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1 同分母分數加減法的教學研究摘要 (2000) 80% N (2003)

2 (2003) (NAGB, 2002;, ; 2001;, 2002; Lesh, 1987) ( )( )() National Assessment of Educational Progress [NAEP](NAGB, 2002) (Mathematical Power) (Mathematical Abilities) (Content Strands) ()

3 NAEP 數力 (Mathematical Power) NAEP(NAGB, 2002) 1996 ( ) NAEP(NAGB, 2002) NAEP(NAGB, 2002)(Reasoning)(Connections) (Communication) National Council of Teachers of Mathematics[NCTM](2000) Principals and Standards for School Mathematics 數力 NAEP(NAGB, 2002) (Conceptual Understanding) (Procedural Knowledge) (Problem Solving) (1994) NAEP(NAGB, 2002) 數 NAEP(NAGB, 2002) () --

4 1. (2000) N-1-7: 1 ()(20 ) (<1)N-2-5: 1 3/4 是任何物件將它平分成四份, 其中三份和整體的比例關係 ; 分數概念也可以是所有物件的 3/4, 例如 3/4 個蛋糕 ( 含一個蛋糕平分成四八份, 其中的三六份 ) 3/4 盒雞蛋 ( 含一盒雞蛋有四八顆, 其中的三六顆 ) 等, 所 (class) 2. (Behr, Harel, Post. & Lesh, ;, 2002) ( ) Bruner,1966Lesh, Post, Behr,1987 Kaput,1987; Dufour-Janvier, Bednarz, & Belanger, 1987; Greeno, 1987; 1994 Lesh 1987 Lesh 1987 (2000) ( ;, 2002)

5 3/ /4 3/4 Cramer & Henry(2002) (Rational Number Project) (2001) Herman, Ilucova, Kremsova, Pribyl Ruppeldtova, Simpson, Stehlikova, Sulista(2004) 探究 19 個六到九年級的學生在解答 2/8+3/8 時, 一開始浮現在他們心中的是什麼研究發現大部份學生的心像是使用圓形的表徵 ( 參見圖二 ) 本研究 ( 李源順,2005) 對 390 名小三學生的實徵調查, 1/2+1/4 30.8% 30.1% (2000) (1991a, 1991b, 1993

6 () (2003) 圖二同分母分加法的表徵 (1987, 1988, 1989; Lukhele, Murray & Olivier, 1999) (1987, 1988, 1989) 3/8 +2/8 =71%11/5 3/5=55% Lukhele, Murray & Olivier(1999)95 7/8 +7/8 13.6% ( 2005)3/6+2/6 = 17/ /50 =65%

7 20% 3/6 +2/6 =2 +3 =5 3/6+2/6= =6.53/6 +2/6 = = 17 () (2002) (2002) (1998) (Fuson, ) 1. Fuson, Combine7 8 Change5 3 Compare3 6 Equalze7 8 Morales, Shute, Pelligrino(1985) (2003) (, 2001) (2005) 2. (, 2003) 2-a-2

8 ()()() (, 2003) (2005) ()() % 46% -- 96

9 四份中的三份的圖像表徵, 帶出 3/4 個蛋糕的符號表徵, 以及口語文字表徵的概念性知 1/4 3/4 7/ /5 20 3/5 23/514/ /25 511/ /8 46/8

10 18/1223/12 133/7 14/ =72 15 Q1~Q7 Q8~Q14 Q15 SPSS10.0 Split half Reliability (Spearman-Brown Formula) 數力

11 0%-10% T T T 58% 52% % 82% * 74% 83% * 36% 30% % 82% * 47% 79% * 55% 55% % 82% * 75% 84% * 60% 50% % 83% % 82% * P<.05 9%-33% 8%-9%32%-33% 14% 32%-33% 80% 82% 80% 50% 71%-75%

12 T Q14: 53/10 74/10 85% 50% 3.100* Q7: 32 又 81/91-5 Q12: 445/ /100 69% 91% * 60% 85% * Q15: 12/4+21/4 Q6: 8-8 又 1/2 Q15: 12/4+21/4 49% 82% * 44% 73% * 47% 79% * * P<.05 Q14 Q14 50% 46.7% 3.3% 84.8% 15.1% Q7Q12 Q15 Q7 90.9% 68.6% 8.6 Q % % 37.1% Q % 40% 52.7%

13 Q14 50%88%33% 85% 97% 12 Q6Q15 Q6 72.7% 概念性迷思 : 只用整數減去帶分數的整數部分 (8 2 又 1/8 =6 又 1/8), 佔 21.2 % 對照組學生的答對率為 43.8%, 主要錯誤型態為概念性迷思 : 也是只用整數減去帶分數的整數部分, 佔 34.4 Q % 6.1%6.1% 43.8% 21.9% 12.5% 80% 80% Q4 51/8 25/8 =Q68 21/8 Q102 32/ /8 Q1348/7913/79 79%80% Q4 15%-27% 3%-9%Q6 (8 21/8 = 61/8)21%-27%

14 Q1 90% 100% 94% Q2 63% 79% 88% Q3 70% 85% 88% Q4* 30% 67% 76% Q5 57% 88% 88% Q6* 23% 64% 73% Q7 53% 91% 85% Q8 63% 82% 82% Q9 Q10* Q11 Q12 Q13* Q14 63% 91% 94% 37% 79% 61% 57% 76% 85% 47% 85% 88% 50% 79% 79% 50% 88% 91% Q15 30% 82% 79% 本教學實驗之前, 本研究曾對學生的解題性知 Q10 18%-33% Q13 15%-18% 理連力 10 3/5 20 3/5( ) 13/5 = 2/5 5 2 ()5 5/53 3/5

15 2 10() 6()4() 2/5 (4 )4 13/5 2/5 T10 xx 3/5 Txx S(xx 13/5 = 2/5 ) T S1 3/52/5 5 2 ()5 5/5 3 3/52 S TS10( )6()4() T S T S4 T4 xx 4 S Txx 4 TS:() T4 2/52/5 4 T S /5 T2/5 T20 S5 T3/5 TS1S2S3 (S =88 4=2 2/5 S =2/52/5 S3 13/5 = 2/5 ) TS1 T S1: 3/5 T3/5 T8 S1 2/5 T2/5 T4 S14

16 TS1 S TS2 S4 S /5 T / S8 TS2 S T S8 T8 S1 TS3 STS3 S3 T3/5 S3 T3/ /52/5 S

17 80% NAEP(2002) () ()()() -- () ()

18 S 王瑞慶 (2002): 國小六年級學童在分數加減法問題的解題研究屏東 : 國立屏東師範學院碩士論文 ( 未出版 ) 民生報(2003) : 國中小數學難度要加深民生報,2003 年 02 月 23 日呂玉琴 (1991a) : 影響分數概念表現的因素臺北市立師範學院學報,4, 呂玉琴(1991b) : 分數概念 : 文獻探討臺北市立師範學院學報,4,587 呂玉琴(1993) : 影響分數二分之一概念的因素國民教育,33,2-11 李源順(2005) : 同分母真分數加減運算的教學建議臺灣數學教師 ( 電子 ) 期刊,(3), 周柏達 (1999): 國民小學數學科新課程實驗班與普通班分數學習表現之比較研究臺北市 : 臺北市立師範學院國民教育研究所碩士論文 ( 未出版 ) 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的分數概念新竹師院學報,4, 林福來(1994): 八十三年度基礎科目數學科試題研發工作計劃臺北市 : 大學入學考試中心張春興 (1989): 張氏心理學辭典臺北市 : 東華書局教育部 (2000): 國民中小學九年一貫數學學習領域課程暫行綱要臺北市 : 教育部教育部 (2003): 國民中小學九年一貫數學學習領域課程綱要臺北市 : 教育部陳竹村林淑君和陳俊瑜 (2001) 國小數學教材分析分數的數概念與運算臺北 : 國立教育研究院籌備處黃俊惟 (2002): 網路擬題練習在解分數加減法問題上之研究嘉義 : 國立嘉義大學碩士論文 ( 未出版 ) 楊壬孝(1987): 國中小學生分數概念的發展行政院國家科學委員會專題研究計劃成果報告 ( 編號 : NSC S ) 執行單位: 國立臺灣師範大學數學系楊壬孝 (1988): 國中小學生分數概念的發展行政院國家科學委員會專題研究計劃成果報告 ( 編號 : NSC S A) 執行單位: 國立臺灣師範大學數學系楊壬孝 (1989): 國中小學生分數概念的發展行政院國家科學委員會專題研究計劃成果報告 ( 編號 :NSC S A) 執行單位: 國立臺灣師範大學數學系蔣治邦 (1994) : 由表徵觀點探討新教材數與計算活動的設計出自臺灣省國民教師研習會 91994): 國民小學數學科新課程概說低年級臺北 : 臺灣省國民教師研習會謝堅蔣治邦和林淑娟 (2002): 國小數學教材分析整數的數量關係臺北 : 國立教育研究院籌備處龐嘉芬 (2001): 國小高年級學童分數概念與能力之研究屏東 : 國立屏東師範學院碩士論文 ( 未出版 ) Behr, M.J.; Harel, G.; Post. T.; & Lesh, R.(1992). Rational number, ratio, and, proportion. In Douglas A. Grouws(ed.) (1992). Handbook of research on mathematics teaching and learning (pp ). Macmillan publishing company,

19 New York. Bruner,J.S.(1966). Toward a theory of instruction. Cambridge, MA:Harvard University. Cramer, K.; Henry, A., (2002) Using manipulative models to build number sense for addition of Fractions. In National Council of Teachers of Mathematics (2002) Yearbook: Making Sense of Fractions, Ratios, and Proportions. (pp ). Reston, VA: National Council of Teachers of Mathematics. Dufour-Janvier, B.; Bednarz, N.; & Belanger, M. (1987).Pedagogical considerations concerning the problem of representations. In C. Janvier(Ed.), Problems of representation in the teaching and learning of mathematics (pp ). Hillsdale, NJ : Lawrence Erlbaum Associates. Fuson, K.C.(1992). Research on whole number addition and subtration. In D.A. Grouws(1992). Handbook of research on mathematics teaching and learning (pp ). Macmillan publishing company, New York. Greeno,J.G.(1987). Instructional representations based on research about understanding. In A.H. Schoenfeld (Ed.), Cognitive Science and Mathematics Education. Hillsdale. Herman, J.; Ilucova, L.; Kremsova, V.; Pribyl,J.; Ruppeldtova, J.; Simpson, A.; Stehlikova,N.; & Sulista, M.(2004). Images of fractions as processes and images of fractions in processes. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 2004, 4, Kaput, J. J. (1987). Representation systems and mathematics. In C. Janvier (Ed.). Problems of representation in the teaching and learning of mathematics (pp.19-26). Hillsdale, NJ: Erbium. Lesh, R.(1987). Representations and Translations among Representations in Mathematics Learning and Problem Solving. Problems of Representation in the Teaching and Learning of Mathematics.33~40. Lesh, R.; Post, T.; & Behr, M. (1987). Representation and translation among representation in mathematics learning and problem solving. In C. Janvier (Ed.), Problem of representation in teaching and learning of mathematics (pp.33-40). Hillsdale,NJ:Erlbaum. Lukhele, R. B.; Murray, H.; & Olivier, A.(1999). Learners understanding of the addition of fractions. Paper presented at the 5th Annual Congress of the Association for Mathematics Education of South Africa (AMESA), Port Elizabeth, 5-9 July Morales, R.V.; Shute, V.J.; & Pelligrino, J.W.(1985). Developmental difference in understanding and solving simple mathematics word problems. Cognition and Instruction, 2, National Assessment Governing Board(2002). Mathematics framework for the 2003 national assessment of educational progress. National Assessment Governing Board U.S. Department of Education National Council of Teachers of Mathematics(2000). Principal and Standards for school mathematics. Reston:VA: NCTM.

20 Teaching Research on Same Denominator Fraction Addition and Subtraction Yuan-Shun Lee Shin-Fub Yu Yung-Yu Lee Department of Mathematics and Computer Science Education,Taipei Municipal University of Education Abstract From literature review, we suggest that the learning of mathematics ability and mathematics power of same denominator fraction addition and subtraction can be help by using representation of conceptual communication, situation structure of fraction conception, contents of unit fraction, semantic structure of whole number addition and subtraction, and operation structure of arithmetic fundamental operation to design teaching materials. Quasi-experimental designs for research, research samples are 69 of grade four. Research data include teaching observation and tests, validity of teaching observation and tests are using expert validity and contents validity, reliability of tests is using split-half reliability, and teaching observation is using triangulation. There are good results in teaching experiment. We suggest that the teaching materials of same denominator fraction addition and subtraction can use our design to help students to connect, to reason and to communicate with relation concepts, to nurture conception understanding, procedure knowledge, and problem solving of students. The teaching action should thought students learning characteristic using multiple optimization teaching methods for cooperation solving, self solving, and communication with schoolmates. The assessment of teaching should use placement assessment, diagnostic assessment, formative assessment, and summative assessment to instantaneously understand and help students learning. Key words: same denominator fraction addition and subtraction, mathematics ability, athematics power

一個以學童數學認知為基礎的小學教師數學專業發展模式

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