國小四年級學生在同分母分數加減的試題概念結構分析

99 90

( ) ( )

Abstract

Constructing basic fraction concepts, such as fraction measurement, unit quantity,

and proper fraction has been carelessly ignored, which has resulted in confusion when

students learn fraction. The changes of concepts of students after learning can be

important messages for the teachers to improve their teaching strategies. Therefore,

the study, sampling ninety fourth-graders in Nantou County in fall semester 2010,

designed a test sheet of additon and substraction of fraction with the same denominator

and further analyzed the written test result by applying “Item Relational Structure

(IRS)” analysis. A concept construction diagram was acquired after the analysis to

present the concepts students had constructed after the learning. Findings from

analyzing the test results were concluded as the following:

I. Through the analysis of the test items being passed by students, it was found that

students performed better in addition and substraction of fraction in comparison

with based concept of fraction.

II. Concepts of proper fraction, fraction measurement, and the combination of fraction

measurement need to be enhanced.

To students,there was no obvious ordinal sequence in learning addition and

substraction of fraction with the same denominator.

III. To solve addition and substration of fraction problems,it is necessary to be familiar

with the transfer among proper fractions,improper fractions and mixed fractions.

IV. From the analysis of test sheet of concepts,the mistakes students often made

were:(1)Confusion of “TWO PHASES MEASUREMENT”and (2)false

identification of unit quantity.

Key words:Fraction Measurement,Unit Quantity,Item Relational Structure(IRS) analysis

... 1 ... 1 ... 2 ... 3 ... 5 ... 7 ... 7 ...11 ... 16 ... 21 ... 31 ... 31 ... 32 ... 32 ... 36 ... 38 ... 39 ... 39 ... 42 ... 59 ... 62 ... 83 ... 83 ... 86 ... 88 ... 88 ... 90 ... 93 ... 102

1-1 92 97 --- 4 2-1 1 23 2-2 2 23 2-3 3 24 2-4 4 24 2-5 i j 27 2-6 28 2-7 0-1 29 4-1 Cronbach s 39 4-2 41 4-3 42 4-4 43 4-5 43 4-6 44 4-7 44 4-8 45 4-9 45 4-10 46 4-11 46 4-12 47 4-13 47 4-14 48 4-15 48

4-16 49 4-17 49 4-18 50 4-19 50 4-20 51 4-21 51 4-22 52 4-23 52 4-24 53 4-25 53 4-26 54 4-27 54 4-28 55 4-29 56 4-30 58 4-31 59 4-32 0-1 60 4-33 61 4-34 ( ) 64 4-35 ( ) 66 4-36 69 4-37 78

2-1 A B 26 2-2 30 3-1 31 3-2 33 3-3 34 3-4 35 3-5 37 4-1 63 4-2 ( ) 65 4-3 ( ) 68 4-4 72 4-5 74 4-6 76 4-7 79

2000 82 90 2003 1997 2004

1997 1991 1990a 1997 1995

1998 1988

2003

Item relational structure analysis IRS

1995 IRS

1 2

(1982) 1 3/4 8/9 a/b .05 1 3 1/4 1/4 a/b 1/b a .05 1/100 .01 Borowski Borwein ( )

m n

m n n 1 ( ) numerator denominator

John Daintith and R.D.Nelson

fraction

a/b

a

92 97 ---92 97 (100 ) N-1-09(3-n-09) N-2-06(4-n-06) N-2-07(4-n-07) N-2-09 N-2-10 (3-n-11) N-2-11(4-n-07) N-2-10(4-n-08) 1-1 92 97 ---92 97 97 3 4

2003

Item relational structure analysis

1980

Orderingtheory

0.5

IRS

IRS IRS

fraction fangere 1980 2001 rational numbers p q p q p 0 2003

Kieren 1976 ( ) part-whole comparisons ( ) ratio ( ) decimals ( ) quotient ( ) operator ( ) measures ( ) ordered pairs

Behr, Lesh, Post, and Silver 1983

( ) part-whole

( ) measure ( ) ratio ( ) quotient

( ) operator

Dickon, Brown,and Gibson 1984 ( ) ( ) ( ) ( ) ( ) Booth 1987

( ) ( ) ( ) 1987 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1988 ( ) ( ) ( ) ( ) 1995 ( ) ( ) / ( )

( ) ( ) ( )

82 1993 6 1 2 3 4 5 6 4/6 4/6 --- ---1 4 除數 被除數÷ = p q p q p 0

p q p q÷

Piaget 1990

(Sensorimotor Stage Preoperational Stage

(Pre-operational Stage)

2 4 6 8

10 2

(Concrete Operational Stage)

(Formal operational or Mathematical logical operational Stage)

Piaget

Piaget

Piaget

Piaget Inhelder

Inhelder

Inhelder

Inhelder &

&

& Szeminska

&

Szeminska

Szeminska

Szeminska

Piaget Inhelder and Szeminska 1960

( ) 1 ( ) ( ) exhaustive ( ) ( ) ( ) ( )

Piaget Inhelder and Szeminska 1960 4 7

( )

( ) ( ) ( ) (Ning) (Ning) (Ning) (Ning)

Ning,1992 1997

1993 1 6 1 6 6 1 12 1 6 1 6 1 6 1 6 6 1 1 6 implicit part-in-whole 1 6 6 1 1 6 1 6

1 5 + 1 5 = 2 5 unit fraction unit 1993 3 4 6 8 3 4 6 8 -3 6 4 8 12 24 3 6 4 8 1 24

1991

(Cramer,Post,& delMas, 2002; Kouba,Zawojewski,and

Strutchens,1997) 24 12 1 4 4 2 4 8 ( ) 36 12 ( ) 36 1 6 6 2 6 12 3 6 18 4 6 24 5 6 30 6 6 36 6 1 2 6 2 12 3 6 3 18 4 6 4 24 5 6 5 30 6 6 6 36 (Mack, 1993)

q qq q p pp p q p q p 2000b Behr, et al., 1984; Cramer, et a., 2002; Kerslake, 1986 Hart,1988; Post et al., 1992

( ) Post (1984) Mack(1993) 3 4 4 4 3 3 3 4× 3=12 ( ) 4 7 3 5 7 5 4 7 3 5 (Behr et al.,1984; Hunting, 1986)

( ) 4 7 8 17 4 8 4 7 8 17 (Behr et al, 1984) 1 7 2 14 1 7 2 14 2 14 1 7

2 14 1991b ( ) 4 7 8 14 4 8 7 14 4 7 8

14 (Behr et al.,1984; Hunting,1986 2000b) q p 1991 ( )Kerslake 1986 13 14 ( )Jeck 1981 90% ( )Tatsuoka 1984

. . . . . . Painter 1989 . 3 5 7 4 10 9 . 4 9 2 3 6 9 . 3 4 + 1 3 = 4 12 . 7 5 + 3 8 = 21 40 ( ) 1995 . . . . × × . . . ( ) 1993 . .

. . .

Airasian Bart 1973

1979

Item relational structure analysis

IRS

1995

S-P

1 0 0 0 0 0 0 1 1 2 1 1 1 1 1 1 1 7 3 0 0 0 0 0 0 0 0 4 0 1 1 1 1 1 1 6 5 0 0 0 0 0 0 1 1 6 1 1 1 1 1 1 1 7 7 1 1 1 1 1 1 1 7 8 0 0 0 0 0 0 0 0 9 0 1 0 0 1 1 1 4 10 0 0 0 0 1 0 1 2 3 5 4 4 6 5 8 1 0 0 0 0 0 0 0 0 2 1 1 1 1 1 1 1 7 3 0 0 0 0 0 0 0 0 4 0 1 0 1 1 1 1 5 5 0 0 0 0 1 0 1 2 6 1 1 0 1 1 1 1 6 7 1 0 1 1 1 1 1 6 8 0 0 0 0 0 0 0 0 9 0 0 1 0 1 1 1 4 10 0 1 1 0 0 0 1 3 3 5 4 4 6 5 8 試 題 B 組 1 0 0 0 0 0 0 1 2 1 1 1 1 1 1 1 3 0 0 0 0 0 0 0 4 0 1 1 1 1 1 1 5 0 0 0 0 0 0 1 6 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 8 0 0 0 0 0 0 0 9 0 1 0 0 1 1 1 學 生 10 0 0 0 0 1 0 1 3 5 4 4 6 5 8

A B 2-1 1 試 題 A 組 1 0 0 0 0 0 0 0 2 1 1 1 1 1 1 1 3 0 0 0 0 0 0 0 4 0 1 0 1 1 1 1 5 0 0 0 0 1 0 1 6 1 1 0 1 1 1 1 7 1 0 1 1 1 1 1 8 0 0 0 0 0 0 0 9 0 0 1 0 1 1 1 學 生 10 0 1 1 0 0 0 1 3 5 4 4 6 5 8 2-2 2-2 2

試 題 B 組 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 學 生 1 1 0 0 0 0 0 8 6 5 5 4 4 3 2-3 2-3 3 S-P ( 1995) 2-4 2-4 4 試 題 A 組 1 1 0 1 0 0 0 0 2 1 1 1 1 1 1 1 3 0 0 0 0 0 0 0 4 1 1 1 1 0 1 0 5 1 1 0 0 0 0 0 6 1 1 1 1 0 1 1 7 1 1 0 1 1 1 1 8 0 0 0 0 0 0 0 9 1 1 0 1 1 0 0 學 生 10 1 0 1 0 1 0 0 8 6 5 5 4 4 3 試 題 B 組 2 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 4 1 1 1 1 1 1 0 9 1 1 1 1 0 0 0 10 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 5 1 0 0 0 0 0 0 3 0 0 0 0 0 0 0 學 生 8 0 0 0 0 0 0 0 3 5 4 4 6 5 8 試 題 A 組 2 1 1 1 1 1 1 1 6 1 1 1 1 0 1 1 7 1 1 0 1 1 1 1 4 1 1 1 1 0 1 0 9 1 1 0 1 1 0 0 10 1 0 1 0 1 0 0 1 1 0 1 0 0 0 0 5 1 1 0 0 0 0 0 3 0 0 0 0 0 0 0 學 生 8 0 0 0 0 0 0 0 8 6 5 5 4 4 3

A 10 B 受試全體學生的人數 受試學生答對人數 試題答對率= 2-1 A B

答對率 A 組結構圖 B 組結構圖 0.3 0.4 0.5 0.6 0.7 0.8 A B A B S-P

1995 2-5 i j A i j B i j C i j D i j 2-5 N i j N=A+B+C+D ( 1995)：

r

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r

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lower concept j upper concept (1991) 0.5

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0.5 i j 試 題 j 1 0 合計 1 A B A+B 0 C D C+D 試 題 i 合計 A+C B+D N （１：表示答對 ０：表示答錯）

2-6 2-6 試 題 試 j 題 i 1 2 3 4 5 6 7 1 1.00 0.29 0.29 0.64* 0.29 0.43 0.18 2 0.44 1.00 0.17 0.58* 0.17 0.33 0.29 3 0.44 0.17 1.00 0.17 0.17 0.33 0.29 4 1.00* 0.58* 0.17 1.00 0.44 0.67* 0.29 5 1.00* 0.38 0.38 1.00* 1.00 1.00* 0.64* 6 1.00* 0.50* 0.50* 1.00* 0.67* 1.00 0.43 7 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00 * 0.5 2-6 i j 0.5

r

舖ij >0.5

r

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2-7 0-1 0.5 0-1
0-1
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試 題 試 j 題 i 1 2 3 4 5 6 7 1 0 0 0 1 0 0 0 2 0 0 0 1 0 0 0 3 0 0 0 0 0 0 0 4 1 1 0 0 0 1 0 5 1 0 0 1 0 1 1 6 1 1 1 1 1 0 0 7 1 1 1 1 1 1 0

答對率 試題關聯結構圖 0.3 0.4 0.5 0.6 0.7 0.8 2-2 7 77 7 5 55 5 2 22 2 3 33 3 4444 6 66 6 1 11 1

(IRS)

3-1 IRS

60 24 90 18 921

92 3-2 3-2 3-3 3-4 25

2-n-10 12 3-n-09 4-n-06 4-n-07 4-n-08

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(Reliability) Cronbach’s α SPSS Cronbach’s α 0.80 4-1 4-1 Cronbach’s α Cronbach's Alpha Cronbach's Alpha Cronbach's Alpha 1 .808 11 .801 21 .813 2 .810 12 .809 22 .808 3 .808 13 .798 23 .805 4 .808 14 .810 24 .814 5 .813 15 .806 25 .815 6 .809 16 .805 7 .802 17 .805 8 .806 18 .808 9 .799 19 .807 10 .810 20 .807

27% 27% PH PL (PH+PL)/2 27% 27% PH (PH-PL) -1.0 ~ +1.0 0 0.25 0.40 4-2

4-2 試題 試題試題 試題 PPP P PHPHPHPH PLPL PLPL (PH+PL)/2(PH+PL)/2(PH+PL)/2(PH+PL)/2 PHPHPHPH----PLPLPLPL x1 0.36 0.50 0.13 0.32 0.37 x2 0.51 0.73 0.26 0.50 0.47 x3 0.67 0.83 0.39 0.61 0.44 x4 0.30 0.50 0.09 0.29 0.41 x5 0.57 0.87 0.48 0.67 0.39 x6 0.43 0.73 0.22 0.48 0.52 x7 0.81 0.97 0.43 0.70 0.53 x8 0.84 1.00 0.57 0.78 0.43 x9 0.63 0.90 0.17 0.54 0.73 x10 0.38 0.60 0.09 0.34 0.51 x11 0.79 1.00 0.43 0.72 0.57 x12 0.47 0.70 0.22 0.46 0.48 x13 0.41 0.80 0.00 0.40 0.80 x14 0.68 0.87 0.39 0.63 0.48 x15 0.77 0.90 0.43 0.67 0.47 x16 0.61 0.87 0.26 0.56 0.61 x17 0.68 0.90 0.35 0.62 0.55 x18 0.72 0.90 0.48 0.69 0.42 x19 0.50 0.77 0.35 0.56 0.42 x20 0.62 0.77 0.35 0.56 0.42 x21 0.62 0.73 0.30 0.52 0.43 x22 0.30 0.47 0.09 0.28 0.38 x23 0.56 0.83 0.22 0.53 0.62 x24 0.21 0.33 0.04 0.19 0.29 x25 0.73 0.80 0.57 0.68 0.23

25% 25% 4-3 4-27
   4-3 1 2 3 4 0.03 0.37 0 0.60 0.05 0.55 0 0.4 0.04 0.09 0 0.87 ＊

33%
77%
 4-4 1 2 3 4 0.33 0.13 0.52 0.02 0.14 0.09 0.77 0 0.57 0.13 0.26 0.04 ＊ 4-5 1 2 3 4 0.13 0.68 0.13 0.06 0.09 0.86 0.05 0 0.22 0.39 0.26 0.13 ＊

   32% 68% 86% 48% 22% 
13%  4-6 1 2 3 4 0.32 0.68 0 0 0.55 0.45 0 0 0.09 0.91 0 0 ＊ 4-7 1 2 3 4 0.09 0.13 0.57 0.21 0.09 0 0.86 0.05 0.13 0.22 0.48 0.17 ＊

73% 27%
22%
48% 81% 43% + 4-8 1 2 3 4 0.44 0.09 0.03 0.44 0.27 0 0 0.73 0.48 0.22 0.09 0.22 ＊ 4-9 1 2 3 4 0.04 0.10 0.81 0.05 0 0.05 0.95 0 0.17 0.26 0.43 0.13 ＊

86% 61% 65% 90% 17% 4-10 1 2 3 4 0.86 0.03 0.06 0.05 1 0 0 0 0.61 0.09 0.17 0.13 ＊ 4-11 1 2 3 4 0.01 0.14 0.20 0.65 0 0.05 0..05 0.9 0.04 0.22 0.57 0.17 ＊

 67% 41% 83% 79% 43% 4-12 1 2 3 4 0.27 0.67 0.02 0.04 0.59 0.41 0 0 0.04 0.83 0.09 0.04 ＊ 4-13 1 2 3 4 0.07 0.79 0.10 0.04 0 1 0 0 0.26 0.43 0.22 0.09 ＊

1 48%  41% 77% 18%  0
4-14 1 2 3 4 0.12 0.31 0.48 0.09 0.09 0.18 0.73 0 0.22 0.43 0.22 0.13 ＊ 4-15 1 2 3 4 0.12 0.41 0.37 0.10 0 0.77 0.18 0.05 0.43 0 0.26 0.3 ＊

4-3 5 2 =1 5 2 67% 86% 35% 35%
77% 35% 4-16 1 2 3 4 0.10 0.14 0.09 0.67 0 0.09 0.05 0.86 0.35 0.17 0.13 0.35 ＊ 4-17 1 2 3 4 0.77 0.08 0.13 0.02 1 0 0 0 0.35 0.17 0.3 0.17 ＊

， 62% 95% 26% 39%
69% 90% 35% 52%   4-18 1 2 3 4 0.20 0.16 0.62 0.02 0.05 0 0.95 0 0.39 0.3 0.26 0.04 ＊ 4-19 1 2 3 4 0.08 0.18 0.05 0.69 0.05 0 0.05 0.9 0.04 0.52 0.09 0.35 ＊

3 6 4 3 6 4 71% 95% 50% 50% 86% 35% 43% _ 4-20 1 2 3 4 0.71 0.09 0.06 0.14 0.95 0 0.05 0 0.43 0.04 0.22 0.3 ＊ 4-21 1 2 3 4 0.10 0.27 0.50 0.13 0.05 0.09 0.86 0 0.17 0.43 0.35 0.04 ＊

65% 91% 35% 43%
60% 73% 26% 39% 4-22 1 2 3 4 0.23 0.10 0.02 0.65 0.09 0 0 0.91 0.43 0.17 0.04 0.35 ＊ 4-23 1 2 3 4 0.07 0.60 0.23 0.10 0 0.73 0.18 0.09 0.09 0.26 0.39 0.26 ＊

  55% 30% 56% 81% 22% 70% _ 4-24 1 2 3 4 0.53 0.13 0.30 0.04 0.18 0.22 0.55 0.05 0.7 0.17 0.09 0.04 ＊ 4-25 1 2 3 4 0.01 0.38 0.05 0.56 0 0.14 0.05 0.81 0.04 0.7 0.04 0.22 ＊

19% 75% 82% 61% 4-26 1 2 3 4 0.21 0.31 0.29 0.19 0.32 0.27 0.32 0.09 0.04 0.17 0.48 0.3 ＊ 4-27 1 2 3 4 0.06 0.10 0.09 0.75 0.14 0 0.05 0.82 0.09 0.17 0.13 0.61 ＊

4-28 1 20 24 0.37 0.65 0.21 7 9 15 19 25 ( ) 0.81 0.65 0.77 0.50 0.75 2 17 0.52 0.69 21 23 0.60 0.56 4 10 0.32 0.27 3 6 8 11 13 14 0.68 0.44 0.86 0.79 0.41 0.67 16 18 0.62 0.71 5 12 22 0.57 0.48 0.30 4-28 21%~65% 50%~81% 52%~69% 56%~60% 27%~32% 41%~86% 30%~57%

4-29 1 20 24 0.55 0.91 0.32 7 9 15 19 25 ( ) 0.95 0.90 1 0.86 0.82 2 17 0.77 0.9 21 23 0.73 0.81 4 10 0.55 0.59 3 6 8 11 13 14 0.86 0.73 1 1 0.77 0.86 16 18 0.95 0.95 5 12 22 0.86 0.73 0.55

4-29 70% 24 32% 4 10 60% 4 10

4-30 1 20 24 0.09 0.35 0.04 7 9 15 19 25 ( ) 0.43 0.17 0.35 0.35 0.61 2 17 0.26 0.35 21 23 0.26 0.22 4 10 0.04 0.04 3 6 8 11 13 14 0.39 0.22 0.61 0.43 0 0.35 16 18 0.26 0.43 5 12 22 0.48 0.22 0.09 4-30 50% 13 0 0 8 11 14

13

4-31 題 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 +0.09 +0.12 +0.25 +0.06 +0.14 +0.08 +0.06 +0.15 +0.20 +0.07 +0.10 +0.20 +0.10 +0.06 +0.05 +0.15 +0.02 +0.14 +0.05 +0.08 +0.05 +0.13 -0.07 +0.03 2 +0.16 +0.15 +0.27 +0.04 +0.11 +0.05 +0.02 +0.10 +0.11 +0.14 +0.14 +0.19 +0.03 +0.07 +0.12 +0.21 +0.04 +0.27 +0.05 +0.02 +0.01 +0.18 +0.09 +0.70 3 +0.38 +0.28 +0.25 +0.18 +0.23 +0.18 +0.07 +0.26 +0.25 +0.16 +0.30 +0.37 -0.03 +0.10 +0.20 +0.08 +0.16 +0.07 +0.30 +0.06 +0.33 +0.22 +0.10 -0.05 4 +0.19 +0.12 +0.06 +0.06 +0.12 +0.07 +0.08 +0.08 +0.05 +0.08 +0.06 +0.12 +0.02 +0.11 +0.04 +0.13 +0.01 +0.16 +0.02 +0.03 +0.14 +0.13 +0.03 +0.10 5 +0.12 +0.05 +0.12 +0.18 +0.35 +0.09 +0.11 +0.07 +0.28 +0.09 -0.02 +0.15 +0.02 -0.02 +0.09 +0.10 +0.10 +0.08 +0.21 -0.03 +0.32 +0.12 +0.23 -0.01 6 +0.17 +0.08 +0.09 +0.18 +0.20 +0.01 -0.31 +0.07 +0.12 +0.08 +0.26 +0.30 +0.07 +0.07 +0.06 +0.05 +0.09 -0.02 +0.12 +0.02 +0.28 +0.15 +0.02 +0.06 7 +0.53 +0.19 +0.38 +0.62 +0.27 +0.05 +0.52 +0.63 +0.34 +0.32 +0.38 +0.72 +0.05 +0.24 +0.62 +0.40 +0.01 +0.53 +0.24 +0.41 +0.80 +0.37 +0.71 +0.12 8 +0.59 +0.10 +0.20 +1.00 +0.05 -0.24 +0.72 +0.78 +0.35 +0.22 +0.20 +0.82 +0.09 +0.11 +0.26 +.021 +.024 +0.69 +0.26 +0.49 +1.00 +0.47 +0.62 +0.48 9 +0.44 +0.17 +0.23 +0.32 +0.09 +0.16 +0.25 +0.22 +0.57 +0.12 +0.24 +0.71 +0.11 +0.14 +0.37 +0.25 +0.19 +0.27 +0.22 +0.24 +0.60 +0.29 +0.55 -0.03 10 +0.18 +0.06 +0.07 +0.06 +0.12 +0.09 +0.04 +0.03 +0.18 +0.04 +0.03 +0.14 +0.11 +0.10 +0.02 +0.07 +0.05 +0.14 +0.09 +0.11 +0.14 +0.10 +0.05 -0.03 11 +0.44 +0.49 +0.29 +0.66 +0.26 +0.39 +0.29 +0.14 +0.25 +0.26 +0.34 +0.50 +0.30 +0.32 +0.49 +0.31 +0.33 +0.47 +0.58 +0.30 +0.47 +0.53 +0.47 +0.07 12 +0.16 +0.13 +0.14 +0.11 -0.01 +0.31 +0.08 +0.03 +0.13 +0.04 +0.08 +0.55 +0.15 +0.10 +0.11 -0.05 +0.13 -0.06 +0.15 +0.04 +0.08 +0.12 +0.15 +0.07 13 +0.24 +0.14 +0.14 +0.20 +0.08 +0.29 +0.12 +0.10 +0.32 +0.19 +0.10 +0.44 +0.23 +0.09 +0.23 +0.05 +0.16 +0.12 +.010 +0.04 +0.23 +0.24 +0.04 +0.08 14 +0.36 +0.06 -0.03 +0.11 +0.03 +0.20 +0.02 +003 +0.13 +0.42 +0.17 +0.35 +0.67 +0.29 +0.11 -0.01 +0.18 +0.24 +0.17 +0.02 +0.08 +0.19 -0.21 0.11 15 +0.34 +0.22 +0.18 +0.84 -0.06 +0.31 +0.20 +0.07 +0.29 +0.58 +0.30 +0.37 +0.41 +0.48 +0.36 +0.20 +0.22 +0.20 +0.12 +0.08 +0.33 +0.19 +0.75 +0.11 16 +0.14 +0.19 +0.16 +0.15 +0.12 +0.12 +0.24 +0.07 +0.35 +0.01 +0.22 +0.20 +0.51 +0.09 +0.17 +0.27 +0.13 +.035 +0.15 +0.12 +0.22 +0.21 +0.41 -0.04 17 +0.53 +0.44 +0.09 +0.66 +0.18 +0.42 +0.21 +0.08 +0.32 +0.30 +0.19 -0.12 +0.15 -0.01 +0.13 +0.37 +0.05 +0.36 +0.14 +0.29 +0.64 +0.16 -0.07 +0.08 18 +0.08 +0.10 +0.19 +0.01 +0.19 +0.29 +0.01 +0.10 +0.27 +0.24 +0.22 +0.36 +0.55 +0.21 +0.16 +0.20 +0.05 +0.23 +0.26 +0.23 +0.23 +0.31 +0.23 +0.11 19 +0.24 +0.26 +0.03 +0.36 +0.06 -0.03 +0.12 +0.12 +0.16 +0.25 +0.13 -0.07 +0.16 +0.12 +0.06 +0.21 +0.16 +0.09 0 +0.15 +0.19 +0.20 +0.22 +0.03 20 +0.14 +0.08 +0.25 +0.06 +0.27 +0.25 +0.09 +0.07 +0.21 +0.26 +0.25 +0.26 +0.23 +0.13 +0.06 +0.15 +0.10 +0.17 0 +0.07 +0.51 +0.15 +0.41 +0.12 21 +0.19 -0.03 +0.04 +0.11 -0.03 +0.04 +0.14 +0.12 +0.21 +0.30 +0.12 +0.07 +0.08 +0.02 +0.04 +0.11 +0.19 +0.14 +0.22 +0.06 +0.17 +0.15 +0.03 +0.02 22 +0.03 +0.04 +0.07 +0.13 +0.10 +0.16 +0.08 +0.07 +0.15 +0.11 +0.05 +0.04 +0.14 +0.02 +0.04 +0.06 +0.12 +0.04 +0.08 +0.13 +0.05 +0.06 -0.03 +0.07 23 +0.27 +0.22 +0.14 +0.36 +0.12 +0.25 +0.11 +0.09 +0.21 +0.23 +0.18 +0.16 +0.41 +0.12 +0.07 +0.16 +0.10 +0.16 +0.25 +0.12 +0.13 +0.17 +0.50 -0.06 24 -0.03 +0.02 0 +0.02 +0.04 +0.01 +0.04 +0.03 +0.08 +0.02 +0.03 +0.04 +0.01 -0.03 +0.05 +0.06 -0.01 +0.02 +0.06 +0.06 +0.01 -0.02 +0.10 +0.03 25 +0.12 +0.19 -0.06 +0.60 -0.03 +0.23 +0.08 +0.22 -0.05 -0.17 +0.05 +0.22 +0.31 +0.14 +0.09 -0.07 +0.09 +0.12 +0.08 +0.20 +0.03 +0.44 -0.13 +0.38

0.5 0.5 0.5 0-1 0.5 0.5 0-1 4-31 4-32 0-1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 8 1 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 9 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 16 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 17 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4-32 0-1 irs 0-1 0-1

1 7 9 15 19 20 24 25 4-33 4-1 4-33 1 3 8 , 2 8 1 8 1_{1 6}5 2 5 3_{1 6}6 45₈ 37%

7 6 8 10 1 8 2 6 3 7 4 1 0 81% 9 1 2 3 4 65% 15 13 15 - 9 15 =( ) ( ) 1 2 3 4 77% 1. 2. 19 1 1 2 13₄ 33₂ 4 0 50% 20 1 1 4+ 1 4+ 1 4+ 1 4 2 1 5+ 1 5+ 1 5+ 1 5 31₆+1 6+ 1 6+ 1 6 4 1 7+ 1 7+ 1 7+ 1 7 64% 24 12 1 1 7 2 6 3_{1 2}7 4_{1 2}6 19% 25 1 2 3 4 75%

4-1

7

9

24

7

19

7

1

答對率 試題關聯結構圖 0.2 0.3 0.4 0.5 0.6 0.7 0.8 20 20 20 20 24 24 24 24 1 11 1 7 77 7 19 1919 19 25 25 25 25 15 15 15 15 9 99 9

15

24

(1)

7

9

24

7 9 24

7

24

1

7

1

9

24

21%

(2)

7

19

7

19

19

(3)

7

1

7

1

1

37%

(4)

15

24

15

24

2 17 21 23 4-34 4-2

4-34 ( ) 2 3 6 9 3 6 8 1 34 2_{1 7} 2 66₈ 36 2₈ 4 3_{1 3}6 51% 1. + 2. 17 22 6 + 1 2 6 =( ) ( ) 1 2 3 4 6 3 1 8 4 6 4 2 2 1 6 68% 1. 2. 21 34 38 115₃₈ 217₃₈ 3_{3 8}4 41₂ 62% 1. 23 94 6 2 2 6 ？ 1 65₆ 2 71₆ 3 43₆ 4 5 56% 1.

4-2 ( )

( )

( )

( )

3 4 6 8 10 11 13 14 16 18 4-35 4-4 4-35 ( ) 3 10 30 12 10 1 2_{1 0}2 68% 1. 2. 答對率 試題關聯結構圖 0.5 0.6 0.7 23 2323 23 21 21 21 21 2 22 2 17 17 17 17

2 2_{1 0 3}7 1 7 1 0 41 2 1 0 4 11 6 2 6 ？ 13₆ 2 13₆ 3 2 4 3 32% 1. + 2. 6 28 112 7 ？ 1 1 65₇ 2 1 72₇ 3 3 72₇ 4 53₇ 44% 1. 8 68 10 1_{1 0 2}8 5 8 1 0 3 2 1 0 41 8 1 0 86% 1. 10 13 7 14 7 ？ 1 21 2_{4 9} 2 3 3 43₇ 4 34₇ 27% 11 72 6 24 6 25 6 ？ 1 22₆ 2 15₆ 3 42₆ 4 64₆ 79% 13 4 32 ？ 41%

1 12₅ 2 24₅ 33₅ 4 13₅ 公 尺 14 57 19 47 19 ？ 1 7_{1 9 2}2 21 7 1 9 33 1 3 1 9 42 7 1 9 67% 16 775 29 3413 29 1 8 11 8_{2 9} 2 2 3_{2 9}8 3 4 22 1_{2 9} 4 5 92 6_{2 9} 62% 18 3 2 3 13 5 13 ？ 1 5_{1 3 2}8 41 0 1 3 35 4 4 5 1 3 71%

4-3 ( )

8 11 14 16 18

答對率 試題關聯結構圖 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 10 10 10 4 44 4 3 33 3 6 66 6 8 88 8 11 1111 11 13 1313 13 14 1414 14 16 1616 16 18 18 18 18

(1)

8

4

8

4

(2)

16

4

16

4

20

1

4

(3)

11

4

11

4

11

1 5 7 9 12 15 19 20 22 24 25 4-36 4-4 4-36 1 3 8 , 2 8 1 8 1_{1 6}5 2 5 3_{1 6}6 45₈ 37%

5 32 9 5 8 9 11 4 9 1 2 05₉ 2 1 41₉ 3 87₉ 4 55₉ 57% 7 68 10 1 8 2 6 3 7 4 1 0 81% 9 _{1 2} 3 4 65% 12 83 15 66 15 4 8 1 2_{1 5}3 2 1 4_{1 5}9 3 1 3_{1 5}9 4 1 41 8_{1 5} 48% 15 13 15 - 9 15 =( ) ( ) 1 2 3 4 77% 1. 2.

19 1 1 2 13₄ 33₂ 4 0 50% 20 1 1 4+ 1 4+ 1 4+ 1 4 2 1 5+ 1 5+ 1 5+ 1 5 31₆+1 6+ 1 6+ 1 6 4 1 7+ 1 7+ 1 7+ 1 7 64% 22 85 6 7 3 6 1 12_{6 2}1 62 6 37 4 6 42 2 6 65% 24 24 12 1 7 2 6 3_{1 2}7 4_{1 2}6 19% 25 1 2 3 4 75%

4-4

答對率 試題關聯結構圖 0.2 0.3 0.4 0.5 0.6 0.7 0.8 24 24 24 24 15 15 15 15 9 99 9 20 20 20 20 1 11 1 7 77 7 19 19 19 19 25 25 25 25 22 22 22 22 12 12 12 12 5 55 5

7

9

22

7(

)

9

22(

)

7

9

22

24

22

20

22

20(

)

22

22

4 7 9 10 15 19 25

4-5

4-5

25

4

25(

)

4(

)

15

4

15(

)

4

答對率 試題關聯結構圖 0.3 0.4 0.5 0.6 0.7 0.8 9 99 9 25 25 25 25 4 44 4 15 15 15 15 7 77 7 10 1010 10 19 1919 19

15

10

15

10

7

4

7

10

7

9

10

7

10

7

9

3 6 7 8 9 11 13

14 15 16 18 19 25

4-6

答對率 試題關聯結構圖 0.4 0.5 0.6 0.7 0.8 0.9 16 1616 16 14 14 14 14 8 88 8 18 1818 18 11 11 11 11 7 77 7 9 99 9 19 19 19 19 13 1313 13 6 66 6 3 33 3 15 1515 15 25 25 25 25

8

7

19

8

7

7

19

8

7

9

13

8

7

9

9

13

8

19

8(

)

19(

)

1

19

4-37

4-37 7 15 25 0.70~1 8 11 18 20 9 19 2 17 21 23 3 6 13 14 16 0.40~0.69 5 12 1 24 4 10 0.01~0.39 22 25 0.5 4-7 7 8 11 12 14 15 17 18 25 7 15 25 8 11 14 18 12 1 4 10 13 19 22 24 1 24 19 4 10 13

0.8

0.7

0.6

0.5

0.4

0.3

0.2

8 88 8 7 77 7 11 11 11 11 15 1515 15 25 25 25 25 18 18 18 18 17 17 17 17 14 14 14 14 12 12 12 12 9 99 9 20202020 16 1616 16 19 1919 19 23 23 23 23 1 11 1 13 1313 13 4 44 4 10 1010 10 22 22 22 22 24 24 24 24 2 22 2 5 55 5 3 33 3 6 66 6 2 22 21111

4-7

4-7 8

10 24 22 24

1

IRS

IRS 70%

1

( )

( )

( )

( ) 2/8 3/8 1/8 5/8 ( )

( ) ( ) 90 ( ) 90

2003 1982 1997 18 18 18 18 136-163 1991 4 44 4 273-314 1991b 1/2vs. 1/4 3131 11,123131 10-21 1997 1990a 4 44 4 295-347 (2003) (2001) 2003 1993 2004 (1989) 1997

1995 (1995) 1998 (1993) ~ ~ 34(3)34(3)34(3)34(3) 45-51 1988 NSC77-0111-S003-009-A 2000b 8(4) 8(4) 8(4) 8(4),379-394 ( 82) 1991 162-194 (1990) 23 23 23 23 265-290 1980 ((((454454454454 455455455455)))) 37-41 2001

Daintith, J. & Neison, R. D. 1997

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concepts and processes(pp. 91-126). New York: Academic Press

Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and equivalence of

rational numbers: A clinical teaching experiment. Journal for Research in

Mathematics Education, 15, 323-341.

Bennett, A. B., Jr. (1987). Fractions-Concepts before Symbols. (ERIC Document

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NFER-Nelso.

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Georgia Center for the Study of Learning and Teaching Mathematics.

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Britain,England School Council Publications

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Mack, N. K. (1993). Learning rational numbers with understanding: The case of

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Research on the teaching, learning, and assessing of rational number concepts (pp.

327-362). Hillsdale NJ: Lawrence Erlbaum Associates.

Piaget, J. (1965). The child’s conception of number. New York: W. W. Norton.

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Geometry ,London: Routledge and Kegan Paul.

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.

Doctoral dissertation,University of Southern

Mississppi,1988

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Silver (Ed.), Mathematics for the middle grades (5-9): 1982 Yearbook (pp.

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Ning, T. C. (1992). Children’s meanings of fractional number words. Unpublished

__________ ______ ______ ______________ 25 ( ) 3 8 , 2 8 1 8 1_{1 6}5 2 5 3_{1 6}6 45₈ 99 11

( ) 3 6 9 3 6 8 1 34 2_{1 7} 2 66₈ 36 2₈ 4 3_{1 3}6 ( ) 10 30 12 10 1 2_{1 0}2 2 2_{1 0}7 3 1_{1 0}7 4 1_{1 0}2 ( ) 11 6 2 6 13₆ 2 13₆ 3 2 4 3

( ) 32 9 5 8 9 11 4 9 1 2 05₉ 2 1 41₉ 3 87₉ 4 55₉ ( ) 28 112 7 1 1 65₇ 2 1 72₇ 3 3 72₇ 4 53₇ ( ) 68 10 1 8 2 6 3 7 4 1 0 ( ) 68 10 1_{1 0}8 2 5_{1 0}8 3_{1 0}2

4 1_{1 0}8 ( ) 1 2 3 4 ( )10 13 7 14 7 1 21 2_{4 9} 2 3 3 43₇ 4 34₇ ( )11 72 6 2 4 6 25 6 1 22₆ 2 15₆ 3 42₆ 4 64₆

( )12 83 15 6 6 15 4 8 1 2_{1 5}3 2 1 4_{1 5}9 3 1 3_{1 5}9 4 1 41 8_{1 5} ( )13 4 32 5 1 12₅ 2 24₅ 33₅ 4 13₅ ( )14 57 19 47 19 1 7_{1 9}2 2 21 7_{1 9} 3 31 3_{1 9} 4 2_{1 9}7

( )15 13 15 - 9 15 =( ) ( ) 1 2 3 4 ( )16 775 29 34 13 29 1 8 11 8_{2 9} 2 2 3_{2 9}8 3 4 22 1_{2 9} 4 5 92 6_{2 9} ( )17 22 6 + 1 2 6 =( ) ( ) 1 2 3 4 6 3 1 8 4 6 4 2 2 1 6

( )18 3 2 3 13 5 13 1 5_{1 3}8 2 41 0_{1 3} 3 5 4 4_{1 3}5 ( )19 1 1 2 13₄ 33₂ 4 0 ( )20 11₄+1 4+ 1 4+ 1 4 21₅+1 5+ 1 5+ 1 5 31₆+1 6+ 1 6+ 1 6 41₇+1 7+ 1 7+ 1 7

( )21 34 38 115₃₈ 217₃₈ 3_{3 8}4 41₂ ( )22 85 6 7 3 6 1 12₆ 2 1 62₆ 3 74₆ 4 22₆ ( )23 94 6 2 2 6 1 65₆ 2 71₆ 3 43₆

( )24 12 1 1 7 2 6 3_{1 2}7 4_{1 2}6 ( )25 1 2 3 4