本研究的目的旨在建立一份具有效度及信度的分數除法試題，並藉由選項分析及試題關聯結構分析來探討國小六年級學童在分數除法上的概念結構，根據研究結果做出以下結論：
一、依通過率由高到低排序，計算題型部分為整數除以整數、分數除以分數、整數除以分數及分數除以整數，文字題部分為等分除、包含除、逆運算、當量數、有餘數題型。
二、低分組學生除了在整數除以整數的計算題外，其他的計算題型及文字題皆表現不佳，而高分組學生除了有餘數的文字題外，其他的計算題型及文字題皆表現很好。
三、分數除法計算概念的教學，以「整數除以整數」-「分數除以分數」-「分數除以整數」-「整數除以分數」的順序來進行教學會比較符合學生的概念結構。
四、要進行分數除法等分除教學時，應先設計「分數除以整數」題型的教學再進行「整數除以整數」題型的教學。
五、進行分數除法倍數關係的教學時，可以先從「同分母且分子可被整除」的題型著手，再來再進行「整數除以分數」或「分數除以整數」的題型，最後再進行「分數除以分數(異分母)」的教學。
六、在進行逆運算題型教學時可以先從面積逆運算的問題作引入，而對於求基準量的問題，學生們的概念是獨立的，也就是說教學時對於這個概念可以獨立教學。
七、在當量除的題型中，求單位當量的題型與求單量數的題型對學生而言是互有關聯的，所以老師在教學時可以同時講解兩種題型，加強學生對這兩題型的關聯性。
八、要進行「分數除法文字題(有餘數)」題型的教學時，教學順序依序為「計算概念」-「分數除法文字題(等分除)」-「分數除法文字題(當量除)」-「分數除法文字題(有餘數)」。
九、文字題教學應先由包含除的題型教起，再來是等分除，再來是逆運算， 然後是當量除的題型，最後才是有餘數的題型。

Establishing the reliability and validity of factional division examination questions is the purpose of the study. The factional division conception of the sixth grade students in elementary schools are discussed with the test of Options Analysis and Item Relational Structure Analysis. According to the result of the study, there are some conclusions.
1.The arrangement of the pass rate is from high to low. The whole questions are divided into two parts. One is calculation problem which is composed of number division, fractions divided by fractions (identical denominators), fractions divided by fractions (different denominators), integrals divided by fractions and fractions divided by integrals. The other is word problem that consists of partitive division, quotative division, division as the inverse of multiplication , extended partitive division and the questions of remainder.
2.The low part students didn’t do well on the word problem and the calculation problem except number division. However, the high part students did well on the word problem and the calculation problem except remainder questions.
3.The best order of factional division teaching that the learners may comprehend easily is from number division, fractions divided by fractions (identical denominators and different denominators), fractions divided by integrals to integrals divided by fractions.
4.The teaching of number division should be introduced before the unit of fractions divided by integrals.
5.The questions of the same denominator and numerator which can be divided with no remainder are regarded as the first method of the introduction of factional division multiple relations. The next one is integrals divided by fractions or fractions divided by integrals. The last is fractions divided by fractions (different denominators).
6.The teaching of division as the inverse of multiplication can be understood by the measure of area teaching. Not only the learners but also the training is independent for the baseline.
7.It’s relative for the students to learn basic quantity and ratio quantityfor students about extended partitive division. Therefore, teachers could introduce them at the same time. It will reinforce the students’ comprehension.
8.The word problem of factional division with remainder which can be taught hsd four methods. The best order is computing concepts, partitive division, extended partitive division and the factional division with remainder.
9.The first item of the word problem teaching is quotative division. The second is partitive division. The third is division as the inverse of multiplication. The fourth is extended partitive division. And then the last one is remainder questions.

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 名詞釋義 3
第四節 研究範圍與限制 5
第二章 文獻探討 7
第一節 分數的意義 7
第二節 分數除法相關研究 10
第三節 國小分數除法教材分析 15
第四節 試題關聯結構分析 21
第三章 研究方法 33
第一節 研究架構 33
第二節 研究對象 34
第三節 研究工具 34
第四節 研究流程 41
第五節 資料處理 42
第四章 研究結果與分析 43
第一節 試題性質分析 43
第二節 選項分析 47
第二節 選項分析 48
第三節 試題關聯結構分析 70
第五章 結論與建議 87
第一節 結論 87
第二節 建議 90
參考文獻 92
中文部分 92
英文部分 93
附錄 96
附錄一 國小學童分數除法試題 96
附錄二 試題檢核表 100
附錄三 國小六年級學童分數除法概念試題專家效度調查問卷 101

中文部分
幼獅數學大辭典編輯小組（1983）。幼獅數學大辭典（下卷）。臺北：幼獅文化事業公司。
朱建正（1997）。國小數學課程的數學理論基礎。國科會成果報告。未出版。
呂玉琴 ( 1991 ) 。分數概念：文獻探討。國立臺北師範學院學報，第4期，753-606。
呂玉琴（1996）。國小教師的分數知識。臺北師院學報，9，427-460。
呂玉琴（1998）。分數的四則運算與等值分數的設計。國立嘉義師範學院86 學年度數學教育研討會論文之10，165- 181
林碧珍（1990）：從圖形表徵與符號表徵之間的轉換探討國小學生分數概念。新竹師院學報，4，295-347。
許天維（1995）。數學試題分析法－以「八十一學年度國民教育階段國小數科學生基本學習成就評量」的分析為例。高雄：大漢唐有限公司。
教育部（2003）：國民中小學九年一貫課程綱要--數學學習領域。教育部。
楊壬孝（1988）。國中小學分數概念的發展。行政院國家科學委員會專題研究計畫成果報告（編號：NSC-77-0111-S-003-09A）。執行單位：國立臺灣師範大學數學系。
楊瑞智(2000)。探究師院生之分數基本概念及分數概念的課室教學。臺北市立師範學院學報，31，357-382。
鄭昭明（1993）。認知心理學-理論與實踐。臺北： 桂冠。
劉秋木（1996）。國小數學科教學研究。臺北： 五南圖書出版公司。
劉祥通、周立勳（2001）。發展國小教師數學教學之佈題能力：以分數乘除法教學為例。科學教育學刊，第九卷第一期，15- 34。
簡茂發、劉湘川、許天維、林原宏(1995)：試題關聯結構分析法在國小高年級學生乘除文字題列式之分析研究。測驗年刊，42，113-156。
英文部分
Airasian, P. W. & Bart, W. M.(1973). Ordering theory: A new and useful measurementmodel. Journal of Educational Technology, Vol.5,pp.56-60.
Aksu , M. (1997). Student performance in dealing with fractions. The Journal of
educational research 90(6), 375-380 .
Ashlock, R. D. (1990). Error patterns in computation. New York:Macmillan.
Barash, A., ＆ Klein, R. (1996). Seventh grades students’ algorithmic intuitive and
formal knowledge of multiplication and division of non negative rational
numbers. In L. Puig＆ A. Gutierrez.(Eds.), Proceedings of the 20th conference of the International Group for the Psychology of Mathematics Education(Vol. 2.
pp.35-42). Valencia. Spain:University of Valencia.
Behr,M.J.& Post,T.R.(1988).Teaching rational number and decimal concepts. In T. R.Post (Eds.),Teaching mathematics in grades K-8,190-229. Boston, MA: Allyn and Bacon.
Behr, M.J., Harel, G., Post, T., & Lesh, R.（1992）. Rational number, ratio, and proportion. In D.A. Grouws（ED）, Handbook of Research on Mathematics Teaching and Learning, 296-333.
Brueckner,L.J.& Melby,E.O.(1931). Diagnostic and Remedial Teaching, Boston: Houghton Miffin Co.
Corwin,R.B.,Russell,S.J.,& Tierney,C.C.(1990).Seeing Fractions: A unit for the upper elementary grades. Developed by TERC (Technical Education Research Centers,Inc.) for the California Department of Education.
Cummins, D. D.（1991）. Children’s Interpretations of Arithmetic Word Problems.
Cognition and Instruction, 8(3), 261-289.
Dickson, L., Brown, M., ＆ Gibson, O. (1984). Child Learning Mathematics: A
Teacher’s Guide to Recent Research (pp.274-298).Oxford, Great Britain,
England: Schools Council Publications.
Fendel, D. M. (1987). Understanding the structure of elementary school mathematics.
Newton, MA: Allyn ＆ Bacon.
Fischbein, E., Deri, M., Nello, M., ＆ Marino, M. (1985). The role of implicit
models in solving verbal problems in multiplication and division. Journal for
research in Mathematics Education, 16, 3-17.
Greer, B. (1992). Multiplication and division as models of situations.In D. A. Grouws (Ed.),Handbook of research on mathematics teaching and learning, 276 – 295. MacmillanPublishing Company.
Hart, K. M. (Ed.). (1981). Children’s understanding of Mathematics, 11-16. London:
John Murray.
Hopkins, K. D. (1998). Educational and psychological measurement and evaluation (8th ed.). MA: Allyn & Bacon
Kieren,T.E.(ed).(1980).The Rational Number Cunstruct-its Elements and Mechanisms,7-15.
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）.
Painter,R.R.（1989）.A comparsion of the procedural error patterns,scores,and other variables,of select ed groups of university and eight-grade studentd in Mississippi on a test involving arithmetic operation on fractions.（ Doctoraldissertation,University of Southern Mississppi,1988）
Payne, J. N. (1976). Review of research on fraction. In R. Lesh(Ed.).Number and
measurement(pp.145-187). Athens: University of Georgia.
Sinicrope, R., Mick, H. W., ＆Kolb, J. R. (2002). Interpretations of Fraction Division.
InB. Litwiller & G. Bright (Ed.), Making Sense of Fractions, Ratios, and
Proportions(pp.153-161). Reston,VA: NCTM.
Van Hiele, P. M.(1986). Structure and insight: A theory of mathematics education.Orlando, FL:Academic Press.
Vergnaud, G (1983). Multiplicative Structures. In R. Lesh & M. Landau (Eds.). Acquisition of mathematics concepts and procedures (pp. 127-174). New York : Academic Press.