臺灣博碩士論文加值系統

本研究之主要目的是從不同學習成就的觀點出發，以探討高中生在空間中的平面單元中之錯誤類型，及其解題歷程之錯誤原因，希望能提供教師往後實施補救教學或教學改進的參考。
本研究是藉由自編之「空間中的平面測驗」來調查與分析學生的錯誤類型，透過學生在試卷上的計算過程，進一步瞭解學生解題時的想法與犯錯原因。本研究之樣本為台中市某私立高中高二學生，研究者分別依學生學習成就高、中、低三群中各抽取一個班級，共70名學生為研究樣本。其研究方法為首先進行預試，接著根據預試結果，針對試題進行分析與修改，最後給予正式施測。根據施測結果可發現答題錯誤之類型與原因分別如下：
一、 錯誤類型
(一) 對空間的平面方程式與直線方程式產生混淆。
(二) 代數運算錯誤。
(三) 向量外積計算錯誤。
(四) 看錯題目或代錯數字所產生的錯誤。
(五) 利用題目的的條件或所得的訊息來猜測答案。
(六) 公式記錯或引用不適當的公式產生錯誤。
(七) 角度判斷錯誤。
(八) 解的個數判斷錯誤。
(九) 常數項位置判斷錯誤。
(十) 四則運算錯誤。
二、 錯誤原因
(一) 對平面的截距式的定義概念不清。
(二) 對空間中點到平面的距離公式與相關性質概念不清。
(三) 新知識與舊經驗的相互干擾。
(四) 先備知識的不足。
(五) 粗心疏忽或考慮不周。
(六) 計算能力不足與計算錯誤。
(七) 沒有依據的推論。

最後，根據研究結果，研究者提出對數學空間中平面教學的七點建議，與對未來研究的三點建議。

The purpose of the study aims to discuss the error types of Planes in Space sections of high students with different academic performance and to analyze the cause of the errors when they solve the plane problems, hoping to provide teachers with reference that can assist their remedial instructions and improve their teaching skills.
The methods of the study are to use self-designed Planes in Space tests to investigate and analyze the error types which students have with the calculation process on the test papers. To understand their thoughts when they solve the questions on the tests and the cause of errors, the subjects are seventy sophomores of certain private high school in Taichung City. The researcher drew 70 samples from three groups of students with high, middle, and low academic performance. The research procedure is the following: First, conduct the test in advance. Second, analyze and revise the questions according to the result. Finally, conduct the test officially. Based on the results, we can find that the error types and the cause of errors of answering the questions are the following:

1. Error types:
(1) misunderstanding about the planes and lines in space.
(2) errors in algebraic calculations.
(3) errors in cross product of the vecyors calculations.
(4) errors because of carelessness in understanding questions,writing the wrong numbers, or reading numbers.
(5) guessing answers based on the problems or the messages when solving problems.
(6) formula mistakes or the errors of using improper formulas.
(7) errors in misjudgement of angles.
(8) errors in misjudgement of numbers of answers.
(9) errors in misjudgement the position of the constant term in plane equation.
(10) errors in misjudgement of the four fundamental operations of arithmetic.

2. The cause of errors:
(1) unclear concepts about the definitions related to intercept form in plane.
(2) unclear concepts about the formulas related to distance from a point to the plane.
(3) interactive interruptions of new knowledge and old experience.
(4) insufficient prior knowledge.
(5) carelessness or negligence.
(6) Insufficient ability of computation and miscalculation.
(7) inference based on nothing.
At last, based on the findings, we propose seven recommendations to teach plane in space of math instructions and three suggestions to conduct the further research in the future.

第一章 緒論--------------------1
第一節 研究背景與動機--------------------1
第二節 研究目的與待答問題 --------------------2
第三節 名詞釋義--------------------3
第四節 研究限制--------------------3
第二章 文獻探討--------------------5
第一節 數學概念的學習--------------------5
第二節 解題歷程的探討--------------------7
第三節 錯誤類型的相關研究--------------------16
第四節 錯誤原因的相關研究 --------------------18
第五節 空間中的平面的相關研究--------------------21
第三章 研究方法--------------------23
第一節 研究對象--------------------23
第二節 研究方法--------------------23
第三節 研究工具--------------------23
第四節 實施步驟--------------------26
第五節 資料處理與統計--------------------29
第四章 研究結果與討論--------------------31
第一節 學生錯誤情形之分析統計--------------------31
第二節 學生錯誤類型與解題歷程之分析--------------------36
第三節 綜合討論--------------------62
第五章 結論與建議--------------------67
第一節 結論--------------------67
第二節 建議--------------------68
參考文獻--------------------69
附錄--------------------74

中文部分
1.田万海(1992)。數學教育學。浙江：教育出版社。
2.林清山、張景媛(1994)。國中生代數應用題教學策略效果之評估。國立台灣師範大學教育心裡與輔導學系教育心理學報，27，35-62頁。
3.林清岳(2001)。高中生無窮等比級數概念與運算錯誤類型之研究。國立高雄師範大學數學系研究所碩士論文。
4.胡炳生(1999)。數學解題思維方法。台北市：九章。
5.郭汾派(1991)。國中生文字符號運算的錯誤類型。數學科教學輔導論文集。國立台灣師範大學中等教育輔導委員會。
6.黃敏晃(1991)。淺談數學解題。教與學，23，2-15頁。
7.?平，馬再鳴(2002)。論數學概念學習。數學傳播，26(2)，89-95頁。
8.陳基正(2007)。綜合高中生在「空間中直線與平面」單元。國立高雄師範大學數學系研究所碩士論文。
9.張鳳燕(1991)。教導心理學微觀。師友月刊，284期，24-29頁。
10.張景媛(1995)。國中生建構幾何概念之研究暨統整式合作學習的幾乎教學策略效果之評估。國立台灣師範大學教育心理與輔導學系，教育心理學報，28期，99-144頁。
11.楊弢亮(1997)。中學數學教學法通論。九章出版社。
12.劉怡蘭(2001)。高雄地區高中生對數運算錯誤類型之研究。國立師範大學數學系碩士班論文。
13.劉錫麒(1997)。數學思考教學研究。台北：師大。
14.劉貞宜(2000)。數學資優生的解題歷程分析─以建中三位不同能力的數學資優生為例。國立台灣師範大學特殊教育研究所論文。
15.Schwarzenberger (1984)。 錯誤的重要性。數學圈，vol. 21，73-80頁。
英文部分
1.Booth, L. R. (1984). Child-method in secondary mathematics. Educational studies in Mathematics, 12, pp. 29-41.
2.Brown, J. S. , &Vanlehn, K. (1980). Repair Theory:A generative theory of bugs in procedural skills. Cognitive Science, 4, pp. 379-426.
3.Cyert, R. M. (1980). Problem solving and educational policy. In D. T. Tuma, & F. Reif (Eds.). Problem solving and education: Issues in teaching and research. Hillsdale, NJ: Erlbaum.
4.Davis, R. B. (1984). Learning mathematics: The Cognitive Science approach to mathematics education. Norwood, New Jersy: Albex Publishing Corporation.
5.Frazier, L. , & Rayner, K. (1982). Making and correcting errors during sentence comprehension: eye movements in the analysis of structurally ambiguous sentences. Cognitive Phychology, 14, pp. 178-210.
6.Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher& J.
7.Kathleen, T. T. (1987). Error Reduction Stractegies for whole number operations in grade four. Doctoral Dissertation, University of Brigham Young.
8.Kilpatrick, J. (1967). Problem Solving in Mathematics.Review of Eduational Research, 39, pp. 523-534.
9.Kilpatrick, J. (1985). A retrospective account of the past twenty-five years of research on teaching mathematical problem solving. In E. A. Silver(Ed.). Teaching and learning mathematical problem solving: Mulitiple research perspectives(pp. 1-15). Hilsdale, N. J. : Lawrence Erlbaum Associates.
10.Klausmeier, H. J. , Ghatala, E. S. , & Frayer, D. A. (1974). Conceptual learning and development. New York: Academic Press.
11.Lester, F. K. (1980). Problem solving: Is it a problem? In M. M. Lindquist(Ed.) , Selected issues in mathematics education, 36. Berkeley, C.A. : McCutchan.
12.Matz, M. (1982). Towards a process model for high school algebra errors. In Sleeman, D. & Brown, J. S.(Ed.), Intelligent Tutoring System. London: Academic Press.
13.Maurer, S. B. (1987). New knowledge about Errors and New Views about Learners: What They Mean to Educators and More Educators Would Like to Know. In A. H. Schoenfeld(Ed.) , Cognitive Science and Mathematics Education(pp. 165-187). N. J. : LEA.
14.Mayer, R. E. (1985). Educational Psychology: Cognition approach. NY: Freeman.

15.Mayer, R. E. (1992). Thinking, problem solving, cognition. New York: W. H. Freeman and Company Press.
16.Movshovitz-Hadar, N. , Zaslavsky, O. , &Inbar, S. (1987). An empircal classificational model for errors in high school mathematics. Journal for Research in Mathematics Education, 18, pp. 3-15.
17.National Council of Supervisors of Mathematics. (1977). Position paper on basic Mathematics skills. Washington, D. C. : National Institute of Education.
18.National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: the author.
19.Polya, G. (1945). How to solve it. New York: Doubleday.
20.Polya, G. (1957). How to solve it. (2nd ED.) New York: Doubleday.
21.Pimm, D. (1987). Speaking mathematically: communication in the mathematics classroom. London: Routledge & Kegan Paul.
22.Resnick, L. B. , Nesher, P. , Leonard, F. , Magone, M. , Omanson, S. , & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20(1), pp. 8-27.
23.Schoenfeld, A. H. (1985). Mathematical problem solving. London: Academic Press.
24.Skemp, R. R. (1971). The psychology of learning mathematics. Harmondsworth: Pengiin.
25.Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, pp. 20-26.
26.Stone, M. E. (1994). Teaching relationships between area and perimeter with the geometer’s sketchpad. The mathemetics Teacher, 87(8), pp. 590-594.
27.Von Glasersfeld, E. (1987). Learing as a constructive activity. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics. Hillsadale, NJ: Lawrence Erlbaum Associates.
28.Webb. Norman. L. (1975). An Exploration of Mathematical Problem Solving Processes. (ERIC Document Reproduction Service NO. ED106148)
29.Whitney (1985): Taking responsibility in school mathematics education in L. Streefland (Ed), Proceedings of the Ninth International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 123-141)