.1.6.6.7.. 8. 25.....44.....49.54.57. 61.63.65. 73
85 97 99...101...103. 105. 108...111...112...113...114...115...116...117...118...119
.....120.....121.....122.....123.....124.....125.....126.....127.....128.....129 2-1 Polya 1945......9 2-2......21 2-3..41 2-4.. 42 2-5.. 51 3-1.. 54
2-1 Polya 1945. 10 2-2 Schoenfeld 1985 14 2-3 Mayer 1992...19 2-4..20 2-5..22 2-6..47 2-7..48 2-8..49 2-9..52 3-1..56 3-2..57 3-3..62 3-4..66 3-5..69 4-1..75 4-2..76 4-3..88 4-4..96
~ 1999 1996 1997 1999 82 1995 1
1989 82 1995 1. 2. 82 1993 -- 1995 2002 2
Kilpatrick, 1987 Krulik Rudnick, 1993 Silver Cai, 1993 1994 Brown Walter, 1993 1994 1999 82 82 2002 2002 3
2000 National Council of Teachers of Mathematics NCTM NCTM, 1989 138 NCTM 1991 95 2003 11 14 94 93 2004a 2004a 4
2004a -- 5
6
.. 1999 2-8.. Mayer 1992 Mayer 7
Polya 1945 Schoenfeld 1985 Mayer 1992 Polya. Polya. Schoenfeld. Mayer.. 8
. Polya Polya 1945 How to solve it 2-1 1. Understand 2. Plan 3. Carry out 4. Look back 2-1 Polya 1945 9
heuristic Polya Polya 2-1 Polya 80 Polya 1999 Polya Polya Polya Polya Polya 2-1 Polya Polya, 1945 1993 12 10
11
. Schoenfeld Schoenfeld 1985 1999 Schoenfeld 1985 resources heuristics control belief system heuristics strategies 12
Schoenfeld 1985 2-2 1. reading 2. analysis 3. exploration well structured 4. planning 5. implement 13
6. verify 2-2 Schoenfeld PI1 PI2 PI3 PI4 PI5 PI6 2-2 transition Schoenfeld 2-2 Schoenfeld Schoenfeld, 1985 1996 301 reading R1 R2 R3 analysis A1 A2 A3 A4 A5 A1-A4 14
exploration E1 E2 E3 E4 -- planning-implementation PI1 PI2 PI3 PI4 PI5 PI6 verification V1 V2 V3 transition T1 T2 T3 T4 15
Schoenfeld Polya Schoenfeld Schoenfeld Schoenfeld metacognition Schoenfeld Polya Schoenfeld thinking aloud 1996 protocol 16
. Mayer Mayer 1992 Mayer 1987, 1992 1. 2. 3. 4. Mayer 17
2-3 1. 2. 1 100 3. 4. 5. Mayer Schoenfeld Polya Mayer Schoenfeld 18
2000 Mayer 2-3 Mayer Mayer, 1992 459 19
. Polya Schoenfeld Mayer 2-4 2-4 Polya 1945 Schoenfeld 1985 Mayer 1992 Schoenfeld Mayer Polya Schoenfeld Mayer 7 2-2 20
1. 2. 3. 4. 5. 2-2 1. 2. 3. 4. 5. Mayer 1 4 21
. 2-5 2-5 22 51 1962 57 1968 64 1975 82 1993 91 2002 92 2004b
1996 1. 2. 3. 4. 5. 6. 7. 8. 1999 1. 2. 3. 4. 5. 23
1999 -- 1. 2. 3. 1. 2. 3. 4. 5. 6. 6 24
.... ( ) ( ) ( ) ( ) 1. Dillon 1982 2. Silver 1994 3. 1994 4. Stoyanova and Ellerton 1996 25
( ) 1994 153 1 idiosyncratic 2 plausible reasoning 3 before during and after problem solving primitive incomplete inplausible insufficient English 1997 26
2002 Leung Silver, 1997 ( ) 1. Brown and Walter 1983 62 0 Choosing a starting point 1 Listing attributes 0 2 What-if-not 1 3 Question asking or problem posing 27
4 Analyzing the problem 2. 1987 1994 164 A. B. C. D. E. F. G. 3. Moses, Bjork and Goldenberg 1993 28
Moses A. B. C. D. 4. Stoyanova and Ellerton 1996 A. structured B. semi-structured C. free 5. Cudmore and English 1998 A. B. C. 29
D. E. F. G. H. I. 6. 1998 A. B. C. D. E. F. G. H. I. J. K. L. 1987 30
31
. ( ) NCTM, 1989 Keil 1965 800 Stover 1982 Brown and Walter 1983 The art of problem posing 32
What-if-not Silver 1993 open approach teaching Nohda, 1984 Hashimoto, 1987 1987 Australian Education Council Stovanova & Ellerton, 1996 Skinner What s your problem Skinner, 1990 33
Winograd 1990 Winograd 8 17 25 Winograd Borba 1994 200 34
Schloemer 1994 What-if-not Silver Mamona-domwns Leung Kenney 1996 53 28 IP Initial Posing PS Problem Posing AP Additional Posing Leung Silver 1997 TAPP Test Arithmetic Problem Posing 63 English 1997 35
English English 1998 54 Cai 1998 181 223 ( ) 1995 65 127 36
1996 104 40 / 1997 1998 1999 37
2000 2002 2003 38
2003 2004 1. 2. 3. 39
4. 40
Polya 1945 How to solve it Understand Plan Carry out Look back 1994 Polya Pose Understand 2-3 Look back Pose Carry out Plan 2-3 1994 159 2-3 41
1994 159 2004 15 Polya 1945 1994 2-4 2004 15 42
English 1997 NCTM, 1989 1987 Brown & Walter, 1983 Stoyanova & Ellerton, 1996 English 1997 43
. 2000 79 82 ( ) 44
( ) 1. 2. 3. 45
. 2000 2-6 Mayer 2-7 -- 3-2 46
2-6 1. N-2-2 1-1 1-2 1-3 2. N-2-17 2-1 2-2 2-3 2-4 3. N-2-17 3-1 3-2 3-3 3-4 4. N-2-15 4-1 4-2 4-3 4-4 5. N-2-15 5-1 5-2 47
48 2-7 1. A-2-1 2. N-2-3 7. N-2-2 1. A-2-1 2. A-2-2 5. N-2-2 6. N-2-2 8. N-2-2 3. 1. A-2-1 2. A-2-2 N-2-2 4. N-2-16 5. N-2-16 1. N-2-2 1-3 N-2-17 2. N-2-17 3. N-2-17
1999 Mayer 1992. 1999 2-8 5 1 5 1 5 ordinal interval ratio 2-8 1999 204 1 2 3 4 5 5 49
2-5 50
1 2 3 4 5 5 2-5 1999 205 51
. 2-2 Mayer 1992 1 5 2-9 2-9 Mayer 1. 2.. 3.. 4.. 5. Mayer 52
( ) ( ) ( ) ( ) 53
8 5. 3-1 54
. 1987 3-1.... 55
3-1 1. 2. 3. 4. 1. 2. 3. 4. 5. 6. 7. 56
. 4 8 3-2 ( ) ( ) ( ) ( ) ( ) 3-2 1. 1. 72 47 21 2. 2. 5 285 7 3. ( ) 8 35 ( ) 12 1 57
4. 3 189 70 1 4. 5. 3 1 324 8 17 6. 1 280 4 400 45 B 7. A 10 B 14 8 A 10 8 10 14 8. 504 12 12 12 12? 58
. 3-2 8 72 47 21. 59
.. 60
S 82 91 91 92 3-3 61
3-3 82 92 91 93 92 91 90 89 88 62
. 93 1 ~93 3. 93 4 1 1 93 4 6 7. 93 5~6 8 8 93 5 3 6 10 13 17 20 24 27 31 6 3 7 10 14 17 21 24 8 63
1 8 8. 93 7 ~93 10. 93 10 ~93 12 64
36 18 p p s m p-s-19 19 p-p-2 2 1 24 336 4.... 65
. ( ) 36 5 p-p-5 ( ) 3-4 1 3-4 3 33 2 34 35 1 1. 1 66
p-p-33 366 25 10 14.64 10 146.4 2 p-p-31 226 2 512 3 p-p-6 10000 100 10 1000 4 p-p-10 6 120 12 6 60 2. 67
p-p-14 3. p-p-27. ( ) 3-5 1. 68
30 83% 6 3 3 2. 3 3-5 33 92% 3 3 0 63 88% 9 6 3 ( ) 3-5 6 1 24 336 4 p-s-10 336 4 84 10 336 4 69
( ) 3 1 1 1 p-p-12 5000 250 150 200 150 30000 12 1 ( ) p-m-15 70
p-m-13 ( ) 5 p-m-18 p-m-25 5 p-m-28 ( ) p-m-1 p-m-3 p-m-9 answer p-m-26 p-m-27 p-m-34 71
. ( ) ( ) ( ) 72
s p m 3-p-20 3 20 8-s-12 8 12. 4-1 5 98.5% 73
4 1 3 3 ( ) 4 4-p-35 ( ) 7 7-p-4 ( ) 8 8-p-30 74
4-1 1 2 3 4 5 5 1 35 2 34 3 35 4 34 5 35 6 35 7 1 33 8 3 32 3 (1.1%) 1 (0.4%) 273 (98.5%) 75
. 4-2 4-2 35 1 3 32 0 35 29 6 34 2 1 33 0 34 28 6 35 3 0 35 0 35 28 7 34 4 2 32 0 34 17 17 35 5 0 35 0 35 29 6 35 6 0 35 0 35 29 6 34 7 1 33 19 15 9 25 35 8 4 31 23 12 24 11 1. 2. 2 4 7 1 4-3 76
7 ( ) 1. 1 72 47 21 1 1-p-3 1-p-12 77
2. 2 5 285 7 2 2-p-28 2-p-35 3. 4 3 189 70 1 4 4-p-21 25000 34 1 8 504 12 12 12 12 78
8 8-p-33 502 15 15 ( ) 7 8 1. 1 72 47 21 1 1-p-21 79
2. 3 8 35 12 1 3 1-p-5 5 3. 1 72 47 21 1 1-p-14 80
4. 3 8 35 12 1 3 3-p-34 2 1 81
( ) 1. 1 72 47 21 1 1-p-12 370000 5 3 1 324 8 17 5 5-p-26 5000 82
2. 2 5 285 7 5 2-p-12 1 18 3. 7 A 10 B 14 8 A B 10 8 10 14 7 7-p-12 83
7 7-p-16 4. 4 3 189 70 1 4 4-P-12 10 500 2 84
.. ( ) ( ) ( ). 85
. 8 8 4-3 2 4 7 1 34 8 3 32 ( ) ( ) ( ) ( ) 4-3 5 6 3 3-2 3 86
( ) 4-1 8 8 9 4 ( ) 4-1 1994 4-3 87
4-3 ( ) 34 (97%) 1 0 1 33 (94%) 2 0 2 30 (86%) 5 1 4 31 (89%) 3 1 2 33 (94%) 2 2 0 32 (91%) 3 0 3 34 (97%) 1 0 1 29 (85%) 5 0 5 28 (80%) 7 5 2 28 (80%) 7 3 4 20 (57%) 15 7 8 24 (69%) 11 7 4 22 (63%) 13 11 2 25 (74%) 9 4 5 16 (46%) 19 18 1 22 (69%) 10 5 5 1. 2. 1 4 5 8 88
. ( ) ( ) ( ) ( ) 1. 5 3 1 324 8 17 5 324 3 17 8 3 3 2. 5 5-p-33 89
450 5 25 5 5 5 3. 5 5-p-34 350 1500 6 6 6 ( ) 1. 7 A 10 B 14 8 A B 10 8 10 14 8 90
10 4 14 2 8 2 2. 7 7-p-4 20 4 14 2 9 ( ) 1. 6 1 280 4 400 45 3 280 4 45 -- 400 2. 6 6-p-33 91
350 3 40 -- 500 92
. 4-4 ( ) 1-p-2 100000 1999 18001 18001 100 18101 ( ) 1 280 4 400 45 6-s-8 280 4 1120 1120 400 720 750 45 16 30 16 1 17 93
( ) 2-p-14 5123555 5 15 60000000 ( ) 4-p-36 200 3 66 2 70-66 4 4-4 2004 94
95
1 1 4-4 1 1 2 2 4 4 1 1 2 0 2 1 3 1 1 2 1 1 1 5 2 2 2 1 1 4 6 1 1 8 2 1 1 4 2 2 3 2 5 2 1 2 5 30(63%) 5(10%) 9(19%) 4(8%) 48 96
. 3-m-16 4-m-7 3-m-27 8-m-11 97
. 1-m-23 2-m-20 5-m-31. 4-m-29 2-m-11 6-m-10 98
. 1-t 4-T 20 3-t. 14 2-t 99
12 1-t 15 7-t. 6 4-t 5-t 24 18 8- t 100
. 98.5%. 101
.. 102
. 83.. 103
.... 104
105 1996 2002 10 135-172 2004 1989 1999 NSC 88-2815-C-023-001-S 2002 1995 1987 1999 62 50-63 2003 2003 2003 1997 NSC86-2815-C-023-005-H 1996 2003
106 2003 1994 152-167 1995 NSC 83-0111-S-023-007 NSC 84-2511-023-001 1997 NSC 84-2511-S-023-006 1999 184-220 1962 1968 1975 1993 2002 2004a 2004 3 27 http://teach.eje.edu.tw/9cc/textbooksource/2004_0205/index.htm 2004b 2002 6 6 19-29 1993-1 1 101-108 1995
2000 1996 37 3 38-44 1999 -- 60-91 1998 1996 247-257 1999 1999 2000 NSC 88-2614-S-153-001 2000 78-97 1999 286-305 Polya 1945/1993 How to Solve It : a new aspect of mathematics method 107
Borba M. C. (1994). High School Students' Mathematical Problem Posing: An Exploratory Study in the Classroom. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA. Brown, S. I. & Walter, M. I. (1983). The Art of Problem Posing. Philadelphia, PA: The Franklin Press. Brown,S. I. & Walter, M. I. (1993). Problem posing: Reflection and application. Hillsdale, NJ: Lawrence Erlbaum Associates. Cai, J. (1998). An investigation of U. S. and Chinese students mathmatical problem posing and problem solving. Mathematics Education Research Journal, 10(1), 37-50. Cudmore, D. H. & English, L. D. (1998). Using Intranets to Foster Statistical Problem Posing and Critiquing in Secondary Mathematics Classrooms. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA. Dillon, J. T. (1982). Problem finding and solving. Journal of Creative Behavior, 16, 97-111. English, L. D. (1997). Promoting a problem-posing classroom. Teaching children Mathematics, 4(3), 172. English, L. D. (1998). Children s problem posing within formal and in formal context. Journal for Research in Mathematics Education, 29(1), 83-106. Hashimoto, Y. (1987). Classroom practices of problem solving in Japanese elementary school. Proceedings of the U.S. Japan Seminar on Mathematical Problem solving. Keil, G. E. (1965). Writing and solving original problems as a means of improving verbal arithmetic problem solving ability. Doctoral dissertation. 108
Kilpatrick,J. (1987). Problem formulation: Where do good problems come from? In A. H. Shoenfeld (Ed.), Cognition science and mathematics (pp. 123-147). Hillsdale, NJ: Lawrence Erlbaum Associates. Krulik, S., Rudnick, J.A.(1993). Reasoning and problem solving: A handbook for elementary school teachers. Boston: Allyn and Bason. Leung, S. S. & Silver, E. A. (1997). The role of task format, mathematics knowledge, and Creative thinking on the arithmetic problem posing of prospective elementary school teachers. Mathematics Education Research Journal, 9 (1), 5-24 Mayer, R. E. (1987). Education psychology: A cognitive approach. Toronto: Little, Brown and company. Mayer, R. E. (1992). Thinking, problem solving, cognition. New York: W. H. Freeman and Company Press. Moses, B., Bjork, E., Goldenberg, E. P. (1993). Beyond peoblem solving: problem posing. In S.I. Brown M. I. Walter (Eds), Problem posing: Reflections and applications (pp. 178-188). Hillsdale, NJ: Lawrence Erlbaum Associates. National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (1991). Professional standards for teaching mathematics. Reston, VA: Author. Nohada, N. (1984). The heart of open approach in mathematics teaching. In T. Kawaguchi (Ed.), Proceddings of ICIM-JSME regional conference on mathematics education (pp. 314-318). Tokyo: Japan society of Mathematics Education. Polya, G. (1945). How to solve it. (2nd ed.). New York: Doubleday. 109
Schloemer, C. G. (1994). Integrating problem posing into instruction in advanced algebra: Feasibility and outcome. Doctoral Dissertation, University of Pittsburgh. Schoenfeld, A. H. (1985). Mathematics problem solving. New York: Academic Press. Silver, E. A. & Cai, J. (1993). Mathematics problem posing and problem solving by middle school students. Paper presented at AERA. Atlanta, GA. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19-28. Silver, E. A., Mamona-Downs, J., Leung,S.S. & Kenney P.A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in mathematics Education, 27(3), 293-309. Skinner, P. (1990). What s your problem: Posing and solving mathematical problem, K-2. Portsmouth, NH: Heinemann. Stover, G. B. (1982). Structural variables affecting mathematical word problem difficulty in sixth graders. Dissertation Abstracts International, 42, 5050A. Stoyanova, E. & Ellerton, N. F. (1996). A framework for research into student s problem posing in school mathematics. In Corwin, R. B. (Ed.), Talking Mathematics: Supporting Children s Voices. Portsmouth, NH. Winograd, K. (1990). Writing, solving and sharing original math story problem: Case studies of fifth grade children s cognitive behavior. Doctoral Dissertation, University of Northern Colorado. 110
1-s-23 1-s-7 111
2-s-17 2-s-34 112
3-s-22 3-s-3 113
4-s-35 4-s-33 114
5-s-31 5-s-24 115
6-s-11 6-s-14 116
7-s-12 7-s-7 117
8-s-1 8-s-36 118
5 1-p-30 5 1-p-29 119
5 2-p-16 5 2-p-29 120
5 3-p-15 5 3-p-4 121
5 4-p-12 10 500 2 5 4-p-35 122
5 5-p-26 5 8 5 5-p-9 2000 123
5 6-p-32 5 6-p-27 124
5 7-p-28 5 7-p-24 125
5 8-p-5 5 8-p-19 126
p-m-15 1-m-16 2-m-28 127
3-m-14 4-m-11 5-m-35 128
6-m-32 7-m-29 8-m-36 129