本研究針對國小數學教科書中幾何單元提供學生推理證明機會之分析比較，主要探討K、Ｈ、Ｎ三版本教科書中提供給學生推理證明機會之多寡。以九年一貫課綱四、五、六年級之國小教科書為研究對象，研究方式採「內容分析法」，研究單位分為示範及學生練習以「數學問題」來分析比較，分別從RP問題的目的、證明類型、證明的目的三方面進行分析。
在RP問題的目的分析結果發現：(1)RP問題的題數以Ｋ版本最高，其次為N版本，最少為Ｈ版本；（2）RP問題的目的以提出主張最高，證明主張次之，最少為評估或發展主張；(3)三版本在學生練習的部分，幾乎沒有相關的RP問題。
在證明類型分析結果發現：（1）三版本提供命題證明的比例相近；(2)三版本提供的證明類型皆以經驗證明為主；（3）在證明類型的四五六年級的安排上，Ｋ版本層次設計較佳，隨著年級降低經驗證明的比例，Ｈ版本三個年級證明層次設計較為平均。
在證明的目的分析結果發現：(1)三版本教科書示範多以形成概念為主要的證明目的，其次為解釋；(2)學生練習多以驗證為主要的證明目的，其次為形成概念；(3)三版本教科書皆未以否證為證明的目的。

This study was conducted to analyze and compare the geometric units in elementary school mathematics textbooks that provide students with opportunities to prove their reasoning. The study was conducted by using the "content analysis method" and the unit of study was divided into demonstrations and student exercises using "mathematical problems" to analyze and compare the purpose of RP problems, the type of proof, and the purpose of proof.
The purpose of the RP questions was analyzed in three ways: (1) the number of questions in the RP questions was highest in the K version, followed by the N version, and least in the H version; (2) the purpose of the RP questions was highest in the formulation of claims, followed by the proof of claims, and least in the evaluation or development of claims; (3) there were almost no relevant RP questions in the student practice section in the three versions.
In the analysis of the types of proofs, it was found that: (1) the three versions provided similar proportions of propositional proofs; (2) all three versions provided mainly empirical proofs; (3) in the arrangement of the types of proofs for grades 4, 5, and 6, the K version had a better hierarchical design, and the proportion of empirical proofs decreased with the grade level, while the H version had a more even hierarchical design for the three grades.
In the analysis of the purpose of proof, it was found that: (1) the three versions of the textbook demonstrated that the main purpose of proof was to form concepts, followed by explanations; (2) students practiced to verify the main purpose of proof, followed by the formation of concepts; (3) none of the three versions of the textbook used disproof as the purpose of proof.

壹、中文

王石番（1996）。傳播內容分析法：理論與實證。臺北市：幼獅文化。
王文科（2002）。教育研究法。臺北市：五南。
王文科、王智弘（2019）。教育研究法。五南圖書出版股份有限公司。
林碧珍、鄭章華與陳姿靜 (2016)。數學素養導向的任務設計與教學實踐── 以發展學童的數學論證為例.。Journal of Textbook Research, 9(1).
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。
徐偉民、曾于珏（2013）。臺灣、芬蘭、新加坡國小數學教科書代數教材之比較。教科書研究，6(2)，69-103。
徐偉民、柯富渝（2013）。臺灣、芬蘭、新加坡國小數學教科書幾何教材之比較。教科書研究，7(3)，101-141。
徐偉民（2013）。國小教師數學教科書使用之初探。科學教育學刊，21(1)，25-48。
教育部（2003）。國民中小學九年一貫課程綱要。臺北市：編者。
教育部（2008）。國民中小學九年一貫課程綱要總綱。臺北市：編者。
歐用生（2000）。教育研究法下冊。臺北：師大。
張景媛、陳萩卿（2003）。促進推理思考的認知策略。課程與教學，6(2)， 79-108。
楊瑞智(2020)。康軒數學五上。康軒文教事業股份有限公司。
楊瑞智(2020)。康軒數學五下。康軒文教事業股份有限公司。
楊瑞智(2020)。康軒數學六上。康軒文教事業股份有限公司。
楊瑞智(2020)。康軒數學六下。康軒文教事業股份有限公司。
楊逸帆(2018)。國小數學教科書提供學生論證機會之比較（未出版碩士論文）。國立清華大學，新竹。
曾瑞怡（2019）國小數學教科書中的論證機會分析——以代數為例（未出版碩士論文）。國立清華大學，新竹。
郝蕾（2021）兩岸國中教科書幾何證明機會的比較研究（未出版碩士論文）。國立清華大學，新竹。
趙旼冠、楊憲明 (2006)。數學障礙學生數學概念理解、數學推理能力與數學解題表現之關係分析研究。特殊教育與復健學報，(16)，73-97。
錢曉建（2013）。談小學數學教學中的推理能力培養。數學教學通訊，13。
揭志勇（2013）。說說幾何證明的教學。湖南教育:下旬 (C)，(12)，42-43。
方瓊婉、徐偉民與郭文金（2017）。運用圖形拆解法提升九年級學生幾何證明的學習表現。臺灣數學教師，38(2)，1-18。

貳、英文
Arzarello, F., Andriano, V., Olivero, F. and Robutti, O. （1998）. Abduction and conjecturing in mathematics. Philosophica, 61(1): 77–94.
Bell, A. W. （1976）. A study of pupil's proof-explanations in mathematical situations. Educational Studies in Mathematics, 7(1): 23–40.
Balacheff, N. (1982). Preuve et démonstration en mathématiques au collège. Recherches en didactique des mathématiques, 3(3), 261-304.
Balacheff, N. (1988a). A study of students' proving processes at the junior high school level. In Second UCSMP international conference on mathematics education. NCTM.
Balacheff, N. (1998b). Construction of meaning and teacher control of learning. In Information and communications technologies in school mathematics (pp. 111-120). Springer, Boston, MA.
Bieda, K. N., Ji, X., Drwencke, J., & Picard, A. (2014). Reasoning-and-proving opportunities in elementary mathematics textbooks. International Journal of Educational Research, 64, 71-80.
Ball, D. L., Bass, H., Hill, H., Sleep, L., Phelps, G., & Thames, M. (2006, January). Knowing and using mathematics in teaching. In Learning Network Conference, Washington, DC.
Ball, D. L., & Bass, H. (2003). Making mathematics reasonable in school. A research companion to principles and standards for school mathematics, 27-44.
Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2003).. The teaching of proof. arXiv preprint math/0305021
Charles, R. I., Chancellor, D., Harcourt, D., Moore, D., Schielack, J., Van de Walle, J., et al. (1999). Scott Foresman-Addison Wesley math: Grade 5 (1 & 2). New York City,NY: Addison Wesley Longman.
Cañadas, M. C. and Castro, E. (2005). “A proposal of categorisation for analysing inductive reasoning”. In Proceedings of the 4th Congress of the European Society for Research in Mathematics Education, Edited by: Bosch, M. 401–408. Spain: Saint Feliu de Guixols.
Davis, P. J. and Hersh, R. (1981). The mathematical experience, Boston: Houghton-Mifflin.
de Villiers, M. (1998).“An alternative approach to proof in dynamic geometry”. In Designing learning environments for developing understanding of geometry and space, Edited by: Lehrer, R. and Chazan, D. 369–393. London: Lawrence Erlbaum Associates.
Duval, R. (1995). Geometrical Picture: Kinds of Representation and Specific Processing. In R.Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with computers inMathematics Education, pp. 142-157. Berlin: Springer.
Duval, R. (1998). Geometry from a cognitive of view: Perspectives on the Teaching of Geometry for the 21st century. An ICMI Study, pp.37-52.
M. G. B., Boero, P.(1998). Geometry from a cognitive of view: Perspectives on the Teaching of Geometry for the 21st century. An ICMI Study, pp.52-62.
de Villiers, M. (1999).“The role and function of proof”. In Rethinking proof with the Geometer's Sketchpad, Edited by: de Villiers, M. 3–10. Key Curriculum Press.
Epp, S. S. (1998). A unified framework for proof and disproof. The Mathematics Teacher, 91(8), 708-713.
Fennell, F. S., Ferrini-Mundy, J., Ginsburg, H., Greenes, C., Murphy, S., & Tate, W. (2001). Silver Burdett Ginn mathematics: Grade 5. Parsippany, NJ: Silver Burdett Ginn.
Gilbert, J. K., Boulter, C., & Rutherford, M. (1998). Models in explanations, Part 1: Horses for courses?. International Journal of Science Education, 20(1), 83-97.
Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1): 6–13.
Hanna, G., & de Bruyn, Y. (1999). Opportunity to learn proof in Ontario grade twelve mathematics texts. Ontario Mathematics Gazette, 38, 180–187.
Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational studies in mathematics, 44(1), 5-23.
Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for research in mathematics education, 31(4), 396-428.
Herbst, P. G. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49, 283–312.

Hurley, P. J. (2006). A concise introduction to logic (9th ed.). Belmont, CA: Thomson Learning.
Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. Second handbook of research on mathematics teaching and learning, 2, 805-842.
Heinze, A., Cheng, Y.-H., Ufer, S., Lin, F.-L., & Reiss, K. (2008). Strategies to foster students’ competencies in constructing multi-steps geometric proofs: Teaching experiments in Taiwan and Germany. ZDM, 40, 443–453.
Knuth, E. J. (2002). Secondary school mathematics teachers' conceptions of proof. Journal for research in mathematics education, 33(5), 379-405.
Kuhn, D., & Udell, W. (2003). The development of argument skills. Child development, 74(5), 1245-1260.
Linda, P. (1999). Supporting mathematical development in the early years.Philadelphia : Open University Press.
Lakatos, I. (2015). Proofs and refutations: The logic of mathematical discovery. Cambridge university press.
Lakatos, I. (2015). Proofs and refutations: The logic of mathematical discovery. Cambridge university press.
Mason, J. and Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15: 277–289.
Miyazaki, M. (2000). Levels of proof in lower secondary school mathematics. Educational Studies in Mathematics, 41(1), 47-68.
Muis, K. R. (2004). Personal epistemology and mathematics: A critical review and synthesis of research. Review of educational research, 74(3), 317-377.
Martin, T. S., McCrone, S. M. S., Bower, M. L. W., & Dindyal, J. (2005). The interplay of teacher and student actions in the teaching and learning of geometric proof. Educational Studies in Mathematics, 60(1), 95-124.
Miyazaki, M., Fujita, T., & Jones, K. (2017). Students’ understanding of the structure of deductive proof. Educational Studies in Mathematics, 94(2), 223-239.
National Council of Teachers of Mathematics. (2009). Focus in high school mathematics:Reasoning and sense making. Reston, VA: Author.
National Governors Association Center for Best Practices & Council of Chief State SchoolOfficers. (2010). Common Core State Standards for Mathematics. Washington, DC.
Otten, S., Gilbertson, N. J., Males, L. M., & Clark, D. L. (2014). The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. Mathematical Thinking and Learning, 16(1), 51-79.
Reid, D. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education, 33(1): 5–29.
Recio, A. M., & Godino, J. D. (2001). Institutional and personal meanings of mathematical proof. Educational studies in mathematics, 48(1), 83-99.
Reid, D. & Knipping, C. (2010). Proof in mathematics education: research, learning, and teaching. The Netherlands: Sense publisher.
Semadeni, Z. (1984). Action proofs in primary mathematics teaching and in teacher training. For the learning of mathematics, 4(1), 32-34.
Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of'well-taught'mathematics courses. Educational psychologist, 23(2), 145-166.
Simpson, A. (1995). Developing a proving attitude. In Conference Proceedings: Justifying and proving in school mathematics (pp. 39-46). University of London, Institute of Education.
Schmidt, W. H., & Prawat, R. S. (2006). Curriculum coherence and national control of education: Issue or non‐issue?. Journal of curriculum studies, 38(6), 641-658.
Stylianides, A. L. (2007). Proof and proving in school mathematics. Journal for research in Mathematics Education, 38(3), 289-321.
Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the learning of mathematics, 28(1), 9-16.
Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical thinking and learning, 11(4), 258-288.
Stylianou, D. A., Blanton, M. L., & Knuth, E. J. (Eds.). (2009). Teaching and learning proof across the grades: A K-16 perspective. New York, NY: Routledge.
Tinto, V. (1988). Stages of student departure: Reflections on the longitudinal character of student leaving. The journal of higher education, 59(4), 438-455.
Toulmin, S. E. (2003). The uses of argument. Cambridge university press.
Thompson, D. R., Senk, S. L., & Johnson, G. J. (2012). Opportunities to learn reasoning and proof in high school mathematics textbooks. Journal for Research in Mathematics Education, 43, 253–295.
University of Illinois-Chicago. (1998). Math trailblazers: Grade 5. Dubuque, IA: Kendall-Hunt Publishing Company.
University of Chicago School Mathematics Project. (2007). Everyday mathematics: Grade 5 teacher’s lesson guides (1& 2). Chicago, IL: McGraw Hill Wright Group.
Van Eemeren, F. H., Grootendorst, R., & Henkemans, A. F. S. (2002). Argumentation: Analysis, evaluation, presentation. Routledge.
Weber, K. (2009). Mathematics majors’evaluation of mathematical arguments and their conception of proof. In Proceedings of the Twelfth sigmaa on rume Conference on Research in Undergraduate Mathematics Education.
Hanna, G., & Yackel, E. (2003). Reasoning and proof. A research companion to NCTM’s Standards, 227-236.
Zack, V. (1999). Everyday and mathematical language in children's argumentation about proof. Educational Review, 51(2), 129-146.
Zack, V., & Reid, D. A. (2003). Good-enough understanding: Theorising about the learning of complex ideas (part 1). For the Learning of Mathematics, 23(3), 43-50.
Zack, V., & Reid, D. A. (2004). Good-enough understanding: Theorising about the learning of complex ideas (part 2). For the Learning of Mathematics, 24(1), 25-28.