本研究抽取吳寶桂教授所蒐集之全國國中及小六在「一元一次方程式情境測驗」作答反應之部份資料，藉以探討年級因素，對學生在一元一次方程式解題概念之差異，並找出學生在一元一次方程式的概念發展趨勢。使用研究工具包括吳寶桂教授所編製之「一元一次方程式概念細圖」及「一元一次方程式情境測驗」。資料分析結果顯示：

(一) 未知數符號的使用：
1.小六對於自己可以處理的部分(不含未知數的部分)會先算，但無法處理的部分(含未知數的部分)，就會忽略而不去處理(或不列式)，而國一與國二較少有這種情況發生。
2.小六遇到題目有未知數時，沒有運用符號假設未知數的習慣，而是先觀察其他的已知數之規律或者經常自己假設未知數為某一特定常數，然後再繼續往下運算，顯示小六對於使用符號假設未知數的想法屬於萌芽階段。
3.國一遇到題目有未知數時，多數學生已經有使用符號假設未知數的習慣，只是國一學生在使用符號時，仍以可填充式圖形如□為主，較少使用英文字母(如 )，且國一學生經常犯「使用同一個□同時表示多種不同事物」的錯誤，顯示國一多數已經會使用可填充式圖形的符號假設未知數，但對於未知數的運算性質與使用，並不能完全瞭解；因此國一對於使用符號假設未知數的想法已漸漸成熟，但對於未知數符號的運算性質與使用屬於過渡期，概念尚未成熟。
4.國二遇到題目有未知數時，多數學生會使用英文字母如 來假設未知數，已逐漸擺脫可填充式圖形如□的使用，而且犯「使用同一個符號同時表示多種不同事物」的錯誤比例有明顯減少的趨勢，顯示國二對於未知數符號的概念發展，已漸趨成熟。

(二) 解題概念：
1.因為小六對於未知數符號的概念發展屬於萌芽階段，對未知數符號的運算性質亦不
瞭解，所以小六的解題概念如下：
(1) 覺得未知數很難處理或不知如何處理，會選擇「不處理未知數」，只處理其他簡單的部分。
(2) 因未知數符號的概念發展不成熟，所以當題目需假設2個未知數解聯立時，基本上，小六沒有能力完整列式，更不用提解題。
2.國一對於未知數符號的運算性質與使用屬於過渡期，概念尚未成熟，其解題概念如下：
(1) 國一對於題目中條件的描述與使用，已能做較有效的串聯。
(2) 將未知數假設成某一固定常數的比例已較小六為低，顯示國一對未知數的概念已有較深的認知。
(3) 當題目需假設2個未知數解聯立時，若題目簡單，則有部分學生能掌握其中一式之意義，即對題目條件轉換成方程式已具備某些能力。若題目較複雜，由於解聯立的概念尚未發展，經常產生將各個線索獨立處理或將各個線索亂混在一起處理，且對未知數符號的運算，經常犯了同一個符號同時表示多種不同的事物的錯誤。

3.國二對於未知數符號的概念發展，已漸趨成熟
(1) 國二對於題目中條件的描述與使用，已能做有效的串聯，
(2) 國二解題過程中，很少出現「同一個符號同時表示多種不同的事物」的錯誤，顯示國二對使用符號代表未知數的概念漸趨成熟。
(3) 當題目需假設2個未知數解聯立時，國二學生基本上是以設定2個未知數解聯立的手法為主(有81％是設定2個未知數解聯立的概念求得正解)，顯示國二學生對於兩個未知數的處理能力已具有一定的水準。

(三) 答對的狀況：整體而言，對於較簡單的問題，各年級答對的比例差異不大，對於較難的問題，學生答對的狀況，國二是優於國一，國一優於小六。

This study reanalyzed a part of the national data of the responses of 288 students in grades 6 to 8 on the “One-Variable Linear Equation Virtual Situational Test” collected by Professor Pao-Kuei Wu from August 1, 2001 through July 31, 2003. The analyses were based on the “One-Variable Linear Equation Conceptual Tables”. The results of the analyses are the following.

I. The use of variables
A. Compared to 7th and 8th graders, 6th graders would first solve the numerical arithmetic and solve the unknown parts next. But if the students could not handle the unknown parts, the 6th graders tended to ignore or even not list the unknown variable in the equations.
B. When encountering the unknown situations, most 6th graders are not accustomed to using symbols to represent unknown variables. Instead, they would observe the numerical components first to try to deduce what the unknown variable would be, and proceed from there. Some students would even set up some constants to represent those unknown variables. These results indicate that the 6th graders’ ability to use symbolic representation is still in the beginning stages.
C. In the unknown virtual situations, the majority of 7th graders were able to use symbolic representations. However, most of them would use pictorial representations such as □, instead of alphabetical representations such as x, y and z. Moreover, many students use the same symbols to represent different variables; this shows that although the 7th graders know to use symbols to represent unknown variables, they still are not able to fully comprehend unknown variables. Hence, the 7th graders’ ability to use symbolic representation is in the transitional stage.
D. When encountering unknown virtual situations, the majority of the 8th graders would able to use the numerical symbols such as x, y and z to represent the unknown variables. The frequency of using pictorial representations such as □ becomes less and less, and the tendency to use the same symbols to represent different variables is decreasing. All these indicate that the 8th graders’ development of the concept of unknown variables is maturing.

II. The concept of problem solving
A. The 6th graders’ ability to use symbolic representation is still in the beginning stages:
1. They only deal with the simple part; for the more complicated part, they chose to ignore.
2. Due to their immature development of symbol representation, when encountering the two variable linear equation problems, they even do not have the ability to write the ‘complete’ equation, not to mention to solve the equations.
B. The 7th graders’ ability to use symbolic representation is in the transitional stage:
1. Compared to the 6th graders, the 7th graders are more able to draw relationships among the different components of the problem.
2. The fact that the substantially decreasing proportion of 7th graders conceiving the unknown variable as a certain numeric compared with 6th graders means that the 7th graders have deeper recognition of unknown variables.
3. When encountering ‘simple’ two-variable linear equation virtual situations, some 7th graders can translate at least one condition into an equation. This result shows that the 7th graders have developed some ability to translate the conditions embedded in the virtual situation into some equations. But when the situation gets more complicated, due to conception immaturity of solving two equations simultaneously, the 7th graders either solve each equation independently, or mess up and tangle the clues of all the conditions together. Moreover, they would use the same symbol to stand for different variables.
C. The 8th graders’ development of the concept of unknown variables is maturing:
1. Most of the 8th graders can use the clues of all the conditions in the virtual situation in a sufficient way.
2. Only a few 8th graders would use the same symbol to stand for different variables during their problem-solving procedure. This result indicates that the ability to use the symbolic way to represent unknown variables is more mature among the 8th grade students.
3. When encountering two-variable linear equation virtual situations, the 8th graders can formulate two independent equations and solve them simultaneously. This result shows that the 8th grade students possess more profound skills to solve two-variable linear equations.

III. Proportion of answering questions correctly:
In general, for simpler virtual problems, there does not exist many differences among grades. Whereas, for the more difficult virtual problems, the 8th graders outperform the 7th graders, and the 7th graders, in turn, outdo the 6th grade students.

第一章 緒論 ……………………………………………………1
第一節 研究動機-----------------------------------1
第二節 研究目的-----------------------------------2
第三節 研究假設-----------------------------------3
第四節 名詞解釋-----------------------------------3
第二章 文獻探討 ………………………………………………5
第一節 概念之定義---------------------------------5
第二節 一元一次方程式之概念-----------------------6
第三節 一元一次方程式之相關研究-------------------8
第四節 情境測驗之探討----------------------------20
第三章 研究方法………………………………………………23
第一節 研究設計----------------------------------23
第二節 研究對象----------------------------------23
第三節 研究工具----------------------------------24
第四節 研究步驟----------------------------------24
第五節 研究限制----------------------------------25
第四章 結果與分析……………………………………………26
第一節 情境 1之分析--------------------------26
第二節 情境9-2之分析--------------------------44
第三節 情境4-2之分析--------------------------58
第四節 情境7-4之分析--------------------------71
第五節 情境5-1之分析--------------------------95
第六節 情境 13之分析-------------------------120
第七節 情境5-2之分析-------------------------150
第八節 情境 12之分析-------------------------182
第五章 結論與建議 …………………………………………205
第一節 結論-------------------------------------205
第二節 建議-------------------------------------218

參考文獻
一、中文部分 --------------------------------------220
二、英文部分 --------------------------------------223

【附錄次】

附錄一 ： 一元一次方程式情境文字題之概念細圖---------227
附錄二 ： 情境資料-樣本缺少之名單--------------------230
附錄三 ： 個人診斷表(樣本)---------------------------231

一、中文部分
王如敏 (2004)。國二學生解一元一次方程式錯誤類型分析研究。高雄師範大學數學系研究所碩士論文。
王瑞慶 (2003)。國小六年級學童在分數加減法問題的解題研究。屏東師範學院數理教育研究所碩士論文。
何俊青 (1995）。國民小學概念教學實驗研究。高雄師範大學教育學系碩士論文。
吳寶桂（2000）。青少年的數學方程式概念發展研究。NSC89-2511-S-110-002。
吳寶桂（2001）。青少年的數學概念學習研究-子計畫十一：青少年的數學方程式
概念發展研究(1/2)。NSC90-2521-S-110-002。
吳寶桂（2002）。青少年的數學概念學習研究-子計畫十一：青少年的數學方程式
概念發展研究(2/2)。NSC91-2521-S-110-002。
林秀吟（2004）。探討情境式S T S 理念教學對國小學童科學創造力之影響。
國立台北師範學院數理教育研究所碩士論文。
林敏雪（1998）。國中二年級數學方程式表徵及解題困難學生之研究。國立高
雄師範大學特殊教育學系研究所碩士論文。
林碧珍 (1990)。新竹師院輔導區國小數學科「怎樣解題」教材實施情況調查與
學習成效研究。新竹師院學報，第4期，363-391。
林麗雯（2000）。國中一年級學生學習二元一次聯立方程式的能力發展與其解題的錯誤分析。國立中山大學教育研究所碩士論文。
邱婉茹 (2001)。以一元一次方程式情境測驗偵測國中一年級學生在一元一次方
程式概念發展之可行性。國立中山大學教育研究所碩士論文。
南一書局（2005）。國民中學數學。第一冊第三章。
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研究所碩士論文。

涂金堂 (1999)。後設認知理論對數學解題教學的啟示。教育研究資訊，7 (1)，
122-137。
國立編譯館（1999）。國民中學數學。第一冊。台北：台灣書店。
張春興、林清山（1985）。教育心理學。台北：東華。
張景媛（1994）：數學文字題錯誤概念分析及學生建構數學概念的研究。國立台
灣師範大學教育心理與輔導系教育心理學報，第27 期，175-200 頁。
許淑玲（2004）。融入情境式的創造思考教學模式對學生學習成效之研究。
國立台北師範學院數理教育研究所碩士論文。
郭汾派 (1988)。國中生文字符號運算的錯誤型態。數學科教學輔導論文集，
11-128。
郭汾派、林光賢及林福來(1989)。國中生文字符號概念的發展。國科會專題研究計畫報告。NSC76-0111-S003-08、NSC77-0111-S003-05A。
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。
陳倩萍（2004）。從真實情境觀點探討學生之解題表現。國立新竹師範學院數理教育碩士班數學組碩士論文。
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。
陳瓊森 (1998)。從建構主義觀點談概念形成及概念轉變。國民中學學生概念
學習學術研討會。
游麗卿 (1999）。Vygotsky 社會文化歷史理論：搜集和分析教室社會溝通活動的對話及其脈絡探究概念發展。國教學報，11，230-258。
湯錦雲（2002）國小五年級學童分數概念與運算錯誤類型之研究。屏東師範學院數理教育研究所碩士論文。
黃郁雯 (2005)。情境式問題導向融入教學對國小六年級學童科學概念及科學態度之影響。國立台北師範學院自然科學教育研究所碩士論文。
黃寶彰（2003）。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。
楊其安 (1989）。利用臨床晤談探究國中學生對力學概念的另有架構。台灣教育學院科學教育研究所碩士論文。
葉玉嬌 (1974）。概念的學習。教育文粹，3，54-56。
趙 寧 (1998)。教學設計之呈現方式在概念學習上的應用。台北：師大書苑。
鄭昭明 (1993）。認知心理學。台北：桂冠圖書公司。
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，3，1491-1521。
謝夢珊（2000）：以不同符號表徵未知數對國二學生解方程式表現之探討。國立
台北師範學院數理教育研究所碩士論文。

二、英文部分
Anderson, C.W., & Smith, E.C. (1986). Childrens’ conceptions of light and color：
understand the role of unseen rayo (Research Series No. 166.) East Lansing：Michigan State University, Institute for Research on Teaching. (ERIC Document Reproduction Service No. ED2703817).
Behr, M., Erlwanger, S. & Nichols, E. (1976). How Children View Equalty Sentences.
Tallahassaa：Florida State University.
Bell, A. W. (1995). Purpose in school algebra . Journal of Mathematical Behavior,
14, 41-73
Berger, D. E.,＆Wilde, J. M.（1984）. Solving algebra word problem. Paper
presented at the Claremont Conference on Applied Cognitive Psychology.
（ERIC Document Reproduction Service No.242 550）
Bishop, J. W. (2001). Promoting algebraic reasoning using students’ thinking.
Mathematics Teaching in the Middle School, 6(7), 508-514.
Brown, D.E. (1988). Students Concept of Force：The importance of understanding
Newton’s third law. Paper presented at the Annual Meeting of the American
Association of Physics Teachers (Crystal City, 1988).
Burton, W. H., Kimball, R. B., & Wing, R. L.(1960). Education for
effective thinking. New York：Appleton Century Crofts.
Clement, J., Lochhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly, 88, 286-290.
Collis, K.F.(1975). The Development of Formal Reasoning. Newcastle. Australin：
University of Newcastle.
Duffiy, T.M., & Jonassen, D.H. (1991). Constructivist：New implication for
instructional technology. Educational Technology, 31(5), 7-11.

Empson, S. B. (2002). Organizing diversity in early fraction thinking. In B. Litwiller
& G. Bright (ED.), Making Sense of Fractions, Ratios, and Proportions
(pp.29-40). Reston, VA：NCTM.
Ernest, P. (2002). A semiotic perspective of mathematical activity. Data cited from pme website：http://www.math.uncc.edu/~sae/dg3/ernest.pdf.
Gilbert, J. K., & Watts, D. M. (1983). Concept misconception and alternative conceptions：Changing perspectives in science education. Studies in Science Education, 10, 61-98.
Hammer, D. I.(1957). Penetrations of mathematical problems by secondary school
students. (Doctoral Disseration, The University of Columbia).
Herscovics, N. , & Kieran, C.(1980). Constructing meaning for the concept of
equation, Mathematics Teacher, 11, 573-580
Ilany, B. S., & Shmueli, N(1998). ’Automatism’ in finding a ‘solution’ among junior
high school students. PME, 22(3), 56-63
Johnson, D. M., & Stratton, R. P.(1966). Evaluation of five methods of teaching
concepts. Journal of Educational Psychology, 57, 48-53.
Kieran, C.(1992). The learning and teaching of school algebra. In Handbook of Research on Mathematics Teaching and Learning by Grouws, D. A. :Macmillan,
390-419.
Kuchemann, D. (1981) Algebra. In K. M. Hart, M. L. Brown, D. E. Kuchemann
,D. Kerslake,G. Ruddock & M. McCartney( Eds.), Childen’s Understanding
of Mathematics：11-16，102-119. London：John Murrary.
Klausmeier, H. J.(1990). Conceptualizing, In Jone, B. F. & Idol, L.(Ed.), Dimensions of thinking and cognitive instruction (pp.93-138). Hillsdale, NJ： Hove and London.
Klausmeier, H. J., & Goodwin, W.(1987). Learning and Human Abilities：Educational
Psychology (4th Ed.) New York: Harper & Row.
Lankford, L. K.（1972）. Final report：Some computational Strategies of seventh grade pupils , Charlottesville, VA：University of Virginia（ ERIC Document Reproduction Service No. ED 069 496）
Leonard A. Marascuilo, Maryellen McSweeney (1977). Nonparametric and distribution
-free methods for the social sciences. Monterey, Calif. : Brooks/ Cole Pub. Co.
Mervis, C. B. & Rosch, E.(1981). Categorization of natural objects . In M. R.
Rosenzweig & L. W. Porter (Eds.), Annual review of psychology, 32, 89-115.
Muth, K .D .（1991）. Effects of cuing on middle-school students’ performance on arithmetic word problems containing extraneous information. Journal of Educational Psychology83(1),173-174.
Newell, A.,& Simon, H. A.（1972）. Human Problem Solving. Englewood
Cliff ,NJ：Prentice-Hall.
Paimter, R. R.（1989）. A comparison of the procedural error patterns, scores,and other variables, of selected groups of university and eight-grade students in Mississippi on a test involving arithmetic operation on fractions.（Doctoral dissertation, University of Southern Mississippi, 1988）
Polya,G.(1945). How to solve it. Princeton：Princeton University Press.
Rosnick, P. (1982). The Use of letters in Precalculus Algebra. (Doctoral Disseration,
The University of Massachusetts).
Sanford, V.(1927). The History and Significance of Certain Standard Problem in Algebra. New York：Bereau of Publication, Teachers College, Columbia Univ.
Simon, H.A.（1980）.Problem solving and education.In Tuma,D.T.＆Reif, F.
（Eds.）,Problem solving and education：Issues in teaching and research. Hillsdale,NJ：Erlbaum.
Thornton, S. (2001). New approaches to algebra: Have we missed the point?
Mathematics Teaching In the Middle School, 6(7), 388-392.
Thorpe, A.J. (1989). Algebra：What should we teach and should we teach it ?
In Wanger, S. and Kieran, C. (Eds). Research Issues in the Learning and Teaching
of Algebra (pp11-24). Reston, VA：NCTM.
Travis, B. P.（1981）. Error analysis in solving algebra word problem.
（ERIC Document Reproduction Service No.209 095）
Vergnaud, G. (1984).Understanding mathematics at secondary school level. In A. Bell
, B. Low, & J. Kilpatrick (Eds), Theory research & practice in mathematical education, 27-35. UK：Shell Center for Mathematical Education.
Verschaffel, L., B., & De Corte, E. (2000). Making sense of word problems.
Lisse, The Netherlands: Swets & Zeitlinger.
Vlassis, Joelle ( 2002 ). Educational Studies in Mathematics, 49(3), 341-359 (EJ656398)
Vygotsky, L. S. (1978). In M. Cole, V. Hohn-Steiner, S. Scribner, & E. Souberman (Eds.), Mind in society：The development of higher psychological processes. Cambridge：Harvard University Press.
Vygotsky, L. S. (1981). The genesis of higher mental functions. In James V.Wertsch
(Ed.), The concept of activity in Soviet psychology(pp.144-188). New York, NY
：M. E. Share.
Wagner, S. (1981). Conservation of equation and function under transformation of variable. Journal for Research in Mathematics Education, 12, 107-118.
Wollman, W. (1983). Determining the sources of error in a translation from sentence to
equation. Journal for Research in Mathematics Education, 4 (3), 169-181