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五年級學童在不同題目表徵之未知數解題表現─基於概念結構的探討

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150

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2( )

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Performance of problem solving with Varied

Representation for Fifth Graders based on Concept

structure.

Abstract

The purpose of this study was to explore the problem-solving performance of 150 fifth graders of elementary school on the unknown number word problem represented formats . The tests of which three forms of testpaper (word problem , line segment drawing , picture problem) were designed by the researcher .

Moreover, this study was to analyze the traits and differences of students’ problem solving performance on the known number word problem represented formats by using the approach of concept advanced interpretive structural modeling (CAISM).

The conclusions as follows.

1. There were significant differences of the fifth graders performance on the problem-solving of the unknown number at the basis of different word problem represented formats. Among them, the students’ problem solving performances on ‘line segment drawing’ and ‘picture problem’ were significantly better than that of ‘word problem.’

2. The same examinee got the same score on the three tests at the basis of different word problem represented formats. CAISM could show that the stages of concepts, proficiency of concepts, and indications among concepts varied with representation of problems, and CAISM also indicates word problem represented formats might affect the master of

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concepts and the problem solving performance.

3. The students’ proficiency of the unknown number word problem of different word problem represented formats might be investigated because CAISM revealed there were certain stages of students’ concepts of word problems and sequences and directions among concepts. Finally, this study was to suggest that teachers could help students understand the semantics of world problems in order to enhance their performance by using ‘line segment’ or ‘picture problem’ when displaying a problem. Besides, teachers could group students based their proficiencies on different word problem represented formats so that teachers could conduct remedial teaching to those who did not master the concepts.

Keyword: concept advanced interpretive structural modeling (CAISM),

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...1 ...1 ...5 ...5 ...7 ...7 ... 10 ... 16 ... 19 ... 23 ... 23 ... 24 ... 24 ... 34 ... 35 ... 37 ... 37 ... 40 ... 67

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... 73 ... 73 ... 74 ... 75 ... 77 ... 77 ... 81 ... 85 ... 85 ... 89 ... 92

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3-2-1 ... 24 3-3-1 ... 25 3-3-2 ... 27 3-3-3 ... 27 3-3-4 ... 28 3-3-5 ... 29 3-3-6 ... 29 3-3-7 ... 31 3-3-8 ... 32 4-1-1 ... 37 4-1-2 ... 38 4-1-3 ... 38 4-2-1 ... 40 4-2-2 ... 47 4-2-3 ... 48 4-2-4 ... 56 4-2-5 ... 57 4-2-6 ... 65 4-3-1 ... 67 4-3-2 ... 71

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3-3-1 ... 23 3-4-1 ... 34 4-2-1 S24 ... 41 4-2-2 S24 ... 41 4-2-3 S24 ... 41 4-2-4 S36 ... 42 4-2-5 S36 ... 42 4-2-6 S36 ... 42 4-2-7 S89 ... 43 4-2-8 S89 ... 43 4-2-9 S89 ... 44 4-2-10 S123 ... 44 4-2-11 S123 ... 45 4-2-12 S123 ... 45 4-2-13 S138 ... 46 4-2-14 S138 ... 46 4-2-15 S138 ... 46 4-2-16 S12 ... 49 4-2-17 S12 ... 49 4-2-18 S12 ... 50 4-2-19 S37 ... 50 4-2-20 S37 ... 51 4-2-21 S37 ... 51 4-2-22 S57 ... 52

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4-2-23 S57 ... 52 4-2-24 S57 ... 52 4-2-25 S69 ... 53 4-2-26 S69 ... 53 4-2-27 S69 ... 53 4-2-28 S90 ... 54 4-2-29 S90 ... 54 4-2-30 S90 ... 55 4-2-31 S17 ... 58 4-2-32 S17 ... 58 4-2-33 S17 ... 58 4-2-34 S74 ... 59 4-2-35 S74 ... 59 4-2-36 S74 ... 60 4-2-37 S93 ... 60 4-2-38 S93 ... 61 4-2-39 S93 ... 61 4-2-40 S103 ... 62 4-2-41 S103 ... 62 4-2-42 S103 ... 62 4-2-43 S143 ... 63 4-2-44 S143 ... 63 4-2-45 S143 ... 64 4-3-1 S34 ... 68 4-3-2 S34 ... 68 4-3-3 S34 ... 68

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4-3-4 S47 ... 69 4-3-5 S47 ... 69 4-3-6 S47 ... 69 4-3-7 S92 ... 70 4-3-8 S92 ... 70 4-3-9 S92 ... 70

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( 2008) (1994) 82

National Council of Teachers of Mathematics [NTCM] 2000

( 1998 1988

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(representation processes) ( ) (solution processes)( 1995)

( 1990 1990 Lewis & Mayer, 1987) ( 1991 1991) ( 1990 1992 1991 Lewis,1989) ( 2003 2007 2010 2000) ( 1989 1991) ( 1998 1990) ( 2002)

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(2008) 82 A ( 2009) ( 2001) ( 2005)

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(classical test theory) (knowledge space theory) (pathfinder) (concept mapping)

(item relational structure, IRS) (ordering theory, OT) (interpretive structural modeling, ISM)

(1994) Warfield (1976) ISM

Lin, Hung and

Huang(2006) (concept

advanced interpretive structural modeling, CAISM)

(

2008 2008

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(Mayer,1992) (representation) ( ) ( 1994) ( ) ( )

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( )

Lin, Hung and Huang (2006) (concept advanced interpretive structural modeling, CAISM)

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( 2008)

NCTM 1980 (problem solving) NCTM 2000

(Principles and Standard for School Mathematics)

Lester(1980)

Kilpatrick(1985)

( 1994)

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( )

( )

(1987)

Mayer(1992) 1. (give stage)2.

(goal stage)3. (obstacles)

(1993) (1996)

(2008)

Polya

Polya(1945) (How to Solve It)

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( ) ( ) ( ) Lester Lester(1980) ( ) (problem awareness) ( ) (problem comprehension) ( ) (goal analysis) ( ) (plan developmant) ( ) (plan implementation)

( ) (procedures and solution evaluation)

Schoenfeld Schoenfeld(1985) ( ) (reading) ( ) (analysis) ( ) (exploration) ( ) (planning) ( ) (implementation) ( ) (verification)

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Mayer Mayer(1992) ( ) (problem representation) 1. (problem translation) 2. (problem integration) ( ) (problem solving)

1. (solution planning and monitoring)

2. (solution execution)

( 1990 1990 Lewis & Mayer,1987)

( 1989 1994 2001 Hegarty,

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( 1989) ( 1994) ( 1995) (Hiebert Carpenter 1992) ( 2005) ( 1995 1994) Bruner(1966) ( ) ( )

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( )

( )

( 2001)

Lesh, Post and Behr(1987)

( ) real-world situations

( ) manipulative aids

( ) static pictorial

( ) spoken symbols

( ) written symbols 80% 2x+y=76

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( 1990 1991) Hart and Sinkinson(1987)

Levin(1981) 4 5 4 5 4 5 1993 10 10 10 ) 23 10 10

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1 1 10 ( 32 15 y 28 32 1993 ( ) 25-10=? 6+7=? ( ) 30-3=? ( ) × ÷ - 18-12=? 10 15 … 10 25 18 12

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(( 2002)

( 1990 2000 2002)

Sowder, L. and Threadgill-Sowder, J. (1982) ( )

( 2002)

Moyer, J. C., Sowder, L., Sowder, J. T. and Moyer, M. B. (1984)

( ) (1997) ( ) (2000) ( ) ( ) (2003)

(28)

( ) (2007) (2010) ( )

(algebra) al-jabr ( 1997) ( 1995) 300 Diophantus

(29)

Usiskin(1988) ( ) 3+7=7+3 a+b=b+a ( ) x+3=8, x+3-3=8-3, x=5 ( ) = ( ) a(b+c) ab+ac Kieran(1992)

( ) (rhetorical algebra stage)

( ) (syncopated algebra stage)

( ) (symbolized algebra stage)

Sfard(1995)

Macgregor and Stacey(1999)

(2001)

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2008 82 : x y ( ) ( 0

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Warfield(1976,1982) (interpretive structural modeling, ISM)

(2005)

ISM

Lin, Hung and Huang (2006) ISM

CAISM (concept

vector matching)

(individualized concept hierarchy structure)

( 2010)

( 2008 Lin, Hung, & Huang, 2006)

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(2008)

(2008)

(2009)

CAISM

(2011)

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(34)
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3-1-1

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86 150 3-2-1 3-2-1 1 17 14 2 15 14 3 16 14 4 15 15 5 16 14 79 71

3-1-1 ( ) 40

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3-3-1 3-3-1 1 1 11 15 2 2 4 12 3 3 14 5 4 4 8 7 5 5 10 14 6 6 13 11 7 7 3 6 8 8 15 8 9 9 2 3 10 10 5 2 11 11 7 16 12 12 12 13 13 13 9 1 14 14 16 10 15 15 6 9 16 16 1 4 ( )

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( ) ( ) SPSS 27% 27% 5 15

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A 3-3-2 3-3-3 3-3-4 1 0 3-3-2 1 2 3 4 3-3-3 (7) (9) (13) (3) (11) (12) (16) (1) (4) (6) (10) (14) (2) (5) (8) (15)

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3-3-4 1 2 3 4 1 0 1 0 0 2 0 0 0 1 3 0 0 1 0 4 0 1 0 0 5 0 0 0 1 6 0 1 0 0 7 1 0 0 0 8 0 0 0 1 9 1 0 0 0 10 0 1 0 0 11 0 0 1 0 12 0 0 1 0 13 1 0 0 0 14 0 1 0 0 15 0 0 0 1 16 0 0 1 0 0 1 1 0 16 ( ) 1 2 6 3-3-5

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3-3-5 31 31 33 25 25 23 26 26 28 82 82 84 27% (PH) (PH) P=(PH+PL)/2 D=(PH-PL) 3-3-6 3-3-6 P D=PH PL 1 .73 .80 .84 .54 .41 .32 .64*** .67*** .61*** 2 .69 .80 .89 .62 .41 .23 .82*** .64*** .68*** 3 .69 .50 .82 .62 1 .36 .82*** .77*** .57*** 4 .67 .84 .77 .67 .32 .46 .84*** .63*** .48*** 5 .75 .78 .77 .50 .45 .46 .85*** .72*** .74*** 6 .61 .75 .68 .79 .50 .65 .75*** .77*** .80*** 7 .77 .87 .73 .46 .27 .55 .86*** .66*** .76*** 8 .71 .53 .59 .58 .95 .82 .69*** .80*** .61*** 9 .65 .87 .66 .71 .27 .68 .84*** .72*** .79*** 10 .73 .84 .68 .54 .32 .64 .82*** .69*** .77*** 11 .67 .50 .59 .67 1 .82 .77*** .79*** .74*** )

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P D=PH PL 12 .69 .55 .57 .62 .91 .86 .77*** .73*** .79*** 13 .65 .62 .64 .71 .77 .73 .75*** .62*** .82*** 14 .61 .82 .57 .79 .36 .86 .76*** .74*** .77*** 15 .61 .57 .57 .79 .86 .86 .77*** .78*** .78*** 16 .63 .59 .57 .75 .82 .86 .56*** .75*** .71*** ***p<.001 .2 .8 .25 .3 ( 2009) 3-3-6 .5 .89 .25 .23 D .25 .3 ( ) .8 ( 2009) 3-3-7 Cronbach's .951 Cronbach's .932 Cronbach's .933 Cronbach's

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3-3-7 Cronbach's 1 .951 .929 .931 2 .947 .929 .930 3 .947 .926 .932 4 .946 .930 .935 5 .946 .927 .928 6 .948 .926 .926 7 .946 .929 .927 8 .950 .925 .933 9 .946 .928 .926 10 .947 .929 .927 11 .948 .926 .928 12 .948 .928 .926 13 .948 .932 .925 14 .948 .927 .927 15 .948 .926 .926 16 .953 .927 .929 ( ) 3-3-3 Cronbach's 16

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( ) 159 150 3-3-8 .50 .82 D .25 .3 3-3-8 P D=PH PL 1 .60 .72 .80 .80 .57 .70 .74*** .83*** .69*** 2 .56 .66 .78 .89 .69 .66 .78*** .74*** .56*** 3 .59 .65 .80 .82 .71 .70 .86*** .75*** .69*** 4 .54 .79 .80 .93 .43 .70 .75*** .77*** .68*** 5 .66 .69 .56 .69 .62 .34 .85*** .77*** .81*** 6 .56 .74 .55 .89 .52 .33 .68*** .78*** .84*** 7 .67 .79 .54 .67 .43 .30 .87*** .78*** .84*** 8 .59 .59 .53 .82 .83 .29 .71*** .73*** .83*** 9 .62 .76 .56 .76 .48 .34 .89*** .76*** .77*** 10 .65 .82 .57 .71 .36 .36 .83*** .79*** .82*** 11 .58 .63 .53 .84 .74 .29 .80*** .73*** .80*** 12 .57 .59 .51 .87 .83 .27 .83*** .73*** .72*** 13 .61 .66 .57 .78 .69 .36 .84*** .75*** .75*** 14 .60 .74 .55 .80 .52 .33 .82*** .79*** .80*** 15 .61 .74 .53 .78 .52 .29 .80*** .77*** .84*** 16 .62 .74 .54 .76 .52 .30 .81*** .74*** .80***

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( )

Cronbach's .962

Cronbach's 0.950 Cronbach's .959

( )

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CAISM

3-4-1

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SPSS 12.0 (

2009)

SPSS12.0

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SPSS 4-1-1 4-1-1 16 11.59 4.51 16 12.64 4.61 16 12.34 4.52 4-1-1

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4-1-2 4-1-2 SS df MS F 88.351 2 44.176 6444.944 149 43.255 2787.649 298 9.355 9320.944 449 4.722 p .05 4-1-2 (F 4.722 p .05) LSD 4-1-3 4-1-3 x 11.59 x 12.64 x 12.34 --- --- --- -1.05 --- --- -.75 -.30n.s. --- p .05 p .01 4-1-3 (p .01) (p .05)

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( 2004 1996 Webb Sherrill, 1974)

( 2006

1990 1992 Lewis, 1989)

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3-3-4 .54 SPSS 27% 27% 5 15 S24 S36 S89 S123 S138 4-2-1 4-2-1 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 S24 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 15 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 S36 12 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 )

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16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 S89 9 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 15 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 13 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 S123 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 10 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 S138 10 0 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 S24 4-2-1 4-2-3 S24 4-2-1 S24 4-2-2 S24 4-2-3 S24 4-2-1 4-2-3 S24 1~4

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1 S24 S36 4-2-4 4-2-6 S36 4-2-4 S36 4-2-5 S36 4-2-6 S36

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1 2( ) 3 4( ) 2 3 4 1 S36 1 2 3 4 1 S36 3 2 .19 S36 2 S89 4-2-7 4-2-9 S89 4-2-7 S89 4-2-8 S89

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4-2-9 S89 4-2-7 4-2-9 S89 1~4 1 2 4 1 3 S89 S123 4-2-10 4-2-12 S89 4-2-10 S123

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4-2-11 S123 4-2-12 S123 4-2-10 4-2-12 S123 2 4 S123 3 2 4 3 S123 1~4 1 S123 2 3 1 S138 4-2-13 4-2-15 S138

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4-2-13 S138 4-2-14 S138 4-2-15 S138 4-2-13 4-2-15 S138 1 2( ) 3 4( ) 1 1 1 2 3 4 1 S138 1

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S 36 3 4-2-2 S123 2 4-2-2 1 2 3 4 16 1 1 1 1 1 16 1 1 1 1 1 S24 16 1 1 1 1 1 15 .59 .59 .62 .62 .61 15 .53 .66 .62 .62 .61 S36 12 .59 .47 .62 .56 .56 16 1 1 1 1 1 15 .59 .66 .56 .62 .61 S89 9 .47 .53 .50 .56 .52 15 .59 .66 .56 .62 .61 13 .59 .66 .50 .56 .58 S123 16 1 1 1 1 1 15 .59 .59 .62 .62 .61 10 .47 .53 .56 .56 .53 S138 10 .59 .53 .56 .44 .53

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S36 B C 1 .06 S24 S36 1 2 3 S 89 1 2 2 S123 2 3 1 S138 1 2 3 2 4 3 4 4 S12 S37 S57 S69 S90 4-2-3 4-2-3 13 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 8 0 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 S12 12 1 0 1 1 0 1 1 1 1 1 0 0 1 1 1 1 )

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14 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 S37 15 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 13 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 15 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 S57 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 15 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 S69 15 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 9 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 S90 6 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 S12 4-2-16 4-2-18 S12 4-2-16 S12 4-2-17 S12

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4-2-18 S12 4-2-16 4-2-18 S12 1 2 3 4 4 2 1 1 3 4 S12 2 2 S12 S37 4-2-19 4-2-21 S37 4-2-19 S37

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4-2-20 S37 4-2-21 S37 4-2-19 4-2-21 S37 2 3 1 4 1 4 1~4 1 2 1 4 3 4 S37 2 4 S57 4-2-22 4-2-24 S57

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4-2-22 S57 4-2-23 S57 4-2-24 S57 4-2-22 4-2-24 S57 1 2 3 4 2 1 4 3 4 1~4 1

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S57 4 S69 4-2-25 4-2-27 S69 4-2-25 S69 4-2-26 S69 4-2-27 S69

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4-2-25 4-2-27 S69 2 1 1 3 4 2 3 4 2 1 S69 2 4 S90 4-2-28 4-3-30 S90 4-2-28 S90 4-2-29 S90

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4-2-30 S90 4-2-27 4-2-30 S90 1 2( ) 3 4( ) 2 1 3 4 2 1 2 1 1 3 3 4 2 1 1 3 4 4 S90 3 4 2 4 4-2-3 2 4 2 4

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4-2-4 1 2 3 4 13 .59 .59 .62 .50 .58 8 .53 .53 .56 .38 .50 S12 12 .59 .66 .50 .50 .56 14 .53 .66 .62 .56 .59 16 1 1 1 1 1 S37 15 .59 .66 .62 .56 .61 13 .59 .59 .62 .50 .58 15 .59 .66 .62 .56 .61 S57 16 1 1 1 1 1 14 .59 .66 .62 .50 .59 15 .59 .66 .62 .56 .61 S69 15 .59 .66 .62 .56 .61 13 .59 .59 .56 .56 .58 9 .59 .66 .44 .38 .52 S90 6 .47 .53 .50 .38 .47 S12 B C 3 S12 2 2 3 S 37 2 1 2 S57 2 2 1 S69 3 2 2 S90 1 4 3

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2 4( S17 S74 S93 S116 S143 4-2-5 4-2-5 7 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 4 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 S17 13 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 8 0 0 1 0 1 0 0 0 1 1 0 0 1 1 1 1 11 0 0 1 1 1 1 1 1 1 1 1 0 0 1 0 1 S74 14 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 10 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 7 1 0 0 1 0 1 1 0 1 1 0 0 0 1 0 0 S93 7 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 10 0 0 1 1 1 0 1 1 0 1 1 1 1 1 0 0 10 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 S103 13 1 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 10 1 1 1 0 1 0 1 1 0 0 0 0 1 1 1 1 14 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 S143 8 0 1 0 1 1 1 1 0 1 1 0 0 0 1 0 0 S17 4-2-31 4-3-33 S17

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4-2-31 S17

4-2-32 S17

4-2-33 S17

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3 4 1 2 2 1 1 4 3 S17 2 2 S74 4-2-34 4-3-36 S74 4-2-34 S74 4-2-35 S74

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4-2-36 S74 4-2-34 4-2-36 S74 2 1 3 4 2 1 4 2 1 4 3 4 2 2 2 S74 S93 4-2-37 4-3-39 S93 4-2-37 S93

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4-2-38 S93 4-2-39 S93 4-2-37 4-2-39 S93 1 2 2 4 3 2 1 3 4 2 2 S93 2 3 4

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S103 4-2-40 4-3-42 S103 4-2-40 S103 4-2-40 4-2-42 S103 4-2-41 S103 4-2-42 S103

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2 2 1 3 4 4 2 1 3 2 1 1 3 1 4 S103 2 2 S143 4-2-43 4-3-45 S143 4-2-43 S143 4-2-44 S143

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4-2-45 S143 4-2-43 4-2-45 S143 3 2 4 1 3 2 1 4 1 4 3 3 S143 3 3 S143 ( ) S 143 1 4-2-6 S93 2

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4-2-6 1 2 3 4 7 .59 .47 .44 .44 .49 4 .53 .47 .38 .38 .44 S17 13 .59 .66 .44 .62 .58 8 .53 .47 .50 .50 .50 11 .53 .59 .56 .50 .55 S74 14 .59 .66 .56 .56 .59 10 .53 .66 .62 .62 .61 7 .53 .66 .38 .38 .49 S93 7 .47 .59 .44 .44 .49 10 .53 .53 .56 .50 .53 10 .53 .66 .50 .44 .53 S103 13 .53 .59 .62 .56 .58 10 .53 .47 .50 .62 .53 14 .53 .66 .56 .62 .59 S143 8 .53 .59 .38 .50 .50 ( ) ( ) S17 2 3 3 S 74 2 2 2 S93 3 3 2 S103 2 3 2 S90 3 2 3 ( )

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2 2( ) 4( ) 2( ) 4( ) 2( )

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S34 S47 S92 4-3-1 4-3-1 15 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 15 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 S34 15 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 14 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 14 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 S47 14 1 2 1 1 1 1 1 1 1 1 1 0 1 1 1 1 15 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 15 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 S92 15 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 S34 4-3-1 4-3-3 S34

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4-3-1 S34 4-3-2 S34 4-3-3 S34 4-3-1 4-3-3 S34 2 1 4 3 4 4 3 1 1 S34 2 1 1 3 4

(81)

S47 4-3-4 4-3-6 S47 4-3-4 S47 4-3-5 S47 4-3-6 S47 4-3-4 4-3-6 S47 3 4 4 1 2

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S47 2 1 1 4 3 2 2 S47 1 2 3 3 S92 4-2-7 4-2-9 S92 4-3-7 S92 4-3-8 S92 4-3-9 S92

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4-3-7 4-3-9 S92 2 3 4 1 1 2 1 4 3 4 4 S92 2 3 1 4 4-2- S34 2 S47 1 S92 2 3 S47 3 .12 4-3-2 1 2 3 4 15 .59 .66 .62 .56 .61 15 .59 .66 .56 .62 .61 S34 15 .53 .66 .62 .62 .61 14 .59 .59 .62 .59 .59 14 .59 .66 .50 .62 .59 S47 14 .59 .66 .56 .56 .59 15 .53 .66 .62 .62 .61 15 .59 .66 .62 .56 .61 S92 15 .59 .66 .62 .56 .61

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4-2-16 4-2-18 S34 2 S46 2 3 2 S92

2

(85)

SPSS

(86)

2( )

SPSS

(87)

150

(2008)

(88)
(89)

(2008) 2008 (2011) 1993 NSC 82-0111-S-002-004 (2006) ( ) (1994) 27 259-279 (2001) 24 263-302 (2008) CAISM TANET 2007 (1990) ( ) (1987) 26 7-20 (2009) SCM CAISM 325-334 :

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(1994) ( ) 18-44 (2004) ( ) (1997) ( ) (2005) 28 161-183 (2009) (2010) 195 121-124 (2006) [ ] (1995) (2003) (2003) (1989) ( ) (2001) 13( ) 23-60 (1991) (1998) CAI ( )

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( ) (2002) ( ) (1989) ( ) (1992) ( ) (2010) ( ) (1988) ( ) (1994) 8 3-56 (1989) (1993) (2008) (2008) 2008

(1994) (Interpretive Structural Modeling)

34 31-35

(1996) 4

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(2002) ( ) : (2000) ( ) (2007) SPSS (2005) 40 48-61 (1995) — 3 31-45 (1995) : (1994) ( ) (1990) ( ) (2001) ( ) (1996) (1991) 14 35-68 (1994) ( ) 60-76 (2000) NSC 89-2413-H-004-005 (2001)

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-(2008) ( ) (2009) (1991) ( ) (1990) (1997)

Hart, K. & Sinkinson, A. (1987). Forging the Link Between Practical and Formal Mathematics. Proceedings of the 11th Annual Conference of the

International Group for Psychology of Mathematics Education, Montreal,

Quebec, Canda University of Montreal.

Hegarty, M., Mater, R. E., & Monk, C.A.(1995). Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. Journal of Educational Psychology, 87, 18-32.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In Grouws,D. A. (Ed.), Handbook of research on

mathematics teaching and learning (pp.65-97). New York, NY:

Macmillan.

Kieran, C. (1992).The learing and teaching of school algebra. In Douglas. A. Grouws(Ed). Handbook of research on mathematics teaching and learning (pp.390-419). New York, NY: Macmillan.

Lesh, R., Post,T., Behr, M. (1987). Representations and translations among representation in mathematics learning and problem solving.In C.Janvier (Ed.). Problems of representation in the teaching and learning of

(94)

Lester, K. F. (1980). Research on mathematical problem solving. In R. J. Shumway (Ed.). Research in mathematics education. The National Council of Teachers of Mathematics. Reston, VA: NCTM.

Levin, J. R. (1981). On functions of pictures in prose. In F. J. Pirozzolo et M. C. Wittrock (Eds.). Neuropsychological and cognitive processes in reading (203-228). New York: Academic Press.

Lewis, A. B.(1989). Training students to represent arithmetic word problems.

Journal of Educational Psychology, 81, 521-531.

Lewis, A. B., Mayer, R. E. (1987). Student,s miscomprehension of relational statements in arithmetic word problems. Journal of Educational

Psychology, 79, 363-371.

Lin, Y. H., & Yih, J. M. (2011). Graphic representation on algebra concepts of university students based on clustering approach. Advanced Materials

Research, Vols. 211-212(pp.866-870).

Lin, Y. H., & Yih, J. M. (2008). An Integration of Concept Structure Analysis and S-P Chart with Application in Equality Axiom Concepts Diagnosis.

Proceedings 2008 International Symposium on Intelligent Information Technology Application (Vol. II , pp.468-472). Shanghai, China.

Lin, Y. H., Hung, K. J., & Huang, K. J. (2006). CAISM software [manual and

software for CAISM]. Taiwan, Taichung, City: National Taichung

University.

Lin, Y. H., Hung, W. L., & Yu, S. C. (2007).Concept Structure Analysis Method based on Integration of FLMP and ISM with Application in Equality Axion Concepts. Proceedings of the 8th WSEAS International Conference on Fuzzy Systems (pp.99-104). Vancouver, British Columbia, Canada, June

(95)

Macgregor, M., & Stacey, K. (1999). Learning the algebraic method of solving problems. Journal of Mathematical Behavior, 18, 149-167.

Mayer, R. E. (1992). Thinking, Problem Solving, Cognition. New York W. H. Freeman and Company.

Moyer, J. C., Sowder, L., Sowder, J. T., & Moyer, M. B. (1984). Story problem

formats draw versus verbal versus telegraphic. Journal for Research in Mathematical Educaton, 15, 342-351.

National Council of Teachers of Mathematics.(2000). Principles and Standards

for School Mathematics.http://www.ntcm.org.

Polya, G. (1945). How to solve it. New York: Doubleday.

Schoenfeld, A. H. (1985). Mathematical Problem Solving. Orlando, FL: Academic Press.

Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. Journal of Mathematical Behavior, 14,15-39. Sowder ,L., & Threadgill-Sowder, J. (1982). Drawn versus verbal formats for

mathematical story problems. Journal for Research in Mathematical

Education,13, 324-331.

Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford(Ed.), The ideas of algebra ,K-12 (pp.8-19). Reston, VA: National Council of Teachers of Mathematics.

Van Amerom, B. A. (2003). Focusing on informal strategies when linking arithmetic to early algebra. Educational Studies in Mathematics, 54, 63-75. Webb, L. F., & Sherrill, J. M. (1974). The effects of differing presentations of

mathematical word problems upon the achievement of preserve elementary teachers. School Science and Mathematics, 74, 559-565.

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