臺灣博碩士論文加值系統

本研究旨在探討不同城鄉和年級的國小高年級兒童在不同數字
型式和語意類型之比例問題的解題表現.本研究分為二個階段進行,第
一階段採測驗調查法,以南投縣(市)國小高年級兒童為研究母群體,
採分層叢集抽樣方式抽取有效樣本441人,以研究者自編<國小高年級
數學比例問題測驗>為工具,了解國小高年級兒童的解題表現;第二階段
以立意抽樣方式訪談10位不同解題層次的兒童,探討兒童的解題策略,
進而歸納有助於兒童解決比例問題的知識和能力基礎.
本研究測驗調查所得的資料,分別以平均數,標準差,百分比和單因
子變異數分析進行統計考驗;訪談部份則以錄音帶記錄,並加以轉譯分
析.本研究主要的結論如下:
一數字型式比例問題的皮題表現:
(1)對國小五年級兒童的困難度(由簡單到困難):第一式(C同時是B和A的
整數倍),第二式(B是A的整數倍),第三式(C是A的整數倍),第四式(C不
是A或B的整數倍).
(2)對國小六年級兒童的困難度(由簡單到困難):第一式,第二式和第三式
,第四式.
(3)解題通過率:第一層次:五年級23%,六年級24%;第二層次:五年級14%,
六年級14%;第三層次:五年級17%,六年級18%;第四層次:五年級5%,六
年級18%.
二,語意類型比例問題的解題表現:
(1)對國小五年級兒童的困難度(由簡單到困難):交換問題和組合問題,
密度問題,母子問題,伸縮問題.
(2)對國小六年級兒童的困難度(由簡單到困難):交換問題和組合問題,
密度問題和母子問題,伸縮問題.
三,兒童的解題策略僅有單價法,倍數法,疊加法,比例關係式和數量分
解等五種策略;解題錯誤者大都以絕對思考方式解題.
四,低層次到高層次兒童之數學知識和能力有其成長的軌跡:層次0的兒
童具備約分和擴分的計算能力,經由提示會用單價法;層次一的兒童
具備約分和擴分的計算能力,會用單價法;層次二的兒童會用單價法
,經由提示會用倍數法;層次三的兒童會用單價法和倍數法,比較習慣
用單價法;層次四的兒童會用單價法和倍數法,經由提示會用倍數法
解相似圖形題.
五,有助於兒童解比例問題之先備知識和能力:
(一)不受數字結構因素的影響:例如:(1)數字較大時解題錯誤,數字變
小則能發現倍數關戲而解題成功.(2)非為整數倍時解題錯誤,整數倍
時則解題成功.
(二)能正確計算多位數乘,除法問題和不受除數小於被除數的錯誤概念
影響.
(三)了解有理數概念的多義性:明確的以分數表示除法的結果和除不盡
的數.
(四)具有考量答案合理性的後設認知能力.
(五)以相對思考解決比例問題.
(六)了解單位量的意義.

The purpose of this study is to investigate urban students'' versus
metropolitan students''ability in solving number type and semantic type
mathematical problems. There were 441 subjects in this experiment. In
the first of twwo stages of this experiment, subjects were asked to answer a
questionnaire. In the second stage of this experiment, subjects were
interviewed, one on one, by a team of researchers.
Statistical analyses techniques such as mean, standard deviation,
percentage, and one-way ANOVA, were used to analyze the data. A tape
recorder wwas used during the interview with each subject in order to collect
data and to analyze protocl afterwards. Conclusions are stated as follows:
i.)Fifth graders'' ability in solving number type problems ranking from
the least difficult to the most difficult was:TypeII(C is integral multiple
of B and A),Type B(B is integral multiple of A),TypeIII(C is integral
multiple of A),Type IV(C is not integral multiple of B and A).
ii.)For Six grader,the level of difficulties ranking from the least
difficult to the most difficult was:Type I,Type II,Type III,Type IV problems.
iii.)Percentage of accuracy:For level III,Fifth graders were correct 17%,
while Sixth graders were correct 18%,For level IV,Fifth graders were
correct 5%,while Sixth graders were correct 18%.
II).Results for solving semantic type problems:
i.)Fifth graders'' ability in solving semantci type problems ranking from
the least difficult to the most difficult was:Exchange problem,
Associated problem, Dense problem, Part-Whole problem, Stretchers &
Shrinkers problem.
ii.)For Sixth grader, the level of difficulties ranking from the least
difficult to the most difficult was:Exchange problem and Associated
problem,Dense problem and Part-Whole problem, Stretchers &
Shrinkers problem.
III.)Five solving strategies:Single unit price, Multiplication, Repeated
addition, Scalar decomposition, and Proportion formula, were used in
this experiment. Researcher found that most of the mistakes were made
by those subjects who tended to apply additive analysis stategies when
they were solving the problem.
IV.)There is a recognizable trace in students'' development of mathematcial
ability ffrom a basic to an advanced. For example, students of level 0
after receiving guidance from the researcher, they were capable of
applying Single Unit Price method for a question. Students of level I
capable of applying Single Unit Price method for a question. Level II
students were capable of applying Single Unit Price method for a
question. After receiving guidance from the research, they were
capable of applying Multiplication for a question. Level III and Level
IV students were capable of applying Single Unit Price and
Multiplication in arriving answers.
V.)The prerequisites needed by students to accelerate their learning in
solving proportion problems:
i.)Students are not restricted by the structure and the component of the
problems. For example: a)they often successfully solve multiplicational
type problems when the numbers are small. However, they tend to make
more mistakes when the number in an problem are larger. b)The
number of correct answers in solving integral numbers is greater than
the number of correct in non-integral numbers.
ii.)Students can correctly calulate multiplication and division. This
success rate is not affected by whether the divider is greater the the
dividend.
iii.)Students can understand the rational number meaning:They are able
to accurately express remainder in a fractional form and quotient
meaning.
iv.)They possess the metacognitive ability of recognizing how an answer
was arrived.
v.)They are capaable of solving the proportion problems by comparative
analysis.
vi.)They understand the meaning of unit in a mathematical sense.

目 次
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…………………………………..3
第三節 名詞釋義………………………………………………..4
第四節 研究範圍與限制………………………………………..8
第二章 文獻探討…………………………………..9
第一節 比、比值與比例的意義………………………………….9
第二節 比例概念的知識和能力基礎……………………………11
第三節 影響兒童解比例問題成敗的因素………………………18
第四節 數學課程「比例」教材之內容分析……………………22
第五節 兒童比例概念發展之相關研究…………………………29
第三章 研究方法………………………………….42
第一節 研究架構………………………………………………..42
第二節 研究對象………………………………………………..43
第三節 研究工具………………………………………………..45
第四節 實施程序………………………………………………..53
第五節 資料處理與分析……………………………………….55
第四章 研究結果與討論………………………… 56
第一節 國小高年級兒童在不同數字型式的比例問題測驗
結果分析與討論……………………………………….56
第二節 國小高年級兒童在不同語意類型的比例問題測驗
結果分析與討論……………………………………….68
第三節 訪談分析的結果與討論……………………………….77
第四節 國小高年級兒童比例問題解題策略之分析與討論….92
第五章 結果與建議…………………………….101
第一節 結論…………………………………………………..102
第二節 建議…………………………………………………..106
參考書目…………………………………………..115
附錄
附錄一 預試試題--數字型式與問題類型細目表…………..124
附錄二 正式試題--數字型式與問題類型細目表………..125
附錄三 施測通知及指導語…………………………………..126
附錄四 國小高年級比例問題測驗…………………………..127
附錄五 解題策略分析表……………………………………..131
附錄六 訪談題目……………………………………………..133
附錄七 訪談原案……………………………………………..135
原案一………………………………………………..135
原案二………………………………………………..138
原案三………………………………………………..142
原案四………………………………………………..147
原案五………………………………………………..152
原案六………………………………………………..155
原案七………………………………………………..159
原案八………………………………………………..163
原案九………………………………………………..168
原案十………………………………………………..173
圖 目 次
圖3-1-1 研究架構圖……………………………………………..42
圖3-3-1 訪談流程圖……………………………………………..51
圖3-4-1 研究流程圖……………………………………………..54
表 目 次
表1-1-1 兒童解題層次劃分標準…………………………………7
表2-4-1 82年數學實驗課程「比例」教材之內容分析……..24
表2-4-2 82年數學實驗課程「比例」問題分析……………..25
表2-4-3 64年數學課程「比例」教材之內容分析…………..26
表2-4-4 64年數學課程「比例」問題分析…………………..27
表2-4-5 82年實驗課程和64年比例問題題數對照表……….28
表2-5-1 語意類型問題對照表………………………………….32
表2-5-2 Noelting比值概念的發展階段……………………..34
表3-2-1 紙筆測驗樣本人數分配表…………………………… 44
表3-2-2 訪談樣本人數分配表………………………………… 44
表3-3-1 國小高年級比例成就測驗試題項目分析之臨界比… 46
表3-3-2 國小高年級比例成就測驗試題分析結果…………… 47
表3-3-3 學童數學先備知識與本研究題目數字設計對照表… 48
表3-3-4 題目呈現方式一覽表………………………………….48
表3-3-5 半結構式訪談大綱…………………………………….52
表4-1-1 國小高年級兒童不同數字型式之比例問題各題答
對率……………………………………………………..57
表4-1-2 五、六年級學童各數字型式平均數與答對率對照
表………………………………………………………..59
表4-1-3 國小五、六年級兒童不同數字型式解題通過率…….60
表4-1-4 國小五年級兒童在不同數字型式之比例問題變異
數分析摘要表…………………………………………..61
表4-1-5 五年級兒童在不同數字型式之比例問題解題表現
事後比較摘要表………………………………………..61
表4-1-6 國小六年級兒童在不同數字型式之比例問題變異
數分析摘要表…………………………………………..61
表4-1-7 六年級兒童在不同數字型式之比例問題解題表現
事後比較摘要表……………………………………….62
表4-1-8 不同城鄉和年級的兒童在數字型式比例問題得分
之平均數和標準差…………………………………….63
表4-2-1 五、六年級兒童在不同語意類型之比例問題答對
率和平均數…………………………………………….69
表4-2-2 五、六年級兒童各語意類型平均數與答對率對照
表……………………………………………………….71
表4-2-3 五年級兒童在不同語意類型比例問題變異數分析
摘要表…………………………………………………..72
表4-2-4 五年級兒童在不同語意類型之比例問題解題表現
事後比較摘要表………………………………………..72
表4-2-5 六年級兒童在不同語意類型之比例問題變異數分
析摘要表………………………………………………..73
表4-2-6 六年級兒童不同語意類型之比例問題解題表現事
後比較摘要表……………………………………………73
表4-2-7 不同城鄉和年級的兒童在五種語意類型的比例問題
得分之平均數和標準差………………………………..74
表4-3-1 各層次受訪兒童解題成敗之因素…………………….78
表4-3-2 受訪兒童的知識和能力區分表……………………….80
表4-3-3 問題情境量數關係表………………………………….86
表4-4-1 各年級使用的解題策略統計表……………………….92
表4-4-2 各層次受訪兒童之解題策略………………………….93

參 考 書 目
一、中文部份：
台灣省國民小學教師研習會(民85)。數學實驗課程教師手冊第九冊。
台灣省國民小學教師研習會(民85)。數學實驗課程教師手冊第十冊。
台灣省國民小學教師研習會(民86)。數學實驗課程教師手冊第十一冊。
台灣省國民小學教師研習會(民87)。數學實驗課程教師手冊第十二冊。
江南青(民73)。比例概念的診斷與補救。師大數學研究所碩士論文。
何意中(民77)。國小三、四、五年級學生比例推理之研究，花蓮師院學報，2期，387-433。
李惠貞(民70 )。兒童比例概念的發展。花蓮師專學報，12期，1-25。
呂玉琴(民80)。國小學生的分數概念。國民教育，31，11.12期，10-16。
林福來(民73)。青少年的比例概念發展。科學教育月刊，73期，7-26。
林福來、林光賢、郭汾派(民74)。國中生的比例念發展，國科會專題研究計劃報告：NSC74-0111-S003-02。
林福來、林光賢、郭汾派(民75)。比和比例的了解層次與認知階段。中華民國七十四年科學教育學術研討會論文彙編，139-150。
林淑美(民79)。國中生反比例概念的了解，師大數學研究所碩士論文。
周立勳、劉祥通(民86)。國小兒童解題策略與數學能力之關係-以解「積木問題」為試探，國民教育研究學報第三期抽印本。
洪碧芳(民75)。國小學生對於「相當問題」之學習情況研究，師大數學研究所碩士論文。
陳英傑(民81)。台南師院學生比例概念的研究，台南師院學報，25期。319-343。
國立台灣師範大學科教中心(民73)。數學科概念發展研究。台北：師大科教中心。
國立編譯館(民75)。國民小學數學科教學指引第十一冊。台北：台灣書店。
國立編譯館(民75)。國民小學數學科教學指引第十二冊。台北：台灣書店。
郭素珍(民72)。比例概念暨學童認知發展關聯之研究。教育學院學報，8期，547-592。
張春興(民82)。教育心理學。東華書局。
教育部(民64)。國民小學課程標準。台北：正中書局。
教育部(民82)。國民小學課程標準。台捷國際文化實業股份有限公司。
甯自強(民83)。新課程對乘法啟蒙教材的處理。八十二學年度數學教育研討會論文暨會議實錄彙編。國立嘉義師範學院，163-173。
黃永和(民85)。國小實習教師數學學科知識之個案研究。新竹師院國民教育研究所碩士論文集，2期，pp.219-247。
劉秋木(民85)。國小數學科教學研究。台北：五南書局。
鄭英豪(民79)。比的參數與比例關係式的瞭解。國立師範大學數學研究所碩士論文。
盧欽銘(民71)。國小數學課程實驗成效評量研究。教育部。
潘耀圭、林軍治(民71)。國小高年級數學科計算能力分析研究。台灣省國小教師研習會。
魏金財(民81)。比例概念學習的電腦情境模擬設計。中華民國第八屆科學教育學術研討會論文彙編，137-163。
二、 英文部份：
Adi, H. & Pulos, S.(1980). Individual differences and formal operational performance of college students. Journal for Research in Mathematics Education. 11(2). 150-156.
Behr, Harel, Post, & Lesh(1992). Rational number, ratio, proportion. In Grouws, D.A.(ed.)：Handbook of research on Mathematics Teaching and Learning. pp.296-333. New York : MacMillan.
Capon, N. & Kuhn, K.(1979). Logical reasoning in the supermarket：Adult femals'' use of a proportional strategy in an everyday context. Developmental Psychology. 15(4). 450-452;6;544-573.
Fischbein, E., Pampu, I., and Manzat, I.(1970). Comparison of ratios and the chance concept in children. Child Development. 41, 377-389.
Greer, B.(1992).Multipilcation and Division as Models of Situations. In Grouws.D.A.(1992.ed.).Handbook of research on Mathematics Teaching and Learning. MacMilian.
Hart, K. M.(1981) . Children''s Understanding of Mathematics 11-16. John Murray Ltd.,London.
Hart, K. M.(1982a) . Children''s Reasoning on Ratio and Proportion Problems. Chelsea College,Univ. of London.
Karplus, R., Pulos, S. and Stage, E. K.(1980). Early Adolescents'' Structure of Proportional Reasoning. In Proceedings of the fourth International Conference for Psychology of Mathematics Education, Berkeley, California. pp.136-142.
Karplus, R., Pulos, S. and Stage, E. K.(1983a). Proportional reasoning of early adolescents, in R. Lesh and M. Landau(eds.). Acquisiton of Mathematics Concepts and Processes. Academic Press, New York.
Karplus, R., Pulos, S. and Stage, E. K.(1983b). Early adolescents'' proportional reasoning on "rate" problems''. Educational Studies in Mathematics. 14. 219-233。
Kieren, T. E. and Nelson, D.(1978). The operator construct of rational numbers in childhood and adolescence-an exploratory study. The Alberta Journal of Educational Research. 24(1). 22-30.
Kieren, T. E. and Sorthwell, B(1979). The development in children and adolescents of construct of rational numbers as operator. The Alberta Journal of Educational Research. 25(4). 234-247.
Kieren, T. E.(1980). Knowing Rational Numbers：Ideas and Symbols. In Selected Issues in Mathematics Education. Mary M. Lindquist ed. Chicage：National Society for the Study of Education. NCTM.
Kulm, G.(1984). The Classification of Problem-Solving Research Variables. In G. A. Goldin & C. E. McClintock (Eds.), Task variables in mathematical problem solving (pp.1-21). Philadelphia, Pennsylvania : The Franklin Institute Press.
Lachance, A., & Confrey, J.(1995). Introducing Fifth Graders to Decimal Notation Tlhrough Ratio and Proportion. ERIC Document Reproduction Service No. ED 389579.
Lamon, J.(1993). Ratio and Proportion：Connecting content and Children''s Thinking。Journal for Research Mathematics Education. 24(1). 41-61.
Lamon, J.(1997). Ratio and Proportion:Elementary Didactical Phenomenology. Chapter 8.
Lawson, A. E., Karplus, R., & Adi. H.(1978). The acquisition of propositional logic and formal operational schemata during the secondary school years. Journal of Research in Science Teaching. 15. 465-478.
Lesh, Behr, & Post(1989). Proportional Reasoning。James Hiebert & Merlyn Behr Eds. Number concepts and operations in the middle grades. NCTM.
Linn, M. C. and Pulos, S.(1983). Aptitude and experience influences on proportional reasoning during adolescence:focus on male-feamale differences. Journal for Research in Mathematics Education. 14(1). 30-46.
Lo, J. J. & Watanabe. T. (1997). Development Ratio and Proportion Schemes：A story of a fifth Grader. American Education Research Association.
Marcia, C., & Pulos, S.(1983). Aptitude and Experience Influences on Proportional Reasoning During Adolescence：Focus on Male-Female Differences. Journal for Research in Mathematics Education. 14(1).30-46.
Marshall, S. P. (1987). Understanding arithemetic story problems. ERIC Document Reproduction Service No. ED 279687.
Marshall, S. P., & Smith, J.D.(1987). Sex differences in learning mathematics：A longitudinal study with item and error analysis. Journal of Educational Psychology, 79, 372-383.
Moyer, J. C.,Sowder, L.,Threadgill-Sowder, J.,& Moyer, M. B.(1984b). Story problem formats：drawn versus verbal versus telegraphic. Journal for Research in Mathematical Education. 15(5), 342-351.
National Council of Teachers of Mathematics(1980). An agenda for action.Reston, VA: The Council.
National Council of Teachers of Mathematics(1989). Curriculum and Evaluation Standards for School Mathematics： Standard 5：Number and Number Relationships. Reston, VA: The Council.
Noelting, G.(1980a). The development of proportional reasoning and the ratio concept：Part I - Differentiation of stages''. Educational Studies in Mathematics. 11, 217-253.
Noelting, G.(1980b). The development of proportional reasoning and the ratio concept：Part II- Problem structure aat successive stages；problem solving strategies and the mechanism of adaptive restructuring''. Educational Studies in Mathematics. 11, 331-363.
Piaget, J., & Inhelder, B.(1958). The Growth of Logical Tlhinking from Childhood to Adolescence. New York:Basic Book.
Piaget, J. & Inhelder, B.(1969). The Psychology of the Child. New York:Basic Books.
Pulos, S., Karplus, R., and Stage, E. K.(1981). Generality of proportional reasoning in early adolescents：Content effects and individual differences. Journal of Early Adolescence. 1. 257-264.
Quintero, A. H. and Schwartz, J. L.(1982). The Development of the Concept of Ratio in Children, DBRE Working Paper Series(WP-15).
Ruply, W. H.(1981).The Effects of Numerical Characteristics on the Difficulty of Proportional Problems. Unpublished doctoral dissertation, University of California, Berkeley.
Scheffler, I. (1965).Conditions of knowledge: An introduction to epistemology and education In J.Hiebert (Ed) Conceptual and Procedural Knowledge in Mathematics: An introductory Analysis(pp.1-27).University of Delaware.
Stanic, G. M. A., & Kilpatrick, J. (1989). Historical perspectives on problem solving in the mathematics curriculum. In R. I. Chales & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp.1-22). Hillsdale, NJ: Erlbaum.
Steffe, L. P.(1988). Children''s construction of number sequences and multiplying schemes.In J.Hiebert & M.Behr (Eds). Number concepts and operations in the middle grades 2,119-140. Reston VA:National Council of Teachers of Mathematics.
Tourniaire, F., & Pulos, S.(1985). Proportional Reasoning：A Review of The Literature. Educational Studies in Mathematics. 16, 181-204.
Tourniaire, F.(1987). Proportions in elementary school. Educational Studies in Mathematics. 18(1). 400-412.
Vergnaud, G.(1983). Multiplicative structures.In R. Lesh and M. Landau(Eds). Acquisiton of Mathematics Concepts and Process. Academic Press.
Vergnaud, G.(1988). Multiplicative structures.In J. Hievert & M. Behr(Eds). Numger concepts and operations in the middle grades. National Council of Teachers of Mathematics and Lawrence Erlbaum Associates.