為探討國小六年級學生在對於「未知結果問題」及「辨別真偽」兩種題目上的表現及思維歷程，本研究將同一種題目設計為兩種臆測命題讓國小六位學生施測，不同成就學生各兩位，施測完後，為了可以更加清楚學生的猜想過程，會再透過20-30分鐘的半結構訪談，最後繪製出不同成就學生的臆測思維歷程圖。
研究結果發現未知結果臆測思維歷程在觀察、猜測、檢驗、相信中循環，高成就學生一旦提出猜測，會因為對於自己所提出的猜想很有信心，而不再加以驗證與修正；中成就學生會因為缺乏自信，出現較多檢驗的情形，因此增加解題成功的機會；低成就學生臆測思維歷程大都停留在觀察。辨別真偽題目，臆測思維歷程自檢驗開始，因為題目已經提出猜想，學生只需要加以檢驗即可，但中、低成就學生會利用觀察取代檢驗，觀察後就大膽的斷定題目為偽，低成就學生可能可以發現題目為偽，但是無法修正偽題目為正確的題目。
另外，本研究結果發現未知結果的臆測思維歷程不一定比辨別真偽複雜，如果是合理性規律，辨別真偽就比未知結果複雜，所以臆測思維歷程與學生的程度以及任務的類型都有相關。
最後，不同成就學生以不同方式形成猜測，學生形成猜想的方式有特殊化、直覺性觀察、擴展、錯誤的類比……等。與學生的訪談中發現，學生在形成猜測的時候最常使用的方法是特殊化與擴展。低成就學生大都以擴展、直覺性觀察的方式形成猜測，中、高成就的學生大都以特殊化為猜測方式。檢驗猜想的方式有帶入數字檢驗、Reid(2010)所提檢驗方式中的種類(或稱類型)(Kinds or types)、舉反例、直覺性觀察、Balacheff（1988）所提出檢驗方式中的思考的實驗(Crucial experiment)……等，其中最常使用的方法就是「帶數字進入算式」檢驗的方式和舉反例兩種方式。

To explore how sixth-graders proceed "guessing an unknown conclusion" and "judging the correctness of a proposition", this study was conducted by designing and employing two types of conjecturing questions based on original mathematical question. Six students with different mathematic performance were purposely selected in this study. In addition to observing how students guessed unknown questions, and how they judged the correctness of proposition, 20-30 minute semi-structural interviews were applied, too. With all these efforts, the models of conjecturing thinking process among students with different academically capacity were created.
The results show that the repeating conjecturing thinking process for the result-unknown questions is comprised of four steps: observation, conjecture, validation and believing. It was found that high-achieving students tended to skip validating and modifying their guesses due to optimistically confidence. However, the medium-achieving students had more occurrence of validation due to lack of confidence. The low-achieving students were found to rely on observation a lot during the entire process. To the second type of question, judging the correctness of a proposition, validating was found as a starting point to initiate the conjecture process. Among this process, the strategy of observation was usually omitted by students because the proposition was embedded in the question itself already. This study found that mid-achieving and low-achieving students would incline to use observation skills rather than validation, and would often claimed the correctness of the proposition confidently; however low-achieving students were possibly just aware of the false proposition, but failed in correcting the false proposition.
In addition, this study found that the “guessing an unknown conclusion” is not always complicated than “"judging the correctness of a proposition” for students. In fact, “judging the correctness of a proposition” is more complicated for students when they encountered with the pattern questions. The associations between conjecturing thinking process, student’s academic achievement, and types of tasks were also confirmed in this study.
The formations of conjecture are varied for different achieving students, such as specialization, intuitively observation, expansion, and false analogy. What can be seen through interviews is that specialization and expansion are the most frequent ways that students would like to use. Low-achieving students were more likely to form their conjectures with intuitively observation, expansion; while the technique of specialization was widely used by high-achieving students.

中文文獻
黃政傑(1987)。課程評鑑。台北:五南。
徐利治、王前（1989）。數學與思維。湖南：湖南教育出版社。
史都華(1996)。 大自然的數學遊戲。葉李華（1996）。台北：天下文化。(原著出版年﹕1995 年)
數學思考（台北市建國高級中學49屆314班全體同學合譯）(2011)。台北：九章。(原
著出版年：1998年)。
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(1998)。數學臆測：思維與能力的研究。未出版博士論文，國立台灣師範大學
科學教育研究所。
劉祥通、周立勳(1999)。國小比例問題教學實踐課程之開發研究。台中師院數理學報，
3(1)。
黃敏晃(2000)。規律的尋求。台北：心理。
教育部(2008)。國民中小學九年一貫課程綱要。台北：教育部。
曹亮吉(2003)。阿草的數學聖杯。台北：天下文化。
吳昭容(2003)。國中小學生平面幾何圖形的概念結構與概念調整歷程之探討。
國科會專題研究計畫成果報告（報告編號：NSC 92-2521-S-152-004）。未出版。
洪明賢(2003)。國中生察覺數形規律的現象初探。國立台灣師範大學數學系教學碩士
班碩士論文，未出版，台北市。
潘淑滿(2003)。質性研究:理論與應用。台北：心理。
呂玉琴、陳瑞發(2003)。直觀規律對國小代課教師數學解題的影響。科學教育研究
與發展，34，66-87。
李佩玟(2005)。國小六年級學童發現數列樣式的推理歷程之分析研究。未出版碩士論
文，國立台北教育大學教育與諮商學系。
陳亮君(2006)。國中小學生數學胚騰覺察能力發展概況之探討。未出版碩士論文，國
立台南教育大學測驗統計研究所。
林福來（2006）。青少年數學論證「學習與教學」理論之研究-總計畫(4/4)。國科會
專題研究計畫成果報告（報告編號：NSC 95-2521-S-003-001）。未出版。
林妙齡(2006)。國小六年級學童平面幾何臆測產生之探究。未出版論文，國立台北
教育大學數學教育研究所。
翁慈青(2007) 。國小五年級學童在數形規律問題解題表現之個案研究。未出版碩士
論文，國立嘉義大學數學教育研究所。
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。
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歷程。未出版
碩士論文，國立彰化師範大學數學系所。
許建銘(2008)。數學好好玩。台北：書泉。
質性研究與評鑑(吳芝儀、李泰儒合譯)(2008)。嘉義：濤石文化。(原著出版年：2002
年)。
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。
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歷程與主動思考能
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。
鄭英豪 (2010)。數學臆測活動的設計、教學與評量－子計畫五：數學臆測與證明學習
連續性的教與學研究。國科會專題研究計畫成果報告（報告編號：NSC 96-2521-S -133 -001-MY3）。未出版。
林碧珍(2011)。數學教學案例-代數篇。台北:師大書苑。

西文文獻
Balacheff, N.(1988).Aspects of proof in pupils’practice of school
mathematics.In D. Pimm(Ed.)Mathematics,teachers and children(pp.
216-235).London: Hodder and Stoughton.
Barkai,R., Tsamir,P., Tirosh,D., ＆ Dreyfus,T.（2002）.Proving or refuting
arithmetic claims:The case of elementary school teachers.In A Cockburn ＆
E.Nardi.(Eds.).Proceedings of the twenty-sixth annual conference of the
International Group for the Psychology of Mathematics
Education(Vol.2,pp.57-64).Norwich,UK.
Bergqvist, T.(2005).How Students Verify Conjectures：
Teachers’Expectations.Journal of Mathematics Teacher Education.8,
171-191.
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.
Chazan,D.(1993).High school geometry students’justification for their views
of empirical evidence and mathematical proof.Educational Studies in
Mathematics.24(4),359-387.
Davis, P.J. and Hersh, R. (1995).The Mathematical Experience
Birkhauser,Boston,Basel,Berlin.
Dural,R. (1990).Pour une approche cognitive de l’argumentation. Annales de
Didactique et Sciences Cognitives.3,195-221.
Healy,L., ＆ Hoyles,C.(2000).A study of proof conceptions in
alegebra.Journal for Research in Mathematics Education,33,176-203.
Furinghetti,F.,Olivero F.,＆Paola,D.(2001).Students approaching proof
through conjectures:snapshots in a classroom.International Journal of
Mathematics Education in Science and Teachnology,32,319-335.
Liu, S. T., & Ho, F. C.（2008）.Conjecture activities for comprehending
statistics terms through Speculations on the functions of imaginary
spectrometers.The Australian Mathematics Teacher,64(3), 17-24.（NSC
95-2521-S-415-004-MY3）。
Lakatos,I.(1976).Proofs and Refutations:The Logic of Mathematical Discovery.
New York:Cambridge University Press。
Lakatos, I. (1987) 證明與反駁(康宏逵譯)。上海市：上海譯文。(原著出版於1976年)
Lin, F. L.(2006). Designing mathematics conjecturing activities to foster
thinking and constructiong actively. Paper presented at the meeting of the
APEC-TSUKUBA International Conference, Tokyo, Japan.
Mason, J., Burton, L., & Stacey, K. (1985). Thinking mathematically. London:
Addison-Wesley Publishing Company.
National Council of Teachers of Mathematics (2002). The principles and
standards for school mathematics.Reston,VA:NCTM.
Owen, A. (1995). In search of the unknown: A review of primary algebra.. In
J. Anghileri (Ed. ), Children’s mathematical thinking in the primary years:
Perspectives on children’s learning. London: Cassell.
Reid, D.A.(2002).Conjectures and refutations in grade 5 mathematics.Journal for Research in Mathematics Education,33,5-29.
Reid,D.A and Knipping,C.(2010). Proof in Mathematics Education.Candada:Acadia University.
Stylianides,A.J.,＆ Ball,D.L.(2008).Understanding and describing
mathematical knowledge for teaching:knowledge about proof for engagibg
students in the activity of proving.Journal of Mathematics Teacher
Education,11,307-332.
Stylianides, A.J. (2007b).Proof and proving in school mathematics. Journal
for Research in Mathematics Education,38,289-321.
Toulmin,S.(1985).The uses of argument. Cambridge: Cambridge University
Press.
Threlfall,J.(1999). Repeating patterns in the early primary years. In A.
Orton (Ed.), Pattern in the teaching and learning of mathematics (pp. 18-30).
Landon: Cassell.
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(pp.469-480).Sant Feliu de Guixols,
Spain.Published online at http://ermeweb.free.fr/CERME4/
Yackel,E.,and Hanna,G.(2003).Reasoning and proof.In J.Kilpatrick,W.G.Martin,＆ D.Schifter(Eds.),A research companion to “Principles and standards for school mathematics”(pp.227-236).Reston,VA:National Council of Teachers of Mathematics.
Popper, K. R. (1959). The logic of scientific discovery. New York: Basic Books.