yuan，2006），本教學實驗也僅是將 AMA 簡報系統和 NLVM 的 Tessellations 運用於 鑲嵌教學中探討虛擬教具環境和傳統實體教具環境的學習成效差異。因此，未來研究者 可以將虛擬教具運用在一般的教學單元或不同的教學環境中，進行實証研究。
5.改良虛擬教具的操作功能
由於本教學實驗中虛擬教具的設計，只是以研究者進行教學的角度去思考學生如何 操作，雖然已請其他教師進行操作並依意見修改調整，然而並沒有找學生試用，所以不 知道學生的操作習慣，因此，可以增加複製、拉近、拉遠等功能和更人性化的操作版面 配合學生的操作習慣，這些都是未來研究者可以努力的方向。
參 考 文 獻
http://ama.nctu.edu.tw/Introduction/AMA-Introduction.mht
曹亮吉（2003）。阿草的數學聖杯阿草的數學聖杯阿草的數學聖杯阿草的數學聖杯－－－－探尋無所不在的胚騰探尋無所不在的胚騰探尋無所不在的胚騰探尋無所不在的胚騰。天下遠見，台北市。
張祖忻、朱純（1995)。教學設計教學設計教學設計教學設計：：：：基本原理與方法基本原理與方法基本原理與方法基本原理與方法。五南，台北市。
張瑜軒（2002）。群論應用於艾雪鑲嵌藝術之對稱構成研究群論應用於艾雪鑲嵌藝術之對稱構成研究群論應用於艾雪鑲嵌藝術之對稱構成研究－群論應用於艾雪鑲嵌藝術之對稱構成研究－－－以多媒體創作為例以多媒體創作為例以多媒體創作為例以多媒體創作為例。中原 大學商業設計學系碩士論文，未出版，中壢市。
彭健彰（2008）。虛擬教具應用於國小四虛擬教具應用於國小四虛擬教具應用於國小四年虛擬教具應用於國小四年年年級重級重級重級重量量量量概概概概念念念念教學之影響研究教學之影響研究教學之影響研究教學之影響研究。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。
數學真魅-密鋪欣賞（2008）。2008 年 7 月 10 日，取自
http://aishuxue.blogspot.com/2008/04/blog-post.html
劉賢建（2006）。虛擬計算機的開發與教學研究虛擬計算機的開發與教學研究虛擬計算機的開發與教學研究 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以探索數字樣式一般化為例以探索數字樣式一般化為例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。
蕭慶利（2007）。運用二維對稱特性建構雙瓦片鋪磚之研究運用二維對稱特性建構雙瓦片鋪磚之研究運用二維對稱特性建構雙瓦片鋪磚之研究運用二維對稱特性建構雙瓦片鋪磚之研究。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。
英文部分
Aiken, L. R. (1986). Sex difference in mathematical ability: a review of the literature.
Educational Research Quarterly, 87, 5, 531-543.
Branson, R. K. (1978). The interservice procedures for instructional systems development.
Educational Technology, March, 11-14.
California Department of Education(1997). Mathematics Content Standards for California Public Schools. Retrieved October 29, 2008, from:
http://www.cde.ca.gov/be/st/ss/documents/mathstandard.pdf.
Cassidy, J. (1998). Tessellations & transitions. Arts and Activities, 123, 5, 42.
Char, C. A. (1989). Computer graphic feltboards: New software approaches for young children’s mathematics exploration. San Francisco American Educational Research Association.
Clauss, J. E. (1991). Pentagonal Tessellations. The Arithmetic Teacher, 38, 5, 52.
Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of reasoning on mathematics teaching and learning (pp.420-464). New York, NY: Macmillan.
Clements, D. H. , & McMillen, S. (1996). Rethinking "concrete" manipulatives. Teaching Children Mathematics, 2(5), 270-279.
Coxeter, H. S. M. , Emmer, M. , Penrose, R. , & Teuber M. L. (editors). (1986). M.C. Escher:
Art and Science, Amsterdam: Elsevier Science. Collection of papers on Escher's work, with analyses of his use of tessellations and polyhedra.
Crawford, C. , & Brown, E. (2003). Integrating internet-based mathematical manipulatives within a learning environment. Journal of Computers inMathematics and Science Teaching, 22(2), 169-180.
Dick, W. (1996). The dick and carey model: Will it survive the decade? . Educational Technology Research and Development, 44(30), 55-63.
Dick, W. , & Carey, L. M. (1978, 1985, 1990, 1996). The systematic design of instruction (editions 1 through 4.). New York: HarperCollins College Publishers.
Dorward, J. (2002). Intuition and research: Are they comparable? . Teaching Children Mathematics, 8, 329-332.
Drickey, N. A. (2000). A comparison of virtual and physical manipulatives in teaching visualization and spatial reasoning to middle school mathematics students. (Doctoral dissertation, Utah State University, 2000). Dissertation Abstracts International, 62(02), 499A.
Durmus, S., & Karakirik, E. (2006). Virtual manipulatives in mathematics education: A theoretical framework. The Turkish Journal of EducationalTechnology, 5(1). Retrieved April 8, 2008, from: http://www.tojet.net/articles/5112.htm.
Dutch, S. (1999). Uniform Tilings. Retrieved August 20, 2008, from:
http://www.uwgb.edu/dutchs/symmetry/uniftil.htm.
Fennema, E. (1972). Models and mathematics. Arithmetic Teacher, 19(Dec.), 633-640.
Furner, J. M. , Goodman, B. , & Meeks, S. (2004). Creating tessellations with pavement chalk:
Implementing best practices in Mathematics. The Australian Mathematics Teacher, 60 ( 2), 25-28.
Giganti, P. J. , & Cittadino, M. J. (1990). The Art of Tessellation. The Arithmetic Teacher, March , 37(7), 6.
Granger, T. (2000). Math is art. Teaching Children Mathematics, 7(5), 10-13.
Grunbaum, B., & Shephard, G.. C. (1986). Tilings and Patterns, New York: W. H. Freeman and Company.
Hale, C. (2003). Create tessellations like M.C. Escher. Instructor, 112(7), 32.
Heath, G. D. (2002). Using applets in teaching mathematics. Mathematics and Computer Education, 36(1), 43-52.
Heinich, R. , Molenda, M. , Russell, J. D. , & Smaldino, S. (2002). Instructional media and technologies for learning(7th ed.). Columbus: Merrill/Prentice Hall.
Izydorczak, A. E. (2003). A study of virtual manipulatives for elementary mathematics.
(Doctoral dissertation, State University of New York at Buffalo, 2003). Dissertation Abstracts International.
Jacobs, K. L. (2005). Investigation of interactive online visual tools for the learning of mathematics. International Journal of Mathematical Education in Science and Technology, 36(7), 761-768.
Johnson, C. D. , & Kashef, A. E. (1996). Tessellations in the technology education classroom.
The Technology Teacher, Journal of the International Technology Education Association, 56(3), 3-7.
Kim, S. (1993). The relative effectiveness of hands-on and computer-simulated manipulatives in teaching seriation, classification, geometric, and arithmetic concepts to kindergarten children. (Doctoral dissertation, University of Oregon, 1993). Dissertations Abstracts International, 54(09), 3319.
Leathrum, T. (2001). Writing mathlets I: A call to math professionals. Journal of Online Mathematics and its Applications. 1(2). Retrieved August 19, 2008,from:
http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=460.
Mathematical Classification of Tessellations (2003). Retrieved July 11, 2008, from:
http://library.thinkquest.org/11750/eschpage/MathClass1.html
Mayer, R. E. (2003). The promise of multimedia learning: using the same instructional design methods across different media. Learning and Instruction, 13, 125-139,
Miller, D. , Brown, A. , & Robinson, L. (2002). Widgets on the web, using computer-based learning tools. Teaching Exceptional Children, 35(2), 24-28.
Molenda, M. (2003). In Search of the Elusive ADDIE Model. Performance Improvemen, Indiana University .
Molenda, M., Pershing, J. A. , & Reigeluth, C. M. (1996). Designing instructional systems. In R. L. Craig(Ed.), The ASTD training and development handbook, 4th ed., (pp. 266-293).
New York:McGraw-Hill.
Morrison, G. R., Ross, S. M., & Kemp, J. E. (2004). Designing effective instruction, 4th edition, New York, NY: John Wiley & Sons Inc.
Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics, 47, 175-197.
Moyer, P. S. , Bolyard, J. J. , & Spikell, M. A. (2002). What are virtual manipulatives? . Teaching Children Mathematics, 8, 372-377.
Moyer, P. S. , & Jones, M. G. (2004). Controlling Choice: Teachers, Students, and
Manipulatives in Mathematics Classrooms. School Science and Mathematics, Jan 2004, 104, 16-31.
Moyer, P. S., Niezgoda, D. , & Stanley, J. (2005). Young children’s use of virtual
manipulatives and other forms of mathematics representations. In P. C. Elliott(Eds.), Technology-supported mathematics learning environments (Sixty-seven yearbook), Reston, VA: NCTM.
National Council of Teachers of Mathematics(2000). Principles and Standards for School Mathematics, Reston, VA: NCTM.
Nute, N. (1997). The impact of engagement activity and manipulatives presentation on intermediate mathematics achievement, time-on-task, learning efficiency, and attitude.
(Doctoral dissertation, University of Memphis, 1997). Dissertations Abstracts International, 58(08), 2988.
Parham, J. L. (1983). A meta-analysis of the use of mamipulative materials and student achievement in elementary school mathematics. (Doctoral dissertation, Auburn University, 1983). Dissertations Abstracts International, 44-01A: 96.
Perl, T. (1990).Manipulatives and the Computer: A Powerful Partnership for Learners of All Ages. Classroom Computer Learning, Dayton: Mar 1990, Vol. 10, Iss. 6, 20-22, 26-29.
Peterson, B. E. (2000). From tessellations to polyhedra. Mathematics Teaching in the Middle School, 5(6), 348-357.
Raphael, D. , & Wahlstorm, M. (1989). The influence of instructional aids on mathematics achievement. Journal for Research in Mathematics Education, 20(2), 173-196.
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.
Riley, G. , Beard, L. A. , & Strain, J. (2004). Assisstive technology at use in the teacher education programs at Jacksonville State University. Tech Trends, 48(6), 47-49.
Sarama, J. , Clements, D. H. , & Vukelic, E. B. (1996). The role of a computer manipulative in fostering specific psychological/mathematical processes. In E. Jakubowski & D.
Watkins & H. Biske (Eds.), Proceedings of the Eighteenth Annual Meeting of the North America Chapter of the Intermational Group for the Psychology of Mathematics
Education(Vol. 2, pp. 567-572). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Schielack, J. F. (1991). Reachingyoungpupilswithtechnology. Arithmetic Teacher, 38(6), 51-55.
Smith, J. P. (1995). The effects of a computer microworld on middle school students’ use and understanding of integers. (Doctoral dissertation, Ohio State University, 1995).
Sobol, A. J. (1998). A formative and summative evaluation study of classroom interactions and student/teacher effects when implementing algebra tile manipulatives with junior high school students. (Doctoral dissertation, St. John's University, 1998). Dissertation Abstracts International, 59(04A), 1103.
Sowell, E. J. (1989). Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics Education, 20(5), 498-505.
Spicer, J. (2000). Virtual Manipulatives: A New Tool for Hands-on Math. ENC Focus, 7(4), 14-15.
Steen, K. , Brooks, D. , & Lyon, T. (2006). The impact of virtual manipulatives on first grade geometry instruction and learning. The Journal of Computers in Mathematics and Science Teaching, 25, 373-391, 4, Academic Research Library.
Stylianou, D. A. , ＆ Grzegorczyk, I. (2005). Symmetry in mathematics and art：an exploration of an art venue for mathematics learning. PRIMUS, 15:1, 30-44.
Suh, M. J. (2005). Third grades’s mathematics achievement and representation preference using virtual and physical manipulatives for adding fractions and balancing equations. Unpublished doctoral dissertation. University of Georgemason. Fairfax, Virginia.
Suh, J. , & Moyer, P. S. (2005). Examining technology uses in the classroom: Developing fraction sense using virtual manipulative concept tutorials. Journal of Interactive Online Learning, 3(4), 1-21.
Suh, J. , & Moyer, P. S. (2007). Developing Students' Representational Fluency Using Virtual and Physical Algebra balances. The Journal of Computers in Mathematics and Science Teaching, 26, 2, 155.
Suydam, M. N. (1984). Attitude toward mathematics, Arithmetic Teacher, 12, 56-59.
Terry, M. K. (1995). An investigation of differences in cognition when utilizing math manipulatives and math manipulative software. (Doctoral dissertation, University of Missouri - Saint Louis, 1996). Dissertation Abstracts International, 56(07), 2605.
Tessellation Tutorials (1994). Retrieved August 31, 2008, from:
http://mathforum.org/sum95/suzanne/tess.intro.html.
The Official M. C. Escher Website. Retrieved August 19, 2008,from:
http://www.mcescher.com/indexuk.htm.
Thompson, P. W. (1992). Notations, conventions, constraints: Contributions to effective uses of concrete materials in elementary mathematics. Journal for research in mathematics education, 23, 123-147.
Totally Tessellated (1998). An Introduction to Tessellations. Retrieved July 10, 2008, from:
http://library.thinkquest.org/16661
Yuan, Y. (2005). Design of virtual manipulatives for mathematical explorations using Flash ActionScript. In Proceedings of the 10th Asia Technology Conference in Mathematics, 182-193.
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.
Yuan, Y. , & Lee, C. I. (2004). Designing instructional tools by Flash MX to teach basic geometry concepts. In proceedings of the TIME-2004 Symposium. Montreal, CA: July, 2004.
Yuan, Y., Lee, C. Y. & Huang, J. R. (2007). Developing Geometry Software for Exploration
—Geometry Player. Journal of the Korea Society of Mathematical Education Series D:
Research in Mathematical Education, 11(3), 209-218.
附錄一 鑲嵌圖形教案設計與學習單
課程名稱 鑲嵌圖形(Tessellations) 設計者 張玉琪老師
能力指標能力指標能力指標
的 Tessellations。
5 分鐘
教師透過 powerpoint 展示各式磁磚圖片，讓 學生欣賞日常生活中幾何圖形的鑲嵌。教師
2.第二節課，45 分鐘
教師利用 powerpoint 簡介說明半規則鑲嵌和 次規則鑲嵌的差異，次規則鑲嵌種類當回家
鑲嵌圖形學習單 鑲嵌圖形學習單 鑲嵌圖形學習單 鑲嵌圖形學習單 1 1 1 1
年年年年 班班班班 座號座號座號座號：：：： 姓名姓名姓名姓名：：：：
看過前面的磁磚圖片，你們認為這些磁磚是如何鑲嵌的？回想一下剛才的圖形，
將想到的畫在下面空白處。
畫完之後，發現磁磚有哪些形狀？圖形如何安排才是鑲嵌？
我們來分析正多邊形的性質，請完成下表。
正多邊形 三 四 五 六 七 八 九 十
每一內角 度數
正多邊形 十一 十二 十三 十四 十五 十六 十七 十八 每一內角
度數
正多邊形 十九 二十 二十四 三十 三十六 四十 四十二
請利用老師給的各種正多邊形教具，實際嘗試看看吧！將能鑲嵌的正多邊形（規
請利用老師給的各種正多邊形教具，實際嘗試看看吧！將能鑲嵌的正多邊形（規