本研究目的為探討在數學臆測教學下引出學生論證元素的教師提問。本研究採個案研究法，研究對象為1位進行多年臆測教學的教師與25位歷經一年臆測教學的四年級學生。本研究資料蒐集的方式是教學觀察影帶及學生的猜想單。資料分析的焦點是教師提問與學生論證，先對師生對話進行編碼，再畫出提問圖，編碼的過程會利用影片與錄音進行來回比對以求信度。
研究結果顯示在數學臆測教學的階段中有以下發現：
（一） 在效化階段中，每個討論主題都能引出資料、證據、支持和修飾，而且經過一連串討論後再回到討論主題能幫助學生思考，還有當猜想寫法不同時修改成相同寫法能幫助學生理解。
（二） 在一般化階段中，因為經過效化階段充分的討論，能使學生容易達成共識，而且教師利用建構結果的提問更容易引出學生的論證。
（三） 在證實階段中，連續的教師提問更能引出學生的論證，而且在討論證實的順序時教師和學生都不會以資料說明想法，在討論證實的策略時教師利用描述做法來徵求學生的策略且用演示做法來說明該策略，在討論證實的做法時教師會使用確認、推測和理由引出資料、證據和支持來說明該做法。
研究結果顯示在數學臆測教學的論證元素中有以下發現：
（一） 大部分的資料能使用單一提問引出，其中確認是最容易引出資料的教師提問，當學生越回答不出來時可以引出資料來說明。
（二） 主張能被所有的教師提問引出，但在討論證實的順序時不容易引出主張，而在合併想法時可以使用以前的結論、術語與建構結果來引出主張。
（三） 證據能被所有的教師提問引出，在效化階段時用單一提問就能引出證據，而當證據不明顯時需要問尋求想法來引出證據，當教師用追問的方式提問更容易引出證據。
（四） 反駁容易被確認、推測和建構結果引出，而且反駁後會伴隨著證據與修飾。
（五） 支持容易被教師提問引出尤其是確認和理由，在討論證實的做法時可以用尋求真實答案、尋求想法與尋求說明引出支持。
（六） 修飾容易被確認、術語與描述做法引出，在證實階段時可以利用推測或建構結果來引出修飾。
（七） 反反駁必然在反駁剛出現就被引出，教師可以用推測來促使反反駁被引出。

The purpose of this study was to explore how teacher’s question to elicit students’ argumentation in mathematically conjecturing teaching. A case study method was adpted. The subject of the research was an experienced teacher who has been teaching conjecturing teaching for 5 years amd 25 fourth-grade students with one year engaging in conjecturing. The main data collection of this study was observation teaching tapes and students’ conjecture. The focus of data analysis was teacher’s questioning and students’ argumentation, and then decodeing, and then drawing the questioning diagrams. When I code, I used to compare the video and the recoding for reliability.
The research result shows the following finding in mathematically conjecturing teaching:
(1) At the stage of validation, every topic can elicit Data, Warrant, Backing and Qualifier; after a series of discussions, going back to discuss the original conjecture can help students understand; when conjecture different ways of writing, you can change the same way of writing to help other students understand.
(2) At the stage of generalization, student easily reach consensus, because they have fully discussed; and teacher ask Constuct Reasult easily elicit student’s argumentation.
(3) At the stage of justification, teacher’s continuous questioning is more able to elicit students’ argumentation; when they discuss to arrange the order of justification, teacher and students are not use Data to elaborate the idea; when they discuss the strategy of justification, teacher use Describe a Method to ask students’ strategy, and use Demonstrate a Method to elaborate the strategy; when they discuss the method of justification, teacher use Identification, Conjecture and Justification elicit Data, Warrant and Backing to elaborate the method.
The research result shows the following finding in the argumentaion of mathematically conjecturing teaching:
(1) If you want to elicit Data, you just need use one questioning; Identification is the easiest questioning to elicit Data; when students haven’t response, teacher can try to elicit Data to elaborate the idea.
(2) If you want to elicit Claim, you can use all teacher’s questioning; when you discuss the order of jusificaion, it’s not easily to elicit Claim; when you combine the conjecture, you can use Previous Result, Term and Construct Result to elicit Claim.
(3) If you want to elicit Warrant, you can use all teacher’s questioning; you can use one questioning to elicit Warrant in the stage of validation; when Warrant is not trivial, you can use Requesting an Idea to elicit Warrant; when teacher continue to ask, it’s easily to elicit Warrant.
(4) If you want to elicit Rebuttal, you can use Identificaion, Conjecture and Construct Result; Rebuttal will be accompanied by Warrant and Qualifier.
(5) If you want to elicit Backing, you can use Identification and Justification; when you discuss the method of justification, you can use Requesting a Factual Answer, Requesting an Idea and Requestin a Elaboration.
(6) If you want to elicit Qualifier, you can use Identification, Term and Describe a method; you can use Conjecture and Construct Result to elicite Qualifier in the stage of justification.
(7) If you want to elicit Anti-rebuttal, you can use Conjecture; Anti-rebuttal must apper after Reburral and be elicited.

第一章、 緒論 1
第一節、 研究背景與動機 1
第二節、 研究目的與待答問題 4
第三節、 名詞釋義 4
第四節、 研究限制 5
第二章、 文獻探討 6
第一節、 數學臆測教學 6
第二節、 數學論證 10
第三節、 教師提問 13
第三章、 研究方法 22
第一節、 個案研究法 22
第二節、 研究架構與研究流程 22
第三節、 研究對象 26
第四節、 資料蒐集與分析 27
第四章、 研究結果 32
第一節、 數學臆測教學中各個階段引出的論證元素與教師提問 32
第二節、 數學臆測教學中引出特定論證元素的教師提問 51
第五章、 結論與建議 70
第一節、 研究結果與討論 70
第二節、 未來研究與建議 73
參考文獻 75

王鵬豪(2020)。高低年級數學臆測教學的規範之比較〔未出版碩士論文〕。國立清華大學數理教育研究所。
石糧豪(2019) 。學生形成論證的教師支持性行為：新竹與浙江數學課堂比較〔未出版碩士論文〕。國立清華大學數理教育研究所。
余姿縈(2018) 。初任教師建立數學臆測規範之行動研究：以高年級為例〔未出版碩士論文〕。國立清華大學數理教育研究所。
吳長恩(2019) 。數學臆測教學下教師提問類型之個案研究〔未出版碩士論文〕。國立清華大學數理教育研究所。
吳長恩(2019) 。數學臆測教學下教師提問類型之個案研究〔未出版碩士論文〕。國立清華大學數理教育研究所。
张万民、石南平（2004) 。简析课堂提问。中国林业教育，62-63。
李宜芬(2006) 。問題式教材之分類架構及其應用〔未出版碩士論文〕。國立臺灣師範大學資訊教育系。
良米(2000) 。善用提問，快樂學習。師友月刊，395，83-84。
周欣怡(2015) 。數學臆測教學課室中國小三年級學生論證結構發展之研究〔未出版碩士論文〕。國立新竹教育大學數理教育研究所。
林勇吉、秦爾聰、段曉林(2014) 。數學探究之意義初探與教學設計實例。臺灣數學教師，35(2)，1-18。
林崇安(2006) 。波普的否證論。國立中央大學太空科學與工程學研究所。http://www.ss.ncu.edu.tw/~calin/science/D2-1.pdf
林碧珍(2015) 。國小三年級課室以數學臆測活動引出學生論證初探。科學教育學刊， 23(1)，83-110。
林碧珍、周欣怡(2013) 。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。發表於第29屆科學教育國際研討會。彰化市：國立彰化師大科學教育研究所。
林碧珍主編(2021)。素養導向數學臆測教學之理論與實務。台北市：師大書苑。
林燕文、洪振方 (2007)。對話論證的探究對促進學童科學概念理解之探討。花蓮教育大學學報，24，139-177。
俞慧霞(2013) 。50招提升孩子的表達力。台灣：品果出版社。
洪神佑(2016) 。在數學臆測教學下一組國小六年級學生論證結構發展之研究〔未出版碩士論文〕。國立新竹教育大學數理教育研究所。
洪誌陽(1999) 。Polya啟發法的哲學面向。數學傳播，23(3)。
洪蘭(2020) 。什麼才是人生最值得的事。台北：天下文化。
秦爾聰、劉致演、張克旭、段曉林(2015) 。數學臆測探究教學對商職學生數學學習成就與動機之影響。數學教育學刊，2(2)，53-83。
康淑娟(2008) 。國小教師提問教學決定之探究〔未出版碩士論文〕。國立嘉義大學數理教育研究所。
康淑娟(2008) 。國小教師提問教學決定之探究〔未出版碩士論文〕。國立嘉義大學數理教育研究所。
康淑娟、劉祥通(2010) 。數學提問教學之探究與應用。科學教育月刊， 333， 2-18。
張玉成(2005) 。發問技巧與學生創造力之增進。教育資料集刊，30，181-200。
張春興(1975) 。小學自然科啟發式教學對兒童推理思考發展的影響。教育心理學報，11-24。
陳英娥(2002)。教師中的數學論證之研究。教育研究資訊，10(6)，111-132。
陳英娥、林福來(1998) 。數學臆測思維模式。科學教育學刊，6(2)，191-218。
陳淑娟(2008) 。國小教師發展數學提問能力之合作行動研究〔未出版博士論文〕國立台南大學教育經營與管理研究所。
陳淑娟、劉祥通(2002) 。國小班級數學討論活動可行方案之探討。科學教育學刊，10(1)，87-107。
黃政傑(2019) 。教師應具備課程素養。台灣教育評論月刊，8(12)，1-5。
黃昭勳(2019) 。教師發問行為之探究。台灣教育評論月刊，8(1)，281-291。
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.
楊育菁(2018) 。高中國文課堂提問教學研究〔未出版碩士論文〕。國立高雄師範大學國文學系。
葉重新(2008) 。教育研究法。台北：心理出版社。
趙曼寧、王明泉、程鈺雄(2008) 。國民小學特教教師教學熱忱之研究。東台灣特殊教育學報，10，121-149。
劉俊明(2015) 。台灣高中數學教育對經濟發展的蝴蝶效應。發布於政策想想。https://www.thinkingtaiwan.com/content/3569
劉致演、秦爾聰(2018) 。探討兩位國中教師發展數學臆測探究教學實務。師資培育與教師專業發展期刊，11(1)，57-89。
潘瑶珍（2010）。基于论证的科学教育。全球教育展望，6(5)。
鄭麗慧(2008) 。有效的教學，來自有效的教師。國教之友，59(4)，62-66。
繆佳燕(2018) 。在數學臆測教學下不同教師引出學生數學論證的介入之比較〔未出版碩士論文〕。國立清華大學數理教育研究所。
簡妙娟(2017) 。Paulo Freire 哲思與教學的反思與實踐。台灣教育評論月刊，6(11)，96-106。
釋自衍(2017) 。以「四層次提問法」開啟希望閱讀。全國新書資訊月刊，226，9-12。
Albergaria-Almeida, P. (2010). Classroom questioning Teachers' perceptions and practices. Procedia - Social and Behavioral Sciences, 2(2), 305-309.
Arslan, M. (2006). The role of questioning in the classroom. Hasan Ali Yücel Eğitim Fakültesi Dergisi, 81-103.
Astrid, A., Amrina, R. D., Desvitasari, D., Fitriani, U., & Shahab, A. (2019).The Power of Questioning Teacher’s Questioning Strategies in the EFL Classrooms. Indonesian Research Journal in Education, 3(1), 91-106.
Balacheff, N. (1988). A study of students' proving processes at the junior high school level. In Second UCSMP international conference on mathematics education. NCTM.
Boaler, J., & Brodie, K. (2004). The importance, nature and impact of teacher questions. In D. E. McDougall, & J. A. Ross (Eds.), Proceedings of the 26th Conference of the Psychology of Mathematics Education (North America, pp. 773-781). Toronto: OISE/UT.Boyd (2016) Relations Between Teacher Questioning and Student Talk in One Elementary ELL Classroom
Cañadas, M. C., Deulofeu, J., Figueiras, L., Reid, D., & Yevdokimov, O. (2007). The conjecturing Process perspectives in theory and implication in practice. Journal of Teaching and Learning, 5(1), 55-72.
Conner, A. M., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Identifying kinds of reasoning in collective argumentation. Mathematical Thinking and Learning, 16(3), 181-200.
Conner, A. M., Singletary, L.M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Teacher support for collective argumentation: A framework for examining how teachers support students’ engagement in mathematical activities. Educational Studies in Mathematics, 86, 401-429.
Correia, C., Nieminen, P., Serret, N., Hähkiöniemi, M., Vilri, J., & Harrison, C. (2016). Informal fonative assessment in inquiry-based science lessons. ESERA 2015.
Davis, D., & Taylor-Vaisey, A. (1997). Two decades of Dixon: The question(s) of evaluating continuing education in health professions. Journal of Continuing Education in the Health Professions, 17(4), 207-213.
Donald, C. O. (1988). Teaching strategies: A guide to better instruction. New York: Houghton Mifflin.
Ericsson, K. A., & Hastie, R. (1994). Contemporary approaches to the study of thinking and problem solving. Handbook of Perception and Cognition, 2, 37-79.
Franke, M. L., Webb, N. M., Chan, A. G., Ing, M., Freund, D., & Betty, D. (2009). Teacher questioning to elicit students’ mathematical thinking in elementary school classrooms. Journal of Teacher Education, 60(4), 380-392.
Gallagher, J. J., & Aschner, M. J. (1963). A preliminary report on analyses of classroom interaction. Merrill-Palmer Quarterly of Behavior and Development, 9(3), 183-194.
Graesser, A. C., & Person, N. K. (1994). Question asking during tutoring. American Educational Research Journal, 31(1), 104-137.
Hähkiöniemi, M. (2013). Probing student explanation. Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, 2, 401-408. Kiel, Germany: PME.
Hähkiöniemi, M. (2016). Student teachers’ questioning behaviour which elicit conceptual explanation from students. Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education, 2, 337-344. Szeged, Hungary: PME.
Harel, G. (2008). A DNR perspective on mathematics curriculum and instruction. Part II: with reference to teacher’s knowledge base. ZDM(40), 893-907.
Hsieh, F. J., Lin, P. J., & Wang, T. Y. (2011). Mathematics-related teaching competence of Taiwanese primary future teachers: Evidence from the TED-M. ZDM The international journal of mathematics education., 44(3), 277-292.
Inam, A., Nomaan, S., & Abiodullah, M. (2016). Parents’ parenting styles and academic achievement of underachievers and high achievers at middle school level. Bulletin of Education and Research, 38(1), 57-74.
Kawanaka, T., & Stigler, J. W. (1999). Teachers' use of questions in eighth-grade mathematics classrooms in Germany, Japan, and the United States. Mathematical Thinking and Learning, 1(4), 255-278.
Kazemi, E., & Stipek, D. (2001). Promoting conceptual thinking in four upper-elementary mathematics classrooms. The Elementary School Journal, 102(1), 59-80.
Ketonen, L., Hähkiöniemi, M., Nieminen, P., & Viiri, J. (2019).Pathways through peer assessment implementing peer assessment in a lower secondary physics classroom. Internaltional Journal of Science and Mathematics Education, 18, 1465-1484.
King, A. (1994). Guiding knoledge construction in the classroom: effects of teaching children how to question and how to explain. American Educational Researcb Journal, 31(2), 338-368.
Kuhn, D. (1991). The skill of argument. Cambridge: Cambridge university press.
Kyeong-Hwa, L., & Bharath, S. (2011). Conjecturing via reconceived classical analogy. Educational Studies in Mathematics, 76, 123-140.
Lakatos, I. (1976). A renaissance of empircism in the recent philosophy of mathematics. The British Journal for the Philosophy of Science, 27(3), 201-223.
Larson, C. E., & Cleemput, N. V. (2017). Automated conjecturing III Property-ralation conjectures.
Lehesvuori, S., Hähkiöniemi, M., Jokiranta, K., Nieminen, P., Hiltunen, J., & Viiri, J. (2017). Enhancing dialogic argumentation in mathematics and science. Studia paedagogica, 22(4), 55-76.
Lehesvuori, S., Viiri, J., Rasku-Puttonen, H., Moate, J., & Helaakoski, J.(2013).Visualizing communication structures in science classroom: Tracing cumulativity in teacher-led whole class discussions. Journal of Research in Science Teaching, 50(8), 912-939.
Lehtinen, A., & Hähkiöniemi, M. (2016). Complementing the guidance provided by a simulation through teacher questioning. Conference: Annual Symposium of the Finnish Mathematics and Science Education Research Association 2015, At: Turku, Finland, 80-89.
Lin, F. L. (2006). Designing mathematics conjecturing activities to foster thinking and constructing actively. APEC-TSUKUBA International Conference, 5-73. Tokyo & Sapporo, Japan.
Lin, F. L. (2015). Designing mathematics conjecturing activities to foster thinking and constructin actively. APEC-TSUKUBA International Conference, Dec. 2~7, 2006, Tokyo & Sapporo, Japan.
Lin, P. J. & Tsai, W. H. (2013).Enhancing pre-service teachers' knowledge of students' errors by using researched-based cases. Proceedings of the 37h Conference of the International Group for the Psychology of Mathematics Education, 3, 273-280. July 28-August 2. Kiel University, Germany.
Lin, P. J. & Tsai, W. H. (2016). Enhancing students' mathematical conjecturing and justification in third-grade classrooms. Journal of Mathematics Education, 9(1), 1-15.
Lin, P. J. (2006). Conceptualizing teachers' understanding of students' mathematical learning by using assessment task. International Journal of Mathematics and Science Education, 4(3), 545-580.
Lin, P. J. (2017). Enhancing preservice teachers' knowledge fo students' errors on fraction by using research-based case. Mathematical thinking and learning, 10(1), 76-91.
Lin, P. J. (2017). Fostering novice teachers' knowledge of students' errors on fractions division by using researched based cases. Journal of Mathematics Education, 10(1), 76-91.
Lin, P. J. (2018). Improving knowledge for teaching mathematical argumentation in primary classroom. Journal of Mathematics Education, 11(1), 17-30.
Lin, P. J. (2018). The development of students’ mathematical argumentation in a primary classroom. Educação & Realidade (Education &Reality), 43(3), 1171-1192.
Lin, P. J. (2018).The development of students' mathematical argumentation in a primary classroom. Educação & Realidade (Education &Reality). 43(3), 1171-1192.
Lin, P. J., & Acosta-Tello, E. (2017).A practicum mentoring model for supporting prospective elementary teachers in learning to teach mathematics. ZDM: the international journal on mathematics education, 49(5), 223-236. DOI: 10.1007/s11858-016-0829-1. (SSCI)
Lin, P. J., & Chang-Liao, P. Y. (2019). Teacher’s Multiple Roles For Students Engaging Mathematically Conjecturing In Elementary Classrooms. Proceeding of culte international conference, (pp. 257-1-257-2). New York.
Lin, P. J., & Horng, S. Y. (2017). The Conjecturing Contributing to the Group Argumentation in Primary Classrooms. Paper presented at the 9th Classroom Teaching Research for All Students Conference. China.
Lin, P. J., & Li, Y. (2009). Searching For Good Mathematics In struction at primary school level valued in Taiwan. ZDM-The international journal of mathematics education, 41(3), 363-378.
Maria C. Cañadas, Jordi Deulofeu, Lourdes Figueiras, David Reid & Oleksiy Yevdokimov. (2007). The conjecturing process: perspectives in theory and implications in practice. Journal of teaching and learning, 5(1), 頁 18.
Mason, J., Burton, L., & Stacey, K. (1985). Thinking mathematically. England: Pearson.
Mata-Pereira, J., & Ponte, J. P. (2018). Teacher’s actions to promote students’ justifications. Acta Scientiae, 20(3).
Morselli, F. (2006). Use of examples in conjecturing and proving: an exploratory study. Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 4, 185-192. Prague: PME.
Rahayu, D. S., Hendayana, S., Mudzakir, A., & Rahmawan, S. (2019).Types and the role of teacher’s questions in science classroom practice. Journal of Physics: Conference, 1157(2).
Ribeiro, L., Rosário, P., Moreira, I., & Cunha, R. S. (2019). First-year law student’s and teacher’s questioning in class. Brief Researcj Report, published: 21 March.
Rozalia, Johar, R., Duskri, M., Khairunnisak, C., & Zubaidah, T. (2020).Teacher’s questions on mathematics learning based on ELPSA framework. Journal of Physics Conference Series, 1470, 25-27.
Osborne, J., Erduran, S., & Simon, S.（2004）. Enhancing the quality of argumentation in school science. Journal of Research in Science Teaching, 41（10）, 994-1020
Passmore, C., & Stewart, J. (2002). A modeling approach to teaching evolutionary biology in high schools. Journal of Research in Science Teaching: The Official Journal of the National Association for Research in Science Teaching, 39(3), 185-204.
Sahin, A., & Kulm, G. (2008). Sixth Grade Mathematics Teachers’ intentions and use of probing, guiding and factual questions. Journal of Mathematics Teacher Education, 11, 221-241.
Shomoossi, N. (2004). The effect of teachers' questioning behavior on EFL classroom interaction a classroom research study. The Reading Matrix, 4(2), 96-104.
Simpson, A., Mercer, N., & Yolanda, J. M. (2015). Editorial: Douglas barnes revisited: If learning floats on a sea of talk, what kind of talk？ And what kid of Learning？English Teaching, 9(2), 1-6.
Singletary, L. M. & Conner, A. M. (2015). Focusing on mathematical arguments.
Trillas, Mas, Monserrat & Torrens (2009) Conjecturing from consequences. Mathematixs Teacher, 109(2), 143-147.
Trillasa, E., Masb, M., Monserratb, M., & Torrensb, J. (2008). Conjecturing from consequences. International Journal of General Systems, 38(5), 567-578.
Vogler, K. E. (2005). Improve your verbal questioning. The Clearing House, 79(2), 98-103.
Wagner, P. A., Smith, R. C., Conner, A. M., Sibgletary, L. M., & Francisco, R. T. (2013). Using Toulmin’s model to develop prospective secondary mathematics teachers’ conceptions of collective argumentation. Mathematics Teacher Educator, 3(1), 8-26.
Watts, M., Alsop, S., Gould, G., & Walsh, A. (1997). Prompting teachers’ constructive reflection: pupils’ questions as critical incidents. International Journal of Science Education, 19(9), 1025-1037.
Yang, N. (2019). On the study of teacher’s question types and students’ answers in primary school English teaching. Journal of Educational Theory and Management, 3(1), 10-16.
Dantonio, M., & Beisenherz, P. C. (2001). Learning to question, questioning to learn: Developing effective teacher questioning practices.
Mason, J. (2000). Asking mathematical questions mathematically. International Journal of Mathematic Educational in Science and Technology, 31(1), 97-111.
Balacheff, N. (1988). A study of students' proving processes at the junior high school level. In Second UCSMP international conference on mathematics education. NCTM.
Toulmin, S. E.（1958）. The uses of argument. Cambridge: Cambridge University Press.