在國際間，數學論證已逐漸成為數學教育的新趨勢。本研究以數學臆測教學為前提，在國科會計畫的參與教師之課堂下，以個案研究法觀察同一教師所帶之班級，經過三年級的臆測教學模式後，學生到了四年級時，透過繪製論證結構圖分析其論證特徵與品質，觀察兩年中的發展情形。
研究結果發現，到了四年級後，學生的論證特徵與論證品質皆有所成長。在證據元素方面，學生提出證據的頻率明顯增加，且以擬證據及真證據為主，內容大都能夠說明猜想與證據之間的因果關係。在論證過程中，部分猜想是遭到學生主動提出反駁的，這些反駁常伴隨著證據的發生，顯示學生已有能力提出有效的反駁，成功推翻或修正原猜想。當學生發現證據仍不足以作為猜想的支撐時，學生便提出支持理論做為更可靠的佐證，說服他人猜想進入一般化與證明一般化階段。而教師的介入主要是協助猜想的修正與完整，但當教師的介入次數減少時，學生反而能主動提出數學語言及引導語的使用。而隨著數學臆測教學的經歷越長，論證元素的組成越完整，在四年級的論證結構圖中，幾乎都能找到猜想、證據、支持理論、反駁、結論，甚至在「三角形」單元中出現反反駁。
因此本研究認為，長期數學臆測教學下，國小四年級學生論證發展是有顯著差異的。學生在論證特徵出現多樣性的元素，使結構更為完整，而在論證品質的得分則有遞增的趨勢，說明學生論證能力有所發展。

The trend of mathematical argumentation is rising internationally.Based on mathematical conjecturing teaching, this research used case study method to observe a fourth grade classroom, led by a teacher who joined the Ministry of Science and Technology's project, which has involved conjecturing teaching model in previous year. This research aims to find development of students' argumentation features and qualities through drawing argumentation structure.
The results suggested great improvement in students' argumentation features and qualities. In the aspect of warrant element, the frequancies of given warrants by student has obviously increased. The main warrant is pseudoevidence and genuine evidence, able to explain the relation of conjecture and warrant. In the process of argumentation, parts of conjectures were refuted actively by students. These refutations often contained warrants, which showed that students had the ability to refute effectively so as to reject or revise the conjectures. When students found warrants not enough to support conjectures, they proposed backing theory to support convincingly, and led conjectures forward to the stages of generalizing and justifying the generalization.The purpose of the interventions from teacher was to help revising and completing conjectures. When teacher tended not to intervene, students could be able to propose mathematical languagea actively. As the length of mathematical conjecturing teaching process extended, the argumentation elements were complete. Conjectures, warrants, backing theories , refutations and conclusions were found in nearly every argumentation structure, even the refutaion of a refutation eas found in triangle chapter.
This research suggested that the develoent of fourth graders' argumentation have obvious difference through long-term mathematical conjecturing teaching. Students showed various elements in argumentation features, which completed the argumentation structure. And the increasement of scores in argumentation qualities represented the development of students' argumentation ability.

參考文獻
中文部份
王文科、王智弘（2010）。教育研究法。臺北市：五南。
方廷榕（2011）。國中學生的解題策略與推理研究－以一個非例行性問題為例。未出版碩士論文。中原大學教育研究所。
李惠如（2010）。在以臆測活動為中心的教學情境下八年級學生臆測思維歷程的展現與其數學解題歷程之研究。未出版碩士論文。國立彰化師範大學科學教育研究所。
林志能（2011）。國小學童在不同學習情境中論證能力表現之研究。未出版碩士論文。國立高雄師範大學科學教育研究所。
林福來（2007）。青少年數學論證「學習與教學」理論之研究：總計畫(4/4)。行政院國家科學委員會專題研究計畫期末報告。(計畫編號：NSC94-2521-S-003-001)，未出版。
林碧珍（2013）。師資培育者幫助教師協助學生學習數學的教師專業發展研究。行政院國家科學委員會專題研究計畫期末報告。（計畫編號：NSC99-2511-S134-005-MY3），未出版。
林碧珍 (2015)。國小三年級課室以數學臆測活動引發學生論證初探。科學教育學刊，23(1)，83-110。
林碧珍、周欣怡（2013年12月）。國小學生臆測未知結果之論證結構:以四邊形沿一對角線剪開為例。「第29屆科學教育國際研討會」發表之論文，國立彰化師範大學。
林碧珍、馮博凱（2013年12月）。國小學生反駁錯誤命題的論證結構-以速率單元為例。「第29屆科學教育國際研討會」發表之論文，國立彰化師範大學。
林碧珍、鍾雅芳（2013）。六年級學生解決數字規律性問題的數學臆測思維。2013年第五屆科技與數學教育國際學術研討會暨數學教學工作坊論文集(pp.100-110)，6月8-9日。國立台中教育大學數學教育學系。
林碧珍、蔡文煥(2014)。數學教師與其師資培育者的專業發展：統整理論建構與實務應用-子計劃一：國小在職教師設計數學臆測活動的專業成長研究(3/3)。行政院科技部補助專題研究計畫(計畫編號：NSC 100-2511-S-134-006-MY3)。
周欣怡 (2015)。在數學臆測教學之下國小三年級論證發展之研究。未出版碩士論文，國立新竹教育大學數理教育研究所。
吳俊德（2007）。不同推理能力學生猜測與檢驗歷程之探討。未出版碩士論文，國立彰化師範大學科學教育研究所。
岳彥汝（2013）。探討高中二年級學生幾何論證的能力—以「圓形的判別」為例。未出版碩士論文。國立彰化師範大學數學研究所。
教育部（2008）。國民中小學九年一貫課程綱要數學學習領域。臺北：作者。
徐利治、王前（1989）。數學與思維。湖南：湖南教育出版社。
陳玉玫（2010）。國三學生幾何論證能力之探討。未出版碩士論文。國立彰化師範大學科學教育研究所。
陳佩卿（2008）。推理方式、形式、正確性對代數論證效力評估的影響：六年級、八年級及成人的比較。未出版碩士論文。國立臺北教育大學心理與諮商研究所。
陳英娥、林福來(1998)。數學臆測的思維模式。科學教育學刊，6(2)，191-218。
陳英娥（2002）。教室中的數學論證之研究。教育研究資訊，10(6)，111-132。
張少偉（2010）。實施以臆測為中心的教學對七年級個案學生數學論證能力影響之研究。未出版碩士論文。國立彰化師範大學科學教育研究所。
黃秋齡（2005）。國二生在論證表現之研究-以商高定理為例。未出版碩士論文。國立高雄師範大學數學研究所。
黃翎斐、張文華、林陳涌（2008）。不同佈題模式對學生論證表現的影響。科學教育學刊，16(4)，375-393。
黃瑞琴（1991）。質的教育研究方法。臺北市：心理。彭淑芬（2013）。探索國小六年級數學學習高成就學童之臆測思維--以比與比值問題為例。未出版碩士論文。國立新竹教育大學數理教育研究所。
馮博凱 (2014)。國小三年級學生論證之比較研究。國立新竹教育大學數理研究所。
廖曉澐（2008）。探討8年級學生在規律問題中形成與檢驗猜想的表現與。未出版碩士論文，國立彰化師範大學數學系所。
蔡忠翰（2011）。高一數理資優班與普通班學生在數列級數單元的解題中所展現的臆測思維與數學素養之比較研究。未出版碩士論文。國立彰化師範大學資賦優異研究所。
蔡俊彥、黃台朱、楊錦潭（2008）。國小學童網路論證能力及科學概念學習之研究。科學教育學刊，16(2)，171-192。
鍾雅芳（2012）。規律性問題下六年級學生臆測思維的探討。未出版碩士論文。國立新竹教育大學數理教育研究所。
蕭淑惠（2010）。實施以臆測為中心的教學探討四年級學生臆測思維與主動思考能力展現之行動研究。未出版碩士論文。國立彰化師範大學科學教育研究所。

西文部份

Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics.In D. Pimm (Ed), Mathematics, teachers and children, (pp.216-235). Stoughton: Hodder education.
Boero, P. (1999). Argumentation and mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education.International Newsletter on the Teaching and Learning of Mathematical Proof,7(8).
Boero, P., Douek, N., & Grauti, R. (2003), Children’s conceptions of infinity of numbers in a fifth grade classroom discussion context, In. N. Pateman, B. Dougherty, & J. Zilliox (Eds.) , Proceedings of the 27th annual conference of the international group for the psychology of mathematics Education (Vol. 2), 121-128, Honolulu, HI: University of Hawaii.
Cañadas, M. C., Deulofeu, J., Figueiras, L., Reid, D., & Yevdokimov, A. (2007). The conjecturing process: Perspectives in theory and implications in practice. Journal of Teaching and Learning, 5(1), 55–72.
Cantlon, D. (1998). Kid+conjecture=mathematics power. Teaching Children Mathematics, 5(2), 108-112.
Davis, P. J., Hersh, R., & Marchisotto, E. A. (1995). The mathematical experience.
Boston：Birkhauser.
Douek, N. (1999). Argumentative aspects of proving: Analysis of some undergraduate mathematics students’ performances. In O. Zaslavsky (Eds.), Proceedings of the 23nd conference of the international group for the psychology of mathematics education(Vol.2,pp. 273-288). Haifa,Israel: PME.
Douek, N., Scali, E., (2000).About argumentation and conceptualisation, Proceedings of the 24th conference of the international group for the psychology of mathematics education(Vol. 2, pp. 249-256). Hiroshima, Japan: PME.
Driver, R., Newton, P., & Osborne, J. (2000).Establishing the norms of scientific argumentation in classroom.Science Education, 84(3), 287-312.
Erduran, S., Simon, S., & Osborne, J. (2004). TAPing into argumentation: Developments in the application of Toulmin’s argument pattern for studying science discourse.Science Education, 88(6), 915-933.
Gold, R. L. (1958).Roles in sociological field observations.Social Forces, 36(3), 217-223.
Hanna, G. (1989). More than formal proof.For the Learning of Mathematics, 9(1), 20-25.
Knipping, C.(2003). Argumentation structures in classroom proving situations. In M. A. Mariotti (Ed.), Proceedings of the third conference of the European society in mathematics education (unpaginated). Retrieved from http://ermeweb.free.fr/CERME3/Groups/TG4/TG4_Knipping_cerme3.pdf
Knipping, C. (2004). Argumentations in proving discourses in mathematics classrooms.In G. Törner, et al. (Eds.), Developments in mathematics education in German-speaking countries.Selected papers from the annual conference on didactics of mathematics, Ludwigsburg, March 5–9, 2001 (pp. 73–84). Hildesheim: Franzbecker Verlag.
Knipping, C.(2008). A method for revealing structures of argumentations in classroom proving processes.ZDM The international journal on mathematics education, 40(3), 427-447.doi: 10.1007/s11858-008-0095-y
Knipping, C., & Reid, D. A., (2013). Revealing structures of argumentations in classroom proving processes. In A. Andrew & I. J. Dove (Eds.), The Argument of mathematics (pp.119-146). Dordrecht: Springer.
Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229-269). Hillsdale, NJ: Lawrence Erlbaum.
Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions. Journal of Mathematical Behavior, 26(1), 60-82.
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. New York: Cambridge University Press.
Mason, J., Burton, L., & Stacey, K. (1985).Thinking mathematically (Rev. ed.). New York: Addison Wesley.
National Council of Teachers of Mathematics(NCTM).Principles and standards for school mathematics. Reston, VA: NCTM, 2000.
National Research Council (1989).Everybody counts. Washington, DC: National Academy Press.
OECD.(2013). Pisa 2015 draft collaborative problem solving framework. Retrieved Apr 28, 2014, from
http://www.oecd.org/pisa/pisaproducts/Draft%20PISA%202015%20Collaborative%20Problem%20Solving%20Framework%20.pdf
Polya, G. (1954). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press.
Reid, D.A. (2002). Conjectures and refutations in grade 5 mathematics.Journal for Research in Mathematics Education, 33(1), 5–29.
Reid, D. A., & Knipping, C. (2010).Argumentation structures.In D. A. Reid & C. Knipping (Eds.), Proof in mathematics education: Research, learning and teaching (pp.179-192). Rotterdam: Sense Publishers.
Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of mathematical behavior, 15(1), 3-31.
Stavy, R., Babai, R., Tsamir, P., Tirosh, D., Lin, F.-L., & McRobbie, C. (2006). Are intuitive rules
universal? International Journal of Science and Mathematics Education, 4(3), 417-436.
Stylianides, A. J. (2007). Proof and proving in school mathematics.Journal for Research in Mathematics Education, 38(3), 289–321.
Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education 11:307-332.
Toulmin, S. E. (1958). The uses of argument. Cambridge: Cambridge University Press.
U. K. Department for Education. (2013). Mathematics programmes of study: key stages 1 and 2. Retrieved May 1, 2014, from
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/239129/PRIMARY_national_curriculum_-_Mathematics.pdf
Yackel, E. (2001). Explanation, justification, and argumentation in mathematics classrooms. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference for the international group of psychology of mathematics education, 1, 9–24, Utrecht, the Netherlands.
Yevdokimov, O. (2005). About a constructivist approach for stimulating students’ thinking to produce conjectures and their proving in active learning of geometry. In M. Bosch (Ed.), Proceedings of the CERME 4 international conference (469-480).Sant Feliu de Guíxols, Spain. Published online at http://ermeweb.free.fr/CERME4/
Yin, R. K. (1981).The case study as a serious research strategy.Science communication, 3(1), 97-114.
Yin, R. K. (2003).Case study research: Design and methods(3rdEdition). California: Sage Publications.