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題名：高中師生對數學歸納法瞭解的情況與教學因應之研究
作者：朱綺鴻
作者(外文)：Chu, Chee-Hong
校院名稱：國立臺灣師範大學
系所名稱：科學教育研究所
指導教授：譚克平
學位類別：博士
出版日期：1999
主題關鍵詞：數學歸納法；良序性；類比；臆測；學習困難；教學法；多元表徵；有意義的學習；mathematical induction；well-ordering principle；analogy；conjecture；learning difficulty；instruction；multiple representation；meaningful learning
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本研究的目的有三：一、探討高中數學教師對數學歸納法教學的各種相關知識與教學上的意見；二、探究高二學生在學過數學歸納法之後，對該法了解的程度；三、發展新的教案進行實驗教學，並分析其是否能增進學生對數學歸納法的了解。
關於探討現職數學教師對教導數學歸納法的意見方面，研究對象為台北區公私立高中數學教師，研究以問卷調查的方式進行，問卷的設計則參考相關文獻、高中教師以及專家的意見而成，共分為教師的個人意見、對課程安排的建議、對數學歸納法的內容知識、及對學生學習該法的困難的了解等四個層面進行探討。調查結果發現大部份教師不但喜歡數學歸納法，且能清楚指出教導該法所需之先備知識及符號等，且對學生學習該法時所遭遇的困難亦有所了解，並可估計出過去所教過的學生中，其對數學歸納法之了解、成功擬題以及解題的程度。這些資料顯示出數學歸納法對高一學生有若干程度上的困難。此外，教師們對該法的內容知識亦有一定程度的掌握，惟有部份教師對歸納法與數學歸納法之間的差別有所混淆，且在符號表徵與推導步驟的用詞上有欠嚴謹的情況。
本研究第二個部份在於探討高二學生在學過數學歸納法之後，對該法了解的程度方面，研究對象為台北市某公立高中剛升上高二的學生，由選讀第一（文、法）、第二（理、工）、及第三(醫、農)等三個類組的學生各一班，共135人，因為他們在過去曾經學習過數學歸納法，仍以問卷調查的部份進行，內容包括對數學歸納法的喜好、感覺難易、以及對該單元的相關內容知識、概念、過程技能等的瞭解到達什麼程度。調查結果顯示，學生在學習數學歸納法之後最深刻保留的概念是該法的固定格式與一成不變的證明流程，同時也發現大多數的學生並不能真正瞭解數學歸納法的精神與本質，只是流於機械化的學習或模仿，這點值得教師們留意。至於對數學歸納法概念的瞭解方面，則發現他們對歸納法與數學歸納法之間在用詞上有混淆、對所使用的n與k等變數角色的疑惑，都是學生的迷思概念產生的根源，在過程技能方面，遞推步驟中由P(k)推論到P(k+1)成立的過程，則是學生最大的困難，這些都是教師在教學時可以多加說明的部份。
在發展新的教案並進行實驗教學方面，研究對象為與上述學生問卷調查對象同校的高一學生，選擇三個班級隨機分派為對照組、多元表徵組、闡述教學組，分別進行六節依照不同教學設計的教學實驗課程，隨後進行成效評估的後測分析，資料顯示學習成就高低不同的學生在接受不同教學法之後，其在後測試卷的表現上並沒有顯著的交互作用，亦即各種教學法對於學習成就高低不同的學生在適用上並沒有顯著的差別。在教學法的主要效果方面，闡述教學組在對數學歸納法證明過程技能的學習，以及判斷數學歸納法的使用時機上，有較佳的平均表現，而多元表徵的方法則較能使學生有更多元化的思考方向，比較能由少數特例中尋找規律或發現反例。因此，如果教學時間許可的話，教師宜對各種概念提供不同的表徵方式，使具備不同先備知識或認知偏好的學生有較多的機會選擇，方能與其原有的認知結構相結合而達到有效學習的目的。
最後整合研究之結果，分別對教師、教材編寫者與學生提供一些教導數學歸納法、編製教科書以及學習該法的建議。
There are three aims of this dissertation. First, to find out the opinions of in-service mathematics teachers regarding issues concerning the teaching of the method of mathematical induction. Secondly, to deal with the 11th grade students'' understanding mathematical induction after they learned it. Thirdly, to advance some new instructions to teaching mathematical induction to improve the understanding of students and to compare the effect of those instruction.
The opinions of in-service mathematics teachers were conducted via a survey on high school mathematics teachers in both public and private sectors in the Taipei area. The survey items were designed in accordance with the literature, and modified from feedback provided by both high school teachers and experts. It contained four major sections, namely, teachers'' personal opinions, curriculum design, subject matter knowledge and the extent of understanding with respect to students'' learning difficulties. It was found that most teachers liked mathematical induction, and could clearly point out the prior concepts and symbols that were necessary for learning this method. They were also capable of pointing out what kind of learning difficulties most encountered by students in this aspect. Furthermore, out of all students they taught in the past, they could estimate the percentage among them who could master the principle behind mathematical induction, as well as the percentage of those who could pose a mathematical induction problem and solved it afterward. The data thus collected revealed that mathematical induction was quite difficult to tenth graders. Besides, most of the teachers surveyed seemed to master the subject matter knowledge to a good extent. Some teachers, however, had rooms for improvement. They might not be aware of the difference between induction and mathematical induction. Also, their use of symbols, and their wordings in terms of the derivation in the implication process was not very rigorous.
The second purpose of this dissertation was to explore the 11th graders'' understanding of mathematical induction after they learned it. It was also conducted via a survey on 135 high school students. The survey items contained the preference, difficulty index, and the extent of understanding of subject matter knowledge, concept, procedural skill about mathematical induction. The results of our investigation support that the retention of memory of students was the fixed form and routine of proof procedure. What has to be noticed were the most students only were rote learning and mimed the procedure of proof and do not really understand the nature of mathematical induction. On the understanding of the concept, the students'' use of symbols, and their wordings in terms was not rigorous, and they have the misconceptions of roles of variables-- n and k. On procedural skill, the induction step, namely, the implication process from P(k) to P(k+1), was the most difficult for students. It seems reasonable to conclude that teacher can explanation more about those part.
Finally, to advance some new instructions to teaching mathematical induction to improve the understanding of students. In the same school as mentions above, 3 classes were chosen, and use different instructions to teach them, namely, general method, multiple-representation method, and expository learning method. After 6 courses of the experiment teaching, they were assessed their effect of learning. The result of the experiment was that the group of expository learning method was the best in behavior skills of proof and in judgement of timing to use mathematical induction. The students of the group of multiple-representation method have multiple thinking and can find rules and counterexamples from some spatial cases. It should be concluded, from what has been said above, to use multiple representations to teaching mathematical concepts, it can help students learning meaningfully. Because students have different prior concepts and cognitive styles, they can chose the adapted ones to connect their cognitive structure and learn effectively.
This dissertation ended with some suggestions regarding the teaching of mathematical induction for both teachers and textbook authors and the learning of it for students.
一、 中文部份
王湘君（1980），高中生對數學歸納法的困惑，數學傳播，六十九年三月。
史久一、朱梧榎(1989)，化歸與歸納、類比、聯想，江蘇教育出版社。
任樟輝(1990) ，數學思維論，廣西教育出版社。
朱敬先（1987），教學心理學，台北：五南。
江新合、張新仁（1991）數學及自然科學師資之培訓研究(I)：能有效運用七種教學模式之物理科教師職前培育研究，行政院國家科學委員會專題研究計畫成果報告，NSC79-0111-S-017-11。
呂發全(1981)，高級中學數學課程實驗教材─數學歸納法的教學心得，科學教育月刊，第40期。
李靜、宋立軍、張大松(1994)，科學思維的推理藝術，台北：淑馨。
李勝利(1982)，談數學歸納法，數學傳播，七十一年十二月。
林曉雯(1994)國中生物教師教學表徵的詮釋性研究，國立台灣師範大學科學教育研究所博士論文。new window
林寶山（1988）教學原理，台北：五南。
林寶山（1990）教學論，台北：五南。
紀惠英(1991)，國小六年級學生數學應用問題表徵類型與同構性之研究，國立台灣師範大學特殊教育研究所碩士論文。
孫名符、呂世虎、傅敏、王仲春(1997)，數學、邏輯與教育，台北：建宏。
徐道寧(1980)，數學歸納法，新竹：凡異出版社。
教育部(1995)，高級中學數學科課程標準。
國立台灣師範大學科學教育中心(1984)，高級中學基礎數學（第一冊）教師手冊，台北：國立編譯館。
國立台灣師範大學科學教育中心(1984)，高級中學基礎數學（第一冊），台北：國立編譯館。
傅偉勳(1984)，西洋哲學史，台北：三民。
黃永和(1997)，教學表徵----教師的教學法寶，國教世紀，86.12。
張巨青、吳寅華(1994)，邏輯與歷史----現代科學方法論的嬗變，臺北：淑馨。
張景中(1996)，數學與哲學，台北：九章。
張新仁(1993)，奧斯貝的學習理論與教學應用，教育研究，82.08 。new window
張瓊，于祺明，劉文君(1994)，科學理論模型的建構，臺北：淑馨。
華羅庚(1972），數學歸納法，香港：商務印書館。
畢家湘(1991)，從科學貫通哲學，台北：台灣商務印書館。
劉大昌(1975)，數學歸納法的應用，科學研習，64.11。
劉正義(1996)，歸納法之基本精神及其在經濟學上之應用，企銀季刊，85.07。
劉福增譯，Kahane, H. & Tidman, P.原著，(1996)邏輯與哲學，台北：心理出版社有限公司。
林清山譯，Mayer, R. E. 著，(1991) 教育心理學----認知取向，台北遠流。
何秀煌譯，Salmon著，(1968)，邏輯，台北：三民。
二、 英文部份
Avital, S. & Hansen, R.T. (1976). Mathematical Induction in the Classroom. Educational Studies in Mathematics, 7, 399-411.
Avital,S. & Libeskind, S.(1978). Mathematical Induction in the Classroom: Didactical and Mathematical Issues. Educational Studies in Mathematics, 9, 429-438.
Baker, J. D. (1996). Students'' difficulties with proof by mathematical induction. Paper presented at the Annual Meeting of the American Educational Research Association (New York, NY, April 8-12, 1996). ED: 369931.
Barker, S. F. (1989). The Elements of Logic. Singapore: McGraw-Hill Book Company.
Blank, A. A. (1963). Mathematical Induction. In Enrichment Mathematics for High School-Twenty-eighth Yearbook of National Council of Teachers of Mathematics. Washington, D. C.
Dickey, E.M. (1993). The Golden Ratio: A Golden Opportunity to Investigate Multiple Representations of a Problem. The Mathematics Teacher, 86(7), 554-557.
Dolan, S. (ed. )(1994). Mathematical Structure：The School Mathematics Project. Great Britain: Cambridge University Press.
Dubinsky, E. D. (1991). Reflective Abstraction in Advanced Mathematical Thinking. In Tall, D. (ed.) (1991). Advanced Mathematical Thinking. Boston: Kluwer Academic Publishers.
Duit, R. (1991). On the Role of Analogies and Metaphors in Learning Science. Science Education , 75(6), 649-672.
Ernest, P. (1984). Mathematical Induction: A Pedagogical Discussion. Educational Studies in Mathematics, 15, 173-189.
For t, K.F.(1998).Teaching Induction : Historical Perspective and Current Views. Thesis (Ph.D.)--The American University at Washington, DC.
Gerstein, L. J. (1996). Introduction to Mathematical Structure and Proofs. New York: Springer ; Sudbury, MA: Jones and Bartlett Publishers.
Greeno, J. G. & Hall, R. P. (1997). Practicing Representation: Learning with and about Representational Forms. Phi Delta Kappa, 97, 361-367.
Hiebert, J. & Carpenter, T. P. (1992). Learning and Teaching with Understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp.65-97). New York: Macmillan.
Hirsch, C.R. (1976). Making Mathematical Induction Meaningful. School Science and Mathematics, 76(1), 27-31.
Hirschfelder, R. (1991). Introduction to Discrete Mathematics. Pacific Grove, Calif.: Brooks/Cole Publishing Company.
Janvier, C. (1984). Problems of Representation in the Teaching and Learning of Mathematics. Lawrense Erlbuam Associates, Inc.
Joyce, B. & Weil, M. (1986). Models of Teaching. N. J.: Prentice-Hall.
Keller, B.A. & Hirsch, C.R. (1998). Student Preference for Representation of Function. International Journal of Mathematical Education in Science and Technology, 29(1), 1-17.
Lowenthal, F. & Eisenberg, T.(1992). Mathematical Induction in School：An Illusion of Rigor ? School Science and Mathematics. 92(5), 233-238.
McLeod, D. (1992). Research on affect in mathematics education：A reconceptualization. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning, (pp.575-596), New York: Macmillan.
Moore, R. C. (1994). Making the Transition to Formal Proof. Educational Studies in Mathematics, 27, 249-266.
Movshovitz-Hadar, N. (1993). Mathematical Induction: A Focus on the Conceptual Framework. School Science and Mathematics, 93, 408-417.
Munro, J. E. (1992). Discrete Mathematics for Computing. London; New York: Chapman & Hall.
National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics, Commission on Standard for School Mathematics, Reston, VA: The National Council of Teachers of Mathematics, Inc.
Piaget, J. (1981). Intelligence and affectivity: their relationship during child development. Palo Alto, Calif.: Annual Reviews Inc.
Pinker, A. (1976). Induction and Well Ordering. School Science and Mathematics, 76(3), 207-214.
Rosen, K.H. (1995). Discrete Mathematics and Its Applications. N.Y.: McGraw-Hill Book Co.
Ross, K. A. & Wright C.R.B. (1985). Discrete Mathematics. Englewood Cliffs. N. J. : Prentice-Hall.
Schoenfeld, A. (l985). Mathematical problem solving. San Diego, CA: Academic Press.
Shaw, B. (1978). n and S. Mathematics Teaching, 82, 6-7.
Shulman, L.S.(1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15(2), 4-14.
Smith, K.E. (1993). Teaching Induction in Discrete Mathematics. In Kenney, M. J. & Hirsch C. R. (ed.). Discrete Mathematics Across the Curriculum, K-12: 1991 Yearbook. Reston, VA: The National Council of Teachers of Mathematics, Inc.
Solow, D. (1990). How to Read and Do Proofs: An Introduction to Mathematical Thought Processes. (Second ed.). NY: John Wiley & Sons.
Trusted, J. (1979). The Logic of Scientific Inference: An Introduction. The Macmillan Press Ltd.
Tsamir, P. & Tirosh, D. (1999). Consistency and Representations: The Case of Actual Infinity. Journal for Research in Mathematics Education, 30(2), 213-219.
Van-Dyke, F. (1995). A Concrete Approach to Mathematical Induction. The Mathematics Teacher, 88(4), 302-318.
Word, K.J. (1988). Instructional Sequence Effects of Recursion and Mathematical Induction in College Algebra. Thesis (Ph.D.)--The University of Texas at Austin.
 
 
 
 
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