國小六年級學童三角形概念之試題編製與分析

Cronbach .888 . 678 . 547 73.94 78.87 69.48 68.59

Abstract

The purpose of this research is to make a competent test of the concept of triangle. With this measurement, the researcher explores the students’ status and their misunderstanding during the test time. The study result will be a reference for teaching and remedy instruction in schools.

The study objects are the sixth grade students from an elementary school in Nantou County, Lugu Township. The study tool is “the Measurement of Triangle Concept”. The study results are as follows:

I. The characteristics of the questions and paper are:

1. This is a good measurement paper with the reliability of Cronbach 0.888. 2. This measurement has good content validity and good construct validity. 3. The average difficulty is 0.678, medium while tend to easy.

4. It’s a good measurement with an average discrimination of 0.547.

II. The features or the item potion characteristic curve of the measurement are:

1. According to the shapes, there are 5 types of correct item potion characteristic curve. 2. All of the distracters have distraction.

. The sixth-graders’ perceptions about the concept of triangle.

The participants performed better and got higher scores in the arena of the recognition concept and interior angle concept, the average rate of completely correct is 73.94% and 78.87%. Most of the children have higher stability on conservation. While few of them were affected by the sense of sight or the shape of the picture; however, they got lower grades in the arena of perimeter concept and the measure of area concept, the average rate of completely correct is 69.48 and 68.59 .High ability students have better performance; medium and lower students have misunderstanding conditions.

1 4 5 7 9 13 16 23 27 35 36 37 44 45 47 52 82 105 109 110 113

115 116 122 123 127 128 129

2-1 15 3-1 37 3-2 38 3-3 Cronbach’s 40 3-4 42 4-1 48 4-2 49 4-3 50 4-4 51 4-5 82 4-6 83 4-7 1 84 4-8 2 85 4-9 3 85 4-10 4 85 4-11 5 86 4-12 6 86 4-13 7 87 4-14 8 87 4-15 9 87 4-16 10 88 4-17 11 88 4-18 12 88 4-19 13 89 4-20 14 89 4-21 15 89 4-22 16 90 4-23 17 90 4-24 18 90 4-25 19 91 4-26 20 91 4-27 21 91 4-28 22 92 4-29 23 92 4-30 24 92 4-31 25 93 4-32 93

4-33 94 4-34 94 4-35 95 4-36 95 4-37 96 4-38 96 4-39 97 4-40 97 4-41 98 4-42 98

2-1 1 29 2-2 2 30 2-3 3 30 2-4 4 31 2-5 5 31 2-6 6 32 2-7 7 32 2-8 8 33 2-9 9 33 3-1 35 3-2 44 4-1 A 52 4-2 B 53 4-3 C 54 4-4 D S 54 4-5 E S 55 4-6 1 56 4-7 2 57 4-8 3 58 4-9 4 59 4-10 5 60 4-11 6 61 4-12 7 62 4-13 8 63 4-14 9 64 4-15 10 65 4-16 11 66 4-17 12 67 4-18 13 68 4-19 14 69 4-20 15 70 4-21 16 71 4-22 17 72 4-23 18 74 4-24 19 75 4-25 20 76 4-26 21 77

4-27 22 78

4-28 23 79

4-29 24 80

4-30 25 81

( Fey, 1984)

Clements and Battista (1992)

Thomas 2000

Piaget Piaget, Inhelder & Szeminska, 1960; Piaget & Inhelder, 1967

topological Projective

Piaget

2003

2002 2003 van Hiele van Hiele ( ) ( ) 2003 Hannibal 1999 2001 2002 2002

45

1992

2001 2002 2002 2003

30% 2002

Stavy and Tirosh 1996

( the theory of intuitive rules )

Fischbein & Schnarch, 1997 2003

acute obtuse right-angle

equiangular isosceles scalene

TestGraf98

IRT TestGraf 98

van Hiele Duval

van Hiele

van Hiele van Hiele van Hiele 1998 van Hiele 1986 ◇

90°

Duval

Duval (1995) 2004 2006 (Perceptual comprehension) (Operative comprehension) ( ) ( ) ( ) (constuction comprehension)

( ) (Discursive comprehension) Duval 1998 1 (visualization) 2 (construction) 3 (reasoning) Duval (1) (2)

(3) Duval 2002 van Hiele Duval vanHiele

2009 1. 2.

3-4 ( ) 2003 1. 2. 3. 2003 2003

2-1 2003 S-1-01 S-1-02 S-1-03 S-1-04 S-1-05 S-1-07 S-2-01 S-2-02 S-2-03 S-2-05 S-2-08 N-2-19 S-3-01 S-3-02

2002 2003 30% 2003 2004 1. 2. 2005 1. 80 30~40 2.

3. 64 ~86 46 ~67 2002 2003 1. 2. 3. 4. 2003 1. 50% 2. 50% 3. 23% 4. 7% 2003

2004 1. 2. 3. 2006 2002

80 180 90 360 2003 1. 37% 2. 80% 2003 1. 2. 3. 2004 1. 2. 1. 2. 3. 4.

5. 1 8 0 2004 270 90 2005 1. 2. 44 3. 70 1995 2 2002 2003

1. 2 2. 3. 4. 5. 2003 70% 2005 2006 2008 1. 2.

3. 2

4. 2

2009

2002

1985

2002

, 2007 selection-type item 2002 2002 ( 2002 1985) 1. 2. 3. 4. 5.

1. 2. 3. 4. 5. 6. 7. 8. 9. 1. 2. 3. 4. 5. 6. 7. 2002

1. 2. 3. 4. 5. 6. 1. 2. 3. 4. 5. OCC

2001

TestGraf 98 item difficulty

index item discrimination index

option characteristic curve OCC 2004

J.O.Ramsay

kernel smoothing approaches to nonparametric item characteristic

curve estimation 1991 Psychometrika kernel

smoothing

1 0

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Ramsay 1991 Ramsay IRT

TestGraf98

2003

2-1 1

2

2-1 1

2-2 2

2-3 1

2

2-4 2 2 1 1

2-4 4

2-5 2

2-6 1 3

2-6 6

2-7 1

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2-8 8

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3-1 3-1 16 1珉0 7 1饋 饋6 13 饋0 饋4 14 饋1 饋恒 30 饋 6 1 3 11 15 × ÷饋 4 饋3 饋7 恒 珉 9 饋饋 10 17 1珉 饋珉 19

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23 71

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3-1 3-1 1 15 7 12 19 26 13 16 24 29 11 20 2 6 9 5 8 22

3-1 3 14 21 25 30 4 18 23 27 28 10 17 3-2 3-2 1. 2. 3. 4. 5. 6. A. 1 21 26 11 14 25 30 2 6 12 B. 4 7 3 20 13 24 29 5 8 16 19 22 C. 28 9 15 10 23 27 17 18 D.

30 5 3 3-1 Bloom 3-2 23 30 40 EXCEL SPSS

Cronbach’s α α .886 α _{lower bound} 3-3 Cronbach’s α 1. α α 1 .879 16 .885 2 .880 17 .878 3 .886 18 .882 4 .887 19 .881 5 .882 20 .882 6 .884 21 .883 7 .883 22 .886 8 .886 23 .875 9 .883 24 .880 10 .876 25 .879 11 .882 26 .884 12 .887 27 .883 13 .884 28 .882 14 .877 29 .887 15 .884 30 .879 Cronbach’s α .886

2. 2002 r_pb 3-4 internal consistency 23 30% 7 30% 7 P_H P_L EXCEL P_H P_L .25 Noll,Scannell &

Craig 1979 .50 Ahmanan and Glock

1981 .40 .70

.20 .80 Aiken 1982

D .40 P .40 .80

3-4 PH PL P=( PH PL)/2 D=PH PL r_pb 1 1.000 0.429 0.714 0.571 .614(**) 2 1.000 0.429 0.714 0.571 .599(**) 3 1.000 0.714 0.857 0.286 .332(*) 4 1.000 0.857 0.929 0.143 .208 5 1.000 0.429 0.714 0.571 .500(*) 6 1.000 0.571 0.786 0.429 .410(*) 7 0.571 0.143 0.357 0.429 .465(*) 8 0.857 0.429 0.643 0.429 .339(*) 9 0.714 0.286 0.500 0.429 .497(*) 10 1.000 0.143 0.571 0.857 .737(**) 11 1.000 0.571 0.786 0.429 .467(*) 12 0.857 0.429 0.643 0.429 .292(*) 13 1.000 0.429 0.714 0.571 .396(*) 14 0.857 0.143 0.500 0.714 .698(**) 15 0.714 0.143 0.429 0.571 .431(*) 16 0.857 0.286 0.571 0.571 .331(*) 17 0.857 0.143 0.500 0.714 .643(**) 18 0.857 0.143 0.500 0.714 .469(*) 19 1.000 0.429 0.714 0.571 .578(**) 20 1.000 0.429 0.714 0.571 .451(*) 21 0.857 0.286 0.571 0.571 .434(*) 22 0.571 0.286 0.429 0.286 .325(*)

3-4 PH PL P=( PH PL)/2 D=PH PL r_pb 23 0.857 0.000 0.429 0.857 .776(**) 24 1.000 0.286 0.643 0.714 .579(**) 25 0.857 0.000 0.429 0.857 .584(**) 26 0.857 0.429 0.643 0.429 .379(*) 27 1.000 0.571 0.786 0.429 .413(**) 28 1.000 0.571 0.786 0.429 .488(*) 29 0.714 0.429 0.571 0.286 .302(*) 30 1.000 0.286 0.643 0.714 .623(**) D .40 P .40 .80 30 5 25 25 4 100 40

3-2

3-2 Bloom SPSS EXCEL TestGraf98 EXCEL

EXCEL

EXCEL

SPSS

SPSS

TestGraf98

Excel Spss/pc r_pb TestGraf98 OCC 10 2011 3 2 3 4 2011 3 2 23 20110302-23

2002 1985 reliability validity 2002 71 SPSS Cronbach α SPSS

4-1 α _.888 4-1 1 .882 11 .886 21 .883 2 .881 12 .879 22 .883 3 .888 13 .886 23 .887 4 .890 14 .882 24 .882 5 .887 15 .884 25 .878 6 .887 16 .885 7 .885 17 .876 8 .886 18 .885 9 .881 19 .884 10 .884 20 .883 25 71 Cronbach α .888 1. Bloom 2. 5 3

71 27% 19 27% 19 EXCEL SPSS 4-2 4-2 PH PL P=( PH PL)/2 D=PH PL rpb 1 1.000 0.579 0.789 0.421 .604(**) 2 1.000 0.579 0.789 0.421 .667(**) 3 0.895 0.684 0.789 0.211 .355(**) 4 0.737 0.368 0.553 0.368 .314(**) 5 0.895 0.368 0.632 0.526 .420(**) 6 0.947 0.684 0.816 0.263 .371(**) 7 1.000 0.316 0.658 0.684 .490(**) 8 1.000 0.684 0.842 0.316 .413(**) 9 1.000 0.263 0.632 0.737 .634(**) 10 0.789 0.211 0.500 0.579 .512(**) 11 0.895 0.474 0.684 0.421 .448(**) 12 0.947 0.263 0.605 0.684 .702(**) 13 0.947 0.316 0.632 0.632 .475(**) 14 1.000 0.474 0.737 0.526 .605(**) 15 1.000 0.684 0.842 0.316 .499(**)

4-2 PH PL P=( PH PL)/2 D=PH PL r_pb 16 1.000 0.579 0.789 0.421 .469(**) 17 1.000 0.053 0.526 0.947 .782(**) 18 0.895 0.158 0.526 0.737 .488(**) 19 1.000 0.579 0.789 0.421 .521(**) 20 1.000 0.526 0.763 0.474 .557(**) 21 1.000 0.579 0.789 0.421 .542(**) 22 0.947 0.211 0.579 0.737 .575(**) 23 0.947 0.316 0.632 0.632 .429(**) 24 0.947 0.316 0.632 0.632 .584(**) 25 1.000 0.158 0.579 0.842 .726(**) .25 Noll,

Scannell, & Craig, 1979, p210 Ebel and Frisbie 1991

2002 4-3 D .40 4-3 0.19 .20~.29 .30~.39 0.40 0 1 3 21 25 0.00% 4.00% 12.00% 84.00% 100.00% 0.547

4-3 .547 .40 .20~.29

1 4 .30 .39 3 12 .40

21 84

.50 .50

Ahmananm and Glock 1981 .40

.70 Chase 1978 .40 .80 .55 .85 2002 4-4 P .40 .80 4-4 0~.19 .20~.39 .40~.59 .60~.79 .80~1 0 0 6 16 3 25 0.00% 0.00% 24.00% 64.00% 12.00% 100.00% 0.678 4-4 .678 .40 .80 4-3 P .4 .79 22 88

1995 A B C D E A 4-1 2 11 12 14 15 16 20 21 24 4-1 A

B A 4-2 4 6 13 4-2 B C 4-3 5 10 22

4-3 C

D S

4-4

1 3 7 8 9 17 19 25

E S

D

4-5 18 23

1

(1) (2)

(3) (4)

2 (1) 24 (2) 36 (3) 40 (4) 50 4-7 2

6 9

10

8

17

3

4 (1) 8 (2) 12 (3) 16 (4) 24

4-8 3

4

3 , 5

(1) 2 (2) 5 (3) 8 (4) 10

5

2 (1) 2 (2) 4 (3) 6 (4 ) 8

6

(1) (2)

(3) (4)

7

2 (1) (2) 2 (3) 4 (4) 6

4-12 7

8

(1) 1 (2) 2 (3) 3 (4) 4

4-13 8

9 BC (1) AB (2) AD (3) AE (4) AF 4-14 9 B D E F C A

10

(1) (2) (3) (4)

11 (1) (2) (3) 360 (4) 60 4-16 11

12

6 8 3

(1) 16 (2) 17 (3) 18 (4) 19

13

2 (1) 2 (2) 4 (3) 6 (4) 8

4-18 13

14 (1) (2) (3) (4) 4-19 14

15

45 45 90

(1) (2) (3) (4)

4-20 15

16

(1) (2) (3) (4) 60

17 10 20 (1) 20 (2) 50 (3) 100 (4) 200 4-22 17 10 20 10 20 20 50 2

底 底 底 18 (1) (2) (3) (4) 底 底

19 (1) 60 (2) 90 (3) 180 (4) 360 4-24 19 90 90 90

20 (1) 24 (2) 36 (3) 48 (4) 60 4-25 20 6 8

6 9

10

8

17

21 A B C ? (1) A (2) B (3) C (4) 4-26 21 C

22 ABC BD (1) AB (2) BC (3) AC (4) CD 4-27 22 AC BC CD A B C D

23 (1) 1 , 2 , 3 (2) 1 , 3 , 5 (3) 2 , 4 , 6 (4) 3 , 4 , 5 4-28 23

24

(1) 0 (2) 1 (3) 2 (4) 3

4-29 24

25 16 8 (1) 1 (2) 2 (3) 4 (4) 8 4-30 25 2 16 8

25 4 100 4-5 72.11 100 12 12 100 72.11 63 4-5 71 100 100 12 72.11 71 4-6 4-1 60 90 4 18 4 18

4-6 1 62 87.32% 14 61 84.51% 2 63 88.73% 15 63 88.73% 3 55 77.46% 16 59 83.10% 4 30 42.25% 17 49 69.01% 5 45 63.38% 18 33 46.48% 6 63 88.73% 19 60 84.51% 7 44 61.97% 20 57 80.28% 8 62 87.32% 21 61 84.51% 9 51 71.83% 22 39 54.93% 10 43 60.56% 23 40 56.34% 11 54 76.06% 24 49 69.01% 12 53 74.65% 25 46 64.79% 13 40 56.34% 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 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 題號 正 確 選 項 答 對 率 4-31

distraction analysis 2002 4-7 4-31 25 75 4 4 5 3 73 4-7 1 * 1* 2 3 4 1.00 0.00 0.00 0.00 0.58 0.32 0.05 0.05

4-8 2 * 1 2* 3 4 0.00 1.00 0.00 0.00 0.11 0.58 0.05 0.26 4-9 3 * 1 2* 3 4 0.00 0.89 0.11 0.00 0.16 0.53 0.21 0.11 4-10 4 * 1 2* 3 4 0.05 0.74 0.11 0.11 0.26 0.21 0.47 0.05

4-11 5 * 1* 2 3 4 0.89 0.00 0.11 0.00 0.37 0.53 0.05 0.05 2 2 2 2 6 4-12 6 * 1* 2 3 4 0.95 0.00 0.05 0.00 0.68 0.05 0.21 0.05

4-13 7 * 1* 2 3 4 1.00 0.00 0.00 0.00 0.32 0.37 0.16 0.16 4-14 8 * 1* 2 3 4 1.00 0.00 0.00 0.00 0.68 0.00 0.26 0.05 4-15 9 * 1 2* 3 4 0.00 1.00 0.00 0.00 0.47 0.26 0.16 0.11

4-16 10 * 1 2 3* 4 0.16 0.05 0.79 0.00 0.21 0.26 0.21 0.32 4-17 11 * 1 2 3* 4 0.00 0.00 0.89 0.11 0.21 0.11 0.47 0.21 180 60 4-18 12 * 1* 2 3 4 0.95 0.05 0.00 0.00 0.26 0.32 0.32 0.11

4-19 13 * 1 2* 3 4 0.05 0.95 0.00 0.00 0.47 0.32 0.16 0.05 4-20 14 * 1 2 3* 4 0.00 0.00 1.00 0.00 0.11 0.05 0.47 0.37 4-21 15 * 1 2* 3 4 0.00 1.00 0.00 0.00 0.21 0.68 0.05 0.05 90 90

4-22 16 * 1 2* 3 4 0.00 1.00 0.00 0.00 0.26 0.58 0.11 0.05 4-23 17 * 1 2 3* 4 0.00 0.00 1.00 0.00 0.05 0.21 0.05 0.68 2 4-24 18 * 1* 2 3 4 0.89 0.11 0.00 0.00 0.16 0.47 0.26 0.11

4-25 19 * 1 2 3* 4 0.00 0.00 1.00 0.00 0.11 0.26 0.58 0.05 90 90 90 4-26 20 * 1 2* 3 4 0.00 1.00 0.00 0.00 0.05 0.53 0.16 0.26 4-27 21 * 1 2 3 4* 0.00 0.00 0.00 1.00 0.00 0.05 0.37 0.58 C

4-28 22 * 1 2 3* 4 0.05 0.00 0.95 0.00 0.58 0.05 0.21 0.16 4-29 23 * 1 2 3 4* 0.05 0.00 0.00 0.95 0.37 0.05 0.26 0.32 1 3 4-30 24 * 1 2 3* 4 0.05 0.00 0.95 0.00 0.11 0.21 0.32 0.37 90 90

4-31 25 * 1 2 3* 4 0.00 0.00 1.00 0.00 0.05 0.42 0.16 0.37 4-32 1 10 7 14 19 8 24 11 15 2 3 5 4 23 6 9 16 18 22 12 25 13 17 20 21 EXCEL

4-33 1 2 3 4 1 0.87 0.08 0.03 0.01 10 0.13 0.10 0.61 0.17 1 87% 10 61% 1 4-34 1 2 3 4 7 0.62 0.18 0.13 0.07 14 0.03 0.01 0.85 0.11 19 0.03 0.07 0.85 0.06 1 14 19 62% 85% 85% 2

4-35 1 2 3 4 8 0.87 0.01 0.10 0.01 24 0.06 0.14 0.69 0.11 8 24 87% 69% 180 3 4-36 1 2 3 4 11 0.06 0.06 0.76 0.13 15 0.07 0.89 0.01 0.03 11 15 76% 89% 1

4-37 1 2 3 4 2 0.03 0.89 0.01 0.07 3 0.04 0.77 0.15 0.03 5 0.63 0.21 0.13 0.03 2 3 89% 77% 5 63% 2 4-38 1 2 3 4 4 0.10 0.42 0.18 0.30 23 0.14 0.14 0.15 0.56 4 23 42% 56%

3 4-39 1 2 3 4 6 0.89 0.01 0.08 0.01 6 89% 1 4-40 1 2 3 4 9 0.15 0.72 0.08 0.04 16 0.10 0.83 0.03 0.04 18 0.46 0.32 0.17 0.04 22 0.27 0.10 0.55 0.08 9 16 18 22 72% 83% 46% 55%

2 4-41 1 2 3 4 12 0.75 0.11 0.11 0.03 25 0.03 0.23 0.65 0.10 12 25 75% 65% 3 4-42 1 2 3 4 13 0.32 0.56 0.10 0.01 17 0.01 0.08 0.69 0.21 20 0.01 0.80 0.11 0.07 21 0.03 0.01 0.11 0.85 13 56% 17 20 21 69% 80% 85%

1 1. (1) (2) (3) (4) T：你認為這四個圖形當中哪一個是三角形？ S20110302-12：第二個圖。 T：你為什麼說它是三角形？ S20110302-12：因為它有三個角、三個邊。 T：那其他三個圖形為什麼不是三角形？ S20110302-12：第一個圖它太長了，太細。第三個圖有一個角沒有接起來。第四個圖有 一個邊不是平的。 2 7. 2 (1) (2) 2 (3) 4 (4) 6 T：根據題目，三個內角的和變為原來的幾倍？ S20110302-14：2 倍 T：為什麼說它變成 2 倍。 S20110302-14：因為邊長變大之後，整個圖形都會變大，所以面積、周長、角度都會變 大。

24. (1) 0 (2) 1 (3) 2 (4) 3 T：一個鈍角三角形有幾個銳角？ S20110302-19：畫出來有一個銳角。 T：你試著畫畫看。 S20110302-19：這個鈍角三角形的三個角裡面，有一個角尖尖的，另外兩個角鈍鈍的， 所以只有一個銳角。 T：一個鈍角三角形有幾個銳角？ S20110303-19：三個角都是銳角。 T：你試著畫畫看。 S20110303-19：這個鈍角三角形的三個角裡面，三個角度都比九十度小。 15. 45 45 90 (1) (2) (3) (4) T：這個三角形是哪種三角形？ S20110303-22：正三角形。 T：為什麼？ S20110303-22：因為有兩個角都一樣是 45 度。 T：為什麼兩個角一樣就是正三角形？ S20110303-22：因為畫出來邊長也會一樣。 T：可以試著畫畫看嗎？ S20110303-22： T：請你再試著說說看這個圖形為什麼是正三角形。 S20110303-22：左邊跟右邊兩個角一樣，兩個邊也一樣，看起來正正的，所以是正三角 形。

3 2. (1) 24 (2) 36 (3) 40 (4) 50 T：請問要怎麼算出塗色部分的周長？ S20110303-16：把所有線段加起來。 T：分別是哪些線段？加起來是多少？ S20110303-16：分別是 8、6、10、9、17，所以 8+6+10+9+17=50 T：請問告訴我塗色部分的面積是哪一塊？【澄清迷思概念】 S20110303-16：這一塊（學童指出邊長 10、9、17 公分的三角形）。 T：那麼它的周長在哪裡？ S20110303-16：這裡（學童指出 10、9、17 公分的三個邊長）。 T：那麼它的周長是多少。 S20110303-16：10+9+17，所以應該是 36 公分才對。 4. 3 , 5 (1) 2 (2) 5 (3) 8 (4) 10 T：哪個長度可能是三角形的第三邊？為什麼？ S20110304-18：8 公分，因為把其中兩個邊 3 公分跟 5 公分加起來，就可以找到第三個邊 是 8 公分。 6. (1) (2) (3) (4) T：哪一個圖形是正三角形？為什麼？ S20110304-13：選項 3 是正三角形，因為左右兩邊看起來一樣長，其他三個圖形看起來 不一樣長。

6 9

10

8

17

4 22. ABC BD (1) AB (2) BC (3) AC (4) CD T：根據題目，請找出對應的底邊，並說說看為什麼？ S20110304-03：線段 AB，因為線段 AB 與線段 BD 相交於一點，所以線段 AB 是底邊， 線段 BD 是高。 T：底邊和高這兩條段應該要互相？【澄清迷思概念】 S20110304-03：垂直。 T：那麼和線段 BD 垂直的有哪些線段？ S20110304-03：線段 AD。 T：選項裡面沒有線段 AD，所以應該是？ S20110304-03：線段 AC 和線段 CD。 T：這兩條線段當中哪一條在三角形的邊上？ S20110304-03：線段 AC。 T：所以線段 BD 當作高的底邊是哪個邊？ S20110304-03：應該是線段 AC 才對。 T：那麼它們有沒有相交於一點？ S20110304-03：沒有。 T：也就是說鈍角三角形的底和高會互相垂直，但不一定相交在一點上，但是延長線會 相交於一點。 25. 16 8 (1) 1 (2) 2 (3) 4 (4) 8 T：算算看三角形的底是幾公分？ S20110304-07：用面積除以高，就可以算出底，所以 16÷8＝2，答案是 2 公分。 13. 2 (1) 2 (2) 4 (3) 6 (4) 8 T：三角形的三邊長分別變為原來的 2 倍時，其面積變為原來的幾倍？為什麼？ S20110304-02：2 倍，因為邊長變得愈大，面積也會變得愈大，邊長變大幾倍，面積就會 A B C D

變大幾倍，所以是 2 倍。

.678 .50 .40 .80 P .4 .79 22 88 .547 .40 Cronbach α .888 .80

Bloom 5 3 A B C D E A 2 11 12 14 15 16 20 21 24 9 B 4 6 13 3 C 5 10 22 3 D S 1 3 7 8 9 17 19 25 8 E S 18 23 2 25 4 100 72.11 100 12 63

4-6 4-1 60 90 4 18 4 18 25 75 4 4 5 3 73 87% 61% 62% 85% 85% 87% 69% 180 76% 89%

89% 77% 63% 42% 56% 89% 72% 83% 46% 55% 75% 65% 56% 69% 80% 85%

2005 2002 2006 -- Duval 78-116 1998 van Hiele 20-61 2001 -2002 2007 2005 1992 van Hiele VHL 2008 2004 2003

2002 2009 2009 2001 2003 2003 1995 1985 2006 2006 2001 2004 Testgraf98 93-104 2003

2000 IRT -Testgraf98 33 23-32 2002 1999 2004 2002 2003 2004 ~ Duval van Hiele 1995 35 7.8

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□ □ 1. 30 2. 3. 4 4. 5

1. (1) (2) (3) (4) 2. (1) 24 (2) 36 (3) 40 (4) 50 3. 4 4 5 (1) (2) (3) (4) 4. (1) ☓ (2) ☓ (3) ☓ ÷ 2 (4) × ÷ 2 5. (1) 1 , 2 , 3 (2) 1 , 3 , 5 (3) 2 , 4 , 6 (4) 3 , 4 , 5 6. 4 (1) 8 (2) 12 (3) 16 (4) 24 7. (1) (2) (3) (4)

6 9

10

8

17

8. 3 , 5 (1) 2 (2) 5 (3) 8 (4) 10 9. 2 (1) 2 (2) 4 (3) 6 (4 ) 8 10. 16 8 (1) 1 (2) 2 (3) 4 (4) 8 11. (1) (2) (3) (4) 12. 2 (1) (2) 2 (3) 4 (4) 6 13. (1) 1 (2) 2 (3) 3 (4) 4 14. BC (1) AB (2) AD (3) AE (4) AF 15. (1) (2) (3) (4) B D E F C A

16. (1) (2) (3) 360 (4) 60 17. 6 8 3 (1) 16 (2) 17 (3) 18 (4) 19 18. 2 (1) 2 (2) 4 (3) 6 (4) 8 19. (1) (2) (3) (4) 20. 45 45 90 (1) (2) (3) (4) 21. (1) (2) (3) (4) 60 22. (1) 2 , 6 , 6 (2) 2 , 2 , 6 (3) 3 , 3 , 6 (4) 23. 10 20 (1) 20 (2) 50 (3) 100 (4) 200 24. (1) 0 (2) 1 (3) 2 (4) 3

底 底 底 25. (1) (2) (3) (4) 26. (1) 60 (2) 90 (3) 180 (4) 360 27. (1) 24 (2) 36 (3) 48 (4) 60

6 9

10

8

17

底 底

28. A B C ? (1) A (2) B (3) C (4) 29. (1) 1 (2) 2 (3) 3 (4) 4 30. ABC BD (1) AB (2) BC (3) AC (4) CD A B C D

α α 1 .879 16 .885 2 .880 17 .878 3 .886 18 .882 4 .887 19 .881 5 .882 20 .882 6 .884 21 .883 7 .883 22 .886 8 .886 23 .875 9 .883 24 .880 10 .876 25 .879 11 .882 26 .884 12 .887 27 .883 13 .884 28 .882 14 .877 29 .887 15 .884 30 .879 Cronbach’s α .886

1. (1) (2) (3) (4) 2. (1) 24 (2) 36 (3) 40 (4) 50 3. 4 (1) 8 (2) 12 (3) 16 (4) 24 4. 3 , 5 (1) 2 (2) 5 (3) 8 (4) 10 5. 2 (1) 2 (2) 4 (3) 6 (4 ) 8 6. (1) (2) (3) (4)

6 9

10

8

17

7. 2 (1) (2) 2 (3) 4 (4) 6 8. (1) 1 (2) 2 (3) 3 (4) 4 9. BC (1) AB (2) AD (3) AE (4) AF 10. (1) (2) (3) (4) 11. (1) (2) (3) 360 (4) 60 12. 6 8 3 (1) 16 (2) 17 (3) 18 (4) 19 13. 2 (1) 2 (2) 4 (3) 6 (4) 8 14. (1) (2) (3) (4) B D E F C A

底 底 底 15. 45 45 90 (1) (2) (3) (4) 16. (1) (2) (3) (4) 60 17. 10 20 (1) 20 (2) 50 (3) 100 (4) 200 18. (1) (2) (3) (4) 19. (1) 60 (2) 90 (3) 180 (4) 360 底 底

20. (1) 24 (2) 36 (3) 48 (4) 60 21. A B C ? (1) A (2) B (3) C (4) 22. ABC BD (1) AB (2) BC (3) AC (4) CD 23. (1) 1 , 2 , 3 (2) 1 , 3 , 5 (3) 2 , 4 , 6 (4) 3 , 4 , 5 24. (1) 0 (2) 1 (3) 2 (4) 3 25. 16 8 (1) 1 (2) 2 (3) 4 (4) 8

6 9

10

8

17

A B C D

1 .882 11 .886 21 .883 2 .881 12 .879 22 .883 3 .888 13 .886 23 .887 4 .890 14 .882 24 .882 5 .887 15 .884 25 .878 6 .887 16 .885 7 .885 17 .876 8 .886 18 .885 9 .881 19 .884 10 .884 20 .883

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

1. 2. 3. 4. 5. 6. E. 1 16 19 6 9 18 22 2 3 7 F. 15 8 24 23 4 11 14 G. 21 5 10 25 17 20 12 13 H.