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博碩士論文 etd-0720105-140614 詳細資訊
Title page for etd-0720105-140614
論文名稱
Title
不同年級學童在具體情境中未知數概念及解題歷程之研究
A Study on the Concept of Unknown and Problem-Solving Process Among Different Graders in Concrete Situations
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
213
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2005-05-30
繳交日期
Date of Submission
2005-07-20
關鍵字
Keywords
代數、解題歷程、半結構式晤談法、未知數
Problem-solving process, Unknown, Algebra, Semi-structured interview
統計
Statistics
本論文已被瀏覽 5666 次,被下載 2608
The thesis/dissertation has been browsed 5666 times, has been downloaded 2608 times.
中文摘要
摘 要
本文旨在研究不同年級學童在具體情境中的未知數概念,以及面對未
知數問題情境之解題歷程。近年來國際間有許多研究 (Carraher, Schliemann,
& Schwartz, in press ; Carraher, Schliemann & Brizuela, 2001 ; Bodanskii, 1991
)發現,經由系統化的教學,國小中低年級學童的代數學習表現,優於同
儕甚至高年級學童。於是研究者以半結構式晤談法(semi-structured interview
)收集個案資料,研究參與對象為國小二年級、國小五年級及國中一年級
學童各乙名,參與本研究時均使用九年一貫課程暫行綱要之現行數學教材
。訪談導引為20 以下的自然數之改變型加減法問題及等組型乘除法問題
:包括單步驟問題、兩步驟混合問題、兩未知數關係問題以及未知數比大
小問題。研究者從訪談逐字稿、學童紙筆解題記錄以及研究者在現場的觀
察記錄等三方面著手進行資料分析。本研究共有六個主要發現,經由訪談
者的引導,個案都能以代數語言轉譯問題的文字語言;三名個案的方程式
解題策略均以「逆運算」為主;國小個案能在具體情境以數的基本運算性
質化簡未知數的式子;小五及國一個案都能檢驗其解的合理性;等號意義
逐漸由「算出答案的指令」發展到「代表相等同類量」;三名個案在「嘗
試錯誤」解題策略有個別差異。以上部分研究結果與Carraher 等人的研究
結果相一致。最後本文對我國現行國中小代數課程提出建議,國小數學教
材若及早加入讓學童學習或討論如何以代數式描述算術文字問題情境之
單元,將有助於他們在國中正式學習代數時,對文字符號概念的認知提升
Abstract
The aim of this study is to explore different graders’ concept of unknown
and performance in solving equations in concrete situations. In recent years
of early algebra research in the United States (Carraher, Schliemann, &
Schwartz, in press), it was found that through systematic teaching, low and
middle graders’ algebra performance was better than the same or even higher
graders without teaching. Therefore, semi-structured interview was adopted
to collect data on three cases: a second-grader, a fifth-grader and a
seventh-grader who were using textbooks that follow Grade one-nine
Integrated Coordinate Curriculum in SY89. The interview questions included
addition and subtraction CHANGE problems, as well as multiplication and
division EQUAL GROUPS problems; with natural numbers below 20, and
given in four types: one-step, two-steps mixed, relating two unknowns and
comparing two unknowns. Data analysis was conducted by referring to three
sources of data: protocols from interviews, children’s problem-solving records
and interviewer’s observation records. Research findings were: all three cases
that received guidance could use equations to express problems; “Undoing”
was the most frequently used problem-solving strategy; both second and fifth
graders could simplify expressions by number properties in concrete situations;
both fifth and seventh graders could check if answers were reasonable; the
meaning of equal sign developed from “finding the results of” to “equality in
measures”; and, individual differences in “trial and error substitution” among
three cases. Such results were consistent to that of Carraher. It is suggested
that, introducing early algebra in the elementary school is helpful to children’s
learning of formal algebra in the junior high school.
目次 Table of Contents
目 錄
第一章 緒論..........................................................1
第一節 研究動機........................................................................1
第二節 研究目的........................................................................3
第三節 名詞釋義........................................................................4
第四節 研究範圍與限制...........................................................6
第二章 文獻探討..................................................9
第一節 未知數概念的背景.......................................................9
第二節 孩童未知數概念的發展情形.....................................12
第三節 未知數學習困難及幼童可學習代數相關研究........16
第四節 代數教材中的未知數概念.........................................26
第三章 研究方法................................................31
第一節 研究設計與流程.........................................................31
第二節 研究對象.....................................................................34
第三節 研究工具.....................................................................36
第四節 晤談方法.....................................................................38
第五節 訪談導引與實施.........................................................39

vii
第六節 資料分析.....................................................................44
第四章 研究結果與分析..................................47
第一節 國小二年級學童個案分析.........................................47
第二節 國小五年級學童個案分析.........................................60
第三節 國中一年級學童個案分析.........................................76
第四節 訪談問題結構的跨個案比較.....................................86
第五節 未知數概念及解題歷程的跨個案比較.....................90
第五章 結論與建議............................................99
第一節 結論.............................................................................99
第二節 與前人研究之比較與反省.......................................103
第三節 建議...........................................................................107
參考書目............................................................ 111
附錄一:訪談導引............................................ 117
附錄二:國小二年級訪談原案........................125
附錄三:國小五年級訪談原案........................153
附錄四:國中一年級訪談原案........................193

viii
附表目次
表2-1 學生文字符號概念層次表……………………………………………….14
表2-2 尚未學習代數的孩童之未知數解題策略……………………………….19
表3-1 訪談問題類型…………………………………………………………….43
表3-2 原案編碼原則.……………………………………………………………45
表4-1 三名個案之解題策略次數統計………………………………………….96
表4-2 三名個案之未知數概念及未知數解題歷程比較……………………….94

ix
附圖目次
圖2-1 「代數」主題之能力指標脈絡圖………………………………………….29
圖3-1 訪談分析設計……………………………………………………………...32
圖3-2 研究流程圖………………………………………………………………...33
參考文獻 References
參考書目
一、中文部分:
王懷權(1987)。數學發展史。新竹:凡異。
朱建正(1997)。國小數學課程的數學理論基礎。未出版。
余文卿、謝暉光(譯)(1997)。John Daintith & R. D. Nelson 著。數學辭典。台北:牛頓。
呂玉琴(1989)。在國小實施代數教學的可能性研究。台北師院學報,2,263-283 。
尚榮安(譯)(2001)。R. K. Yin 著。個案研究。台北:弘智。
林光賢、郭汾派、林福來(1989)。國中生文字符號概念的發展。國科會專題研究計畫報告。計畫編號:NSC 77-0111-S004-001-A
袁媛(1993)。國中一年級學生的文字符號概念與代數文字題的解題研究。國立高雄師範大學數學教育系碩士論文,未出版,高雄市。
教育部(1993)。國民小學課程標準。台北:教育部。
教育部(2000)。國民教育九年一貫課程暫行綱要:數學學習領域。台北:教育部。
教育部(2003)。國民中小學九年一貫課程綱要:數學學習領域。台北:教育部。
陳李綢(1996)。個案研究。台北:心理。
陳盈言(2001)。國二學生變數概念的成熟度對其函數概念發展的影響。國立台灣師範大學數學系碩士論文,未出版,台北市。
陳維民(1998)。兒童的未知數概念研究—一個國小六年級兒童的個案研究。國立高雄師範大學數學系碩士論文,未出版,高雄市。
陳慧珍(2001)。南投縣國一男女生對文字符號概念與代數文字題之解題研究。
國立高雄師範大學數學系教學碩士論文,未出版,高雄市。
黃志賢(2000)。原住民學生利用代數方法解題之研究。原住民教育季刊,21,17-38。
黃瑞琴(1991)。質的教育研究方法。台北:心理出版社。
黃寶彰(2002)。六、七年級學童數學學習困難部分之研究。國立屏東師範學院數理教育研究所碩士論文,未出版,屏東縣。
甯自強(1993)。「建構式教學法」的教學觀-由根本建構主義的觀點來看。載於詹志禹(主編),建構論-理論基礎與教育應用(頁286-294)。台北市:正中書局。
趙文敏(1985)。數學史。台北市:協進。
戴文賓、邱守榕(1999)。國一學生由算術領域轉入代數領域呈現的學習現象與特徵。科學教育,10,頁148-174。
謝和秀、謝哲仁(2002)。國一學生文字符號概念及代數文字題之解題研究。九十一年度師範院校教育學術論文發表會論文集,3,頁1491-1521。
二、英文部分:
Bodanskii, F. (1991). The formation of an algebra method of problem solving in primary school children. In V. Davydov (Ed.), Soviet Studies in Mathematics Education (vol. 6, pp. 275-338). Reston, VA: NCTM.
Booth, L. R. (1984). Algebra: Children’s strategies and errors. Windsor, United Kingdom: NFER-Nelson.
Booth, L. R. (1988). Children’s difficulties in beginning algebra. In A. F. Coxford & A. P. Shulte (Eds.), The Ideals of Algebra, K-12 (pp.20-32). Reston, VA: NCTM.
Brito-Lima, A. P. & da Rocha Falcao, J. T. (1997). Early development of algebraic representation among 6-13 year old children: the importance of didactic contract. In Proceedings of the 21th International Conference Psychology of Mathematics Education (vol. 2, pp 201-208), Lahti, Finland.
Brizuela, B. M., & Schliemann, A. D. (2003). Fourth graders solving equations. In Proceedings of the 27th International Conference Psychology of Mathematics Education (vol. 2, pp 201-208), Lahti, Finland.
Carraher, D., Schliemann, A. D., & Schwartz, J. L. (in press). Early Algebra is not the same as algebra early. In J. Kaput, D.W. Carraher, & M. Blanton (ed). Algebra in
the Early Grades. NJ: Lawrence Erlbaum Associates.
Carraher, D., Schliemann, A.D. (2000).Bringing out the Algebra Character of Arithmetic: Instantiation Variables in Addition and Subtraction. In Proceedings of the 24th conference of the International Group for the PME (vol. 2, pp 145- 152).
Carraher, D., Schliemann, A.D., & Brizuela, B.M. (2001). Can Young Students Operate on unknowns. In Proceedings of the 25th conference of the International Group for the PME (vol. 1, pp 130-140). Utrecht, The Netherlands: Freudenthal Institute.
Chaiklin, S., & Lesgold, S. (1984). Prealgebra students’ knowledge of algebraic tasks with arithmetic expressions. Paper present at the annual meeting of the American
Educational Research Association.
Collis, K. F. (1975). The Development of Formal Reasoning. Newcastle, Australia: University of Newcastle.
Davis, R. (1985). ICME-5 Report: Algebraic thinking in the early grades. Journal of Mathematical Behavior, 4, pp.195-208
Davis, R. (1989). Theoretical considerations: Research studies in how human think about algebra. In S. Wagner & C. Kieran (Eds.) Research Issues in the Learning
and Teaching of Algebra, vol. 4. (pp.266-274)Restion, VA: NCTM/ NJ: Lawrence Erlbaum Associates.
Freudenthal, H. (1974). Soviet research on teaching algebra at the lower of the elementary school. Educational Studies in Mathematics, 5, 391-412.
Fuson, K. C. (1992). Research on Whole Number Addition and Substraction. In D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp.243-275). New York: Macmillan Pub.
Greeno, J. G. (1982). A cognitive learning analysis of algebra. Paper presented at the annual meeting of the American Educational Research Association, Boston, MA.
Harper, E. (1987). Ghosts of Diophantus. Educational Studies in Mathematics, 18(1), 75-90.
Herscovics, N., & Kieran, C. (1980). Constructing Meaning for the Concept of Equation. Mathematics Teacher, 73(8), 572-580.
Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59-78.
Kaput, J., Carraher, D.W. & Blanton, M. (in press). Algebra in the Early Grades. NJ: Lawrence Erlbaum Associates.
Kieran, C. (1984). A comparison between novice and more-expert algebra students on tasks dealing with the equivalence of equations. In J. M. Moser (Ed.),
Proceedings of the Sixth Annual Meeting of PME-NA (pp. 93-91). Madison: University of Wisconsin.
Kieran, C. (1989). The early learn of algebra: a structural perspective. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra(pp.
33-56). Hillsdale, NJ: Lawrence Erlbaum Associates; Reston, VA: NCTM.
Kieran, C. (1992). The learning and teaching of school algebra. In D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning. (pp.390-419).
New York: Macmillan Pub.
Kuchemann, D. ( 1978). Children's understanding of numerical variables. Mathematics in School, 7(4), 23-26.
Kuchemann, D. (1981). Algebra. In K. Hart (Ed.), Children’s Understanding of Mathematics: 11-16 (pp 102-119). London: John Murray.
Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics. NJ: Lawrence Erlbaum Associates.
MacGregor, M., & Stacey, K. (1993). Cognitive models underlying student’s formulation of simple linear equations. Journal for Research in Mathematics
Education, 24(3), 217-232.
Malara, N. A. (2003). Dialectics Between Theory and Practice: Theoretical Issues and Aspects of Practice from an Early Algebra Project. In Proceedings of the 27th
International Conference Psychology of Mathematics Education (vol. 2, pp 201-208), Lahti, Finland.
Martin, M. O., Mullis, I. V. S., & Chrostowski, S. J. (Eds.) (2004). TIMSS 2003 Technical Report. Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.
Mayer, R.E. (1982). Different problem-solving strategies for algebra word and equation problems. Journal of Experimental Psychology: Learning, Memory and
Cognition, 8(5), 448-462.
NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.
Neuman, W. L. (2000). Social Research Methods: Qualitative and Quantitative Approaches. MA: Boston.
Schliemann, A.D., Carraher, D.W., & Brizuela, B. (2005). Bringing Out the Algebraic Character of Arithmetic: From Children’s Ideas to Classroom Practice. NJ: Lawrence Erlbaum Associates.
Sfard, A. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in
Mathematics, 22(1), 1-36.
Sowder, L. K. (1980). Concept and Principle Learning. In Shumway, R.J. (Ed.) Research in Mathematics Education. Reston, VA: NCTM.
Stake, R. E. (1995). The art of case study research. Thousand Oaks, CA: Sage.
TERC (2003). Early Algebra, early arithmetic - Class materials. Retrieved from http://www2.earlyalgebra.terc.edu/Materials/index.html
Van den Heuvel-Panhuizen, M. (1996). Assessment and realistic mathematics education. Utrecht, The Netherlands: Center for Science and Mathematics Education.
Wagner, S. (1981). Conservation of equation and function under transformations of variable. Journal for Research in Mathematics Education, 12, 107-118.
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