研究者準備設計動態幾何環境來幫助學生學習線性規劃的原理，經過分析各版本有關於線性規劃的教材，再透過 Dubinsky 的 APOS 認知理論提出起源分解（genetic decomposition）的概念，假設出學生學習線性規劃的軌道，藉此探討學生學習線性規劃的認知結構，說明一個特定的數學概念要如何學習以及應該如何教，才能對學生的學習有所助益，而起源分解並不是唯一的，甚至在同一個主題下也有可能會發生不同的起源分解，探討學生對一數學概念之起源分解的認知特性，有助於瞭解學生概念建構歷程，因而提供研究者動態幾何教材設計的參考，接著再探討學生透過動態幾何教學後，其不同環境下學習者的學習表現。本研究採用量化分析和質性分析，透過動態幾何環境來輔助學生學習，分為實驗組與對照組，探討不同學習方式的同學，在線性規劃的原理上其學習表現的差異，得到 p=0.001 有顯著差異，再經過半結構式的訪談，探討兩個組別高中低在學習過程中是怎麼學、怎麼想，並重新建構起源分解結構圖，從個案學習過程與跨個案分析去探討學習困難的地方以及動態幾何有甚麼幫助。

Researchers are prepared to design dynamic geometric environments to help students learn the principles of linear programming, after analysis of the domestic version of the textbook on the linear programming and then through Dubinsky's APOS cognitive theory put forward the concept of decomposition (genetic decomposition). Suppose that students learn the proper track of linear programming, so as to explore the cognitive structure of students learning linear rogramming, It is helpful to explain how a particular mathematical concept is to learn and how to teach it, and the genetic decomposition is not unique, and even under the same subject, The cognitive characteristics of the genetic decomposition of the mathematical concept can help to understand the course of student concept construction, thus provide the reference of the researcher's dynamic geometric teaching design, and then discuss the learners' learning performance in different environments after dynamic geometry teaching.
This study uses quantitative analysis and qualitative analysis，assist students in learning through dynamic geometry，divided into experimental group and control group，students who study different ways of learning have different learning performance in the principle of linear programming，there was a significant difference for p = 0.001，and after a semi-structured interview, we discuss how the two groups learned in the learning process and how they think. And reconstruct the original decomposition structure map, from the case study process and cross-case analysis to explore where learning difficulties and dynamic geometry help.

參考文獻
[1] 蘇俊鴻 線性規劃 科學Online 科技部高瞻自然科學教學資源平台 2014/1/21
http://highscope.ch.ntu.edu.tw/wordpress/?p=51290
[2] 李國偉、黃文璋、楊德清、劉柏宏 (2013)，教育部 提昇國民素養實施方案─數學素養研究計畫結案報告，數學素養向度建議文P.5,10
[3] 謝佩君(2006) 高職線性規劃單元之動態電腦教材設計與補救教學研究p.120-p121
[4] 國民教育階段九年一貫課程總綱綱要，教育部(1998) P,4
[5] 李政蒲 、許銘津 、陳承泰 E 化專科教室發展創新教學模式之初探 中華民國第 25 屆科學教育學術研討會(2009) p.324
[6] 左台益(2002)：淺談資訊科技融入數學實驗教學。數學新天地，2 期，13-15 頁。
[7] 江紹祥(1999)，資訊科技數學教育，《數學教育》9，頁 33-45。
[8] 謝哲仁(2004) 數學關鍵學習之電腦設計研究 九十二學年度國科會數教學門結案報告
[9] 單維彰(100)科學月刊【數‧生活與學習】專欄‧百年 3 月 線性規劃是怎樣進入高中教材的？
[10] 普通高級中學必修科目「數學」課程綱要(99課綱)
[11] 十二年國民基本教育課程綱要 國民中小學暨普通型高級中等學校數學領域 （草案）(108課綱)
[12] 十二年國民基本教育課程綱要 數學領域課程手冊初稿，(106)
[13] 沈湘屏(2016) 高中生建構平面向量線性組合概念之個案研究，P9
[14] 周霆(2004)，商職二元一次不等式與線性規劃問題錯誤類型之分析研究
[15] 左台益(2012)，動態幾何系統的概念工具
[16] 左台益、蔡志仁(2001)，高中生建構橢圓多重表徵之認知特性。科學教育學刊，9(3)，281-297
[17] 許技江(2010)，動態鏈結多重表徵視窗環境下複數乘法學習之研究。
[18] Dalton, D. W. & Hannafin,M. J. （1988）。The Effects of Computer-Assisted and Traditional Mastery Methods on Computation Accuracy and Attitudes. Journal of Education Research, 82（1）,pp.27-33.
[19] Moyer, P. S., Bolyard, J. J., & Spikell, M. A. (2002). What Are Virtual Manipulatives?Teaching Children Mathematics, v8(n6), p372.
[20] Suh, J., & Moyer, P. S. (2007). Developing students' representation fluency using virtual and physical algebra balances. Journal of Computers in Mathematics and Science Teaching, 26(2), 155-173.
[21] Dunhan, P.H., & Thomas P.D.(1994).Research on Graphing Calculators. Mathmatics Teacher 86:P440-45
[22] Sheets, C.(1993).Effects of Computer Learning and Problem-Solving Tools on the Development of Secondary School Students’ Understanding of Mathematical Functions. Ph.D. Diss., University of Martland, Department of Curriculum and Instruction.
[23] Rojano, T. (1996). Developing Algebraic Aspects of Problem Solving within a Spreadsheet Environment. In Approaches to Algebra: Perspectives on Resrarch and Teaching, edited by Nadine Bednarz, Carolyn Kieran, and Lesley Lee. Dordrecht, Netherlands: Kluwer.
[24] Roy D. Pea (1985) Beyond Amplification: Using the Computer to Reorganize Mental Functioning 167-182
[25] Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014). APOS theory: A framework for research and curriculum development in mathematics education. Springer Science & Business Media
[26] Vinner, S. (1983).Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3),293-305.
[27] Tall, D., &Vinner, S.(1981).Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational studies in mathematics, 12(2), 151-169.
[28] Lesh, R., Post, T., & Behr, M. (1987). Representations and Translations among Representations in Mathematics Learning and Problem Solving. Problems of Representation in the Teaching and Learning of Mathematics, 21, 33-40.
[29] Kaput, J. J. (1989). Linking representations in the symbol systems of algebra. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 167–194). Hillsdale, NJ: Erlbaum.
[30] Hiebert, J., & Lefevre, P. (1986). Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis (pp. 1-27). Hillsdale, NJ: Erlbaum.
[31] Skemp, R. (1987). The Psychology of Learning Mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.
[32] Winn, W.(1999) Learning in Virtual Environments:A Theoretical Framework and Consideraton for design.虛擬環境中學習：理論架構與設計考慮。翻譯；單文經。校閱：邱貴發。 教學科技與媒體 24-32
[33] Nicolas Balacheff,James J. Kaput,(1996). Computer-Based Learning Environments in Mathematics
[34] Schuman, H. (1993), Continuous Variation of Geometric Figures: Interactive Theorem Finding and Problem in proving, Pythagoras, 31, 9-20.
[35] Hershkowitz, R.(1989), Visualisation in Geometry-Two Sides of the Coin, Focus on Learning Problems in Mathematics.
[36] Bishop, A. J. (1983). Space and Geometry , in R. Lesh and M. Landau(Eds), Acquisition of Mathematics Concepts and Processes , Academic Press Inc.
[37] Goldin, G. A.(2002). Representation in mathematical learning and problem solving. Handbook of international research in mathematics education, 197-218.