近年來各國教育政策多朝向數學素養導向的改革，關注學生如何在快速變遷的社會中，將所學得的數學知識、技能、思維等靈活地應用在生活當中解決問題。因此對於「數學素養」一詞的見解，各國專家學者們出現各自表述的狀況。為避免「數學素養」定義的採用有所偏頗，本研究使用國際學生能力評估計畫(Programme for International Student Assessment，簡稱PISA)中對於數學素養的定義，以Toulmin論證模式作為主架構，以PISA建模過程、Duval幾何活動認知過程兩個面向分別探討在紙本環境下、提供動態幾何環境下，不同程度的九年級學生在數學論證歷程發展及特色。

本研究立意取樣台北市公立國中18位九年級學生，其中只有紙本環境下有9人、提供紙本與動態幾何環境下有9人，各分成高、中、低程度三組不同學習成就的學生進行質性資料分析，並詮釋其論證歷程。研究工具包括論證測驗題目單、動態幾何環境、半結構性晤談記錄、錄影檔、錄音檔。

本研究發現Toulmin論證模式在PISA建模過程中─形成過程探討由證據資料(D)到主張(C)的歷程；應用過程探討以論據(W)或支持理論(B)來說明主張成立的歷程；解釋與評估過程探討以限定修飾詞(Q)或反駁(R)來潤飾主張成立或不成立的歷程。

論證歷程的特色：
1.論證歷程會因為題目設計而有所不同，其中論證元素─「支持理論」、「反駁」、「限定修飾詞」不一定會在論證歷程中出現。
2.「限定修飾詞」是論證歷程中出現次數最少的論證元素。
3.論證歷程是動態的，每個「主張」皆有可能是下一個「主張」的中介主張，並非如原始Toulmin論證架構是靜態的。
4.驗證的動作發生在學生臆測答案或是無法確定自己的主張時，其中又以高程度學生發生驗證的比例較高。

中文部分：
蔡清田 (2011)。課程改革中的核心素養之功能。教育科學期刊，10(1)，203-217。.
洪萬生 (2008)。從國際教育評比淺論當前數學教育的得失。科學月刊，39(2)，84-85.
黃哲男 (2002)。於動態幾何環境下國中生動態心像建構與幾何推理之研究。國立臺灣師範大學數學研究所碩士論文，台北市。
吳德邦、馬秀蘭 (2003)。九年一貫數學圖形與空間課程學生在知覺、操弄性、作圖性、論說性了解之研究---從Mesquita和van Hiele的觀點。行政院國家科學委員會，NSC91-2521-S142-004
鄭勝鴻 (2005)。於動態幾何巨集環境下國中生證明概念與技能發展之研究。國立臺灣師範大學數學研究所碩士論文，台北市。
左台益 (2007)。動態心像與幾何學習之研究(3/3)。行政院國家科學委員會專題研究計畫成果報告，NSC 95-2521-S-003-004-
左台益 (2012)。動態幾何系統的概念工具。中等教育，63(4)，6-15。

英文部分：
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Duschl, R. A., & Osborne, J. (2002). Supporting and Promoting Argumentation Discourse in Science Education. Studies in Science Education, 38(1), 39–72.
Duval, R. (1995). Geometrical Pictures: Kinds of Representation and Specific Processings. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education, 138, 142–157. Berlin, Heidelberg: Springer Berlin Heidelberg.
Duval R. (1998). Geometry from a Cognitive Point of View. In C. Mammana and Villani (Eds), Perspectives on the Teaching of Geometry for the 21st century:an ICMI study. Dordrecht: Kluwer
Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning.
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Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), Studies in mathematical thinking and learning series. The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229-269). Hillsdale, NJ, US: Lawrence Erlbaum Associates, Inc.
Kuhn, D. (1992). Thinking as argument. Harvard Educational Review, 62(2), 155–178.
Kuhn, D. (1999). A Developmental Model of Critical Thinking. Educational Researcher, 28(2), 16–46.
Kuhn, D., & Udell, W. (2003). The Development of Argument Skills. Child Development, 74(5), 1245–1260.
Laborde, C. (1994). Enseigner la geometrie, Bulletin de l'A.P.M.E.P., 396,523-548, 1994.
Langsdorf, L. (2011). Argumentation as contextual logic: An appreciation of backing in Toulmin’s model. Cogency: Journal of Reasoning and Argumentation, 3(1), 51–78.
Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 173–204). Rotterdam, The Netherlands: Sense Publishers.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Niss, M. (2003). Mathematical competencies and the learning of mathematics: The Danish KOM project.Third Mediterranean conference on mathematics education (pp. 115–124).
Niss M. (2015) Mathematical Competencies and PISA. In: Stacey K., Turner R. (eds) Assessing Mathematical Literacy. Springer, Cham
Niss, M., W. Blum and P. Galbraith (2007), “Introduction”, in W. Blum, P.L. Galbraith, H.W. Henn and M. Niss (eds.), Modelling and Applications in Mathematics Education (The 14th ICMI Study), Springer, New York, pp. 3-32.
OECD (2017), PISA 2015 Assessment and Analytical Framework: Science, Reading, Mathematic, Financial Literacy and Collaborative Problem Solving, revised edition, PISA, OECD Publishing, Paris.
Schwarz, Baruch & Asterhan, Christa. (2010). Argumentation and reasoning. International Handbook of Psychology in Education. 137-176.
Simpson, A. (2015). The anatomy of a mathematical proof: Implications for analyses with Toulmin’s scheme. Educational Studies in Mathematics, 90(1), 1–17.
Stacey K., Turner R. (2015) The Evolution and Key Concepts of the PISA Mathematics Frameworks. In: Stacey K., Turner R. (eds) Assessing Mathematical Literacy. Springer, Cham
Toulmin, S. (1958). The use of argument. Cambridge: Cambridge University Press.
Toulmin, S. (2003). The Uses of Argument. Cambridge: Cambridge University Press.
van Eemeren, F. H., Grootendorst, R., Henkenmans, F. S., Blair, J. A., Johnson, R. H, Krabb, E. C., Plantin, C., Walton, D. N., Willard, C. A., Woods, J. & Zarefsky, D. (1996). Fundamentals of Argumentation theory: A handbook of historical background and contemporary developments. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.