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1 Chinese Journal of Science Education 2003,, , 11(1), Greene, Herman, & Haury, 2000; Mullis et al., Edwards, 1994; Grouws & Cebulla, 2000a; Kilpatrick, 1992

2 28 (7) 1989 NCTM, 1989;, NCTM, 2000 MathCAL MathCAL, 2002a 84, 1997;, 1997, 1998; Lin, Juang & Sun, 1999; Lin, Juang & Sun, 2000a, 1999; Lin et al., 2000b (1), 2000; Lin et al., 2001 (2) (3) (4), 2001 (5) MathCAL (6)

3 29 MathCAL (5) (6) Pólya Pólya, 1971 NCTM, 2000 technology equity curriculum teaching learning assessment, 1989;, 1984;, 1990;, 1992; Campione et al., 1989; Garofalo & Mtetwa, 1990 (1) (1) (2) (3) (2) (4) (3)

4 30 (4) Liu2000 PCLogo (5) (6) CAI CAI, 2002b Software to Support Mathematics TeachingBASS, 2002, Math University, 2002, The Kieran Geometry Center University of Minnesota, , ESC Software LibraryESC, 2002 Cardelle- CAI Elawar Wetzel1995 IDEA CAI CAI self-regulated Abidin Hartley1998 FunctionLab 1999 CAI Nakano Murakami, Hirashima Takeuchi basic feedbackcollins, 2000 Carnine, & Gersten, 1987 ILE Chen, Horng Yang 2000 concept map Logo Collins,

5 31 Carnine Gersten CAI elaborated MathCAL corrective feedback Collins, Carnine, & Gersten, , 1990;, / Burton 1982 Help binary tree 2 assistance Empowering Reactive Learning Modeling Coaching Tutor postorder (1) MathCAL (2) MathCAL 1 (3) Ma thcal MathCAL

6 32 linked recursive lists node NULL (1) MathCAL 3 (2) a x = b MathCAL 1 a x = b b (3) x = a MathCAL (4) Petri Nets (3) Carl Adam Petri 1960 (5) automata theory concurrently (6) occurring events (7) x + 2x 1 x x = x 2 3

7 33, 1996; Petri 5 Nets World, 2001 Peterson Peterson, 1981, p.11 G (Bipartite Directed Graph)G =V, A V = {v 1,v 2,...,v s } vertices A = {a 1, a 2,, a r } directed arcsa i = {v j, v k } 6 v j, v k VV P T V = P T P T = a i A a i =v j, v k v j P v k Tv j T v k P AB C P = {p 1, p 2,, p n n 0 } places CAB T = {t 1, t 2,., t m m 0 } transitions DCB = θcd A P T T P CAB A = θ AB = 26 AB CAB Cosθ = tokens 12/13 CAB AC = 24 enabled Cosθ = 12/13 Sinθ = 5/13 4 1/2 * AB * AC * Sinθ CAB = bipartite directed graph BDGPN

8 34 (Places) (Transitions) (Arcs) (Tokens) 4 Set of Places Set of Transitions 5 AB O CD C DCB = θcosθ = 12/13 AB = 26 CAB C θ A O B D 6 3, 1998

9 35 P1 1 P2 2 P4 4 P3 3 P5 5 C AB=26 Cosθ =12/13 DCB=θ CD P6 ACB P7 CAB C T18 P4 CAB A=θ 2 3 T8 3 T3 AC=24 P9 Sinθ =5/13 2 T11_1 P10 =120 7

10 36 1 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 2 T3 T8 T11 T11_11/2* * * Sinθ T17 T18 T11_2 * /2 4 MathCAL T0_1 T0_ m x n m n p i It j p i O t j p, t, I, O MathCAL

11 37 3 T0 T0_1 T1 T2 T3 T0_2 T4 T4_1 T5 T6 T7 T8 T9 T10 T4_2 T11 T11_11/2* * * Sin θ T12 T T11_2 * /2 T14 T14_1 T14_2 T14_3 T14_4 T15 T15_1 T16 T17 T18 T19 T20 T15_2

12 sin A* cos A 23. sin A + cos A P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 T T T11_ T T MathCAL BDGPN BDGPN 5 BDGPN Entity-Relationship Diagrams Date, (1) relational database (2) Structured Query Language (3) (4)

13 39 (5) (6) (7) 8 entity sets 8 MathCAL normalization MathCAL 8 MathCAL relation entity Pólya1971 MathCAL (1) (2) (3) (4)

14 40 (1) K (2) U (3) T ) K U T 8MathCAL

15 CD C ACB CAB T (1) (2) (3) (1) (2) MathCAL

16 AB O 2 CD C AB CAB CD O C ACB 3 DCB = θ DCB θ 4 Cos θ = 12/13 = 12/13 5 AB = 26 CAB AB = A 3 B B A A B 1 B B 2 A B 3 A B A B A MathCAL 9 Client Java applets MathCAL Java application application 1 server Java Socket

17 43 MathCAL thread input stream output Socket Client MathCAL 10 MathCAL 11 9MathCAL JDBC-ODBC SQL Bridge Client Server Client Server Client Client Client Server Server Encode Client Server ( ) Client Client ❶ 1314 ❷ ❸ ❹ 15 ]

18 44 11MathCAL MathCAL MathCAL MathCAL MathCAL 17

19 45 MathCAL MathCAL MathCAL MathCAL () 14 () 15

20 46 15 MathCAL NSC S NSC S S MathCAL CAI, 7(3),

21 47 [BASS] (2002). Software to Support Mathematics Teaching. Retrieved September 20, 2002 from MathCAL Booch, G. (1994). Object-oriented analysis and design with applications. (2 nd Ed.). New York: Addison-Wesley Publishing Company. 22. Burton, R. R., & Brown, J. S. (1982). An investigation of computer coaching. In D. H Sleeman & J. S. Brown (Eds.), Intelligent tutoring systems (pp.79-98). New York: Academic Press a Cardelle-Elawar, M., & Wetzel, K. (1995) /k-mainframe.htm b (1990) assessing of mathematical problem solving (pp ). Hillsdale, NJ: Erlbaum Publishing company. 25. Chen, Y., Horng, T. A., & Yang, D. (2000, November). A Java-based interactive learning system of junior high school level geometry. txt 19. Abidin, B., & Hartley, J. R. (1998). Developing mathematical problem solving skills. Journal of Computer Assisted Learning, 14(4), Birmingham Advisory & Support Service Students and computers as partners in developing students problem-solving skills. Journal of Research on Computing in Education, 27(4), Campione, J.C., Brown, A.L., & Connell, M.L. (1989). Metacognition: On the importance of understanding what you are doing. In R. I. Charles & E. A. Silver (Eds.), The teaching and Paper presented at the 8 th International Conference on Computers in Education, Taipei, Taiwan Ruric 26. Collins, M., Carnine, D., & Gersten, R. (1987). E.Wheeler Elaborated corrective feedback and the acquisition of reasoning skills: A study of computer assisted instruction. Exceptional Children, 154 (3), Date, C. J. (1995). An introduction to database systems (6th Ed.). New York: Addison-Wesley Publishing Company. 28. Drexel University (2002). The Math

22 48 Drexel. Retrieved September 20, 2002 from Edwards, T. G. (1994). Current reform efforts in mathematics education. ERIC Digest. (ERIC Document No. EDO-SE-94-5). 30. Educational Software Cooperative [ESC] (2002). ESC Software Library. Retrieved September 20, 2002 from org/library/index2.shtml#math. 31. Garofalo, J., & Mtetwa, D. K. (1990). Implementing the standards: Mathematics as reasoning, Arithmetic Teacher, 37(5), Greene, B. D., Herman, M., & Haury, D. L. (2000). TIMSS: What have we learned about math and science teaching? ERIC Digest. (ERIC Document No. EDO-SE-00-05) 33. Grouws, D. A., & Cebulla, K. J. (2000a). Improving student achievement in mathematic part1: Research findings. ERIC Digest. (ERIC Document No. EDO-SE-00-09) 34. Grouws, D. A., & Cebulla, K. J. (2000b). Improving student achievement in mathematic part2: Recommendations for the classroom. ERIC Digest. (ERIC Document No. EDO-SE ) 35. Kieran, C. (1996). The changing face of school algebra. Invited lecture given at the 7th International Congress on Mathematical Education. Retrieved December 20, 2001 from Kilpatrick, J. (1992). A history of research in mathematics education. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp.3-38). New York: Macmillan Publishing Company. 37. Lin, J. M.-C., Juang, J.-Y., & Sun, P. (1999, March). An Internet-based CAL software for solving trigonometric problems. Paper presented at the International Conference on Mathematics/Science Education and Technology, San Antonio, Texas. 38. Lin, J. M.-C., Juang, J.-Y., & Sun, P. (2000a, February). Representation of problem-solving procedures in MathCAL. Paper presented at the International Conference on Mathematics / Science Education and Technology, San Diego, CA. 39. Lin, J. M.-C., Hsu, C.-Y., Huang, K.K., & Juang, J.-Y. (2000b, November). MathCAL s diagnostic sub-system. Paper presented at the 8 th International Conference on Computers in Education, Taipei, Taiwan. 40. Lin, J. M.-C., Huang, L.-H., Huang, K.K., & Juang, J.-Y. (2001, June). MathCAL and its database design. Paper presented at the World Conference on Educational Multimedia, Hypermedia & Telecommunications, Tampere, Finland. 41. Liu, L. (2000, February). PCLego and mathematical thinking processes. Paper presented at the International Conference on Mathematics/Science Education and Technology, San Diego, CA. 42. Mullis, I. V. S., Martin, M. O., Beaton, A. E., Gonzalez, E.J., Kelly, D.L., & Smith, T.A. (1998). The mathematics and science achievement in the final years of secondary school. Retrieved September 20, 2002 from timss.bc.edu/timss1995i/mathsciencec.html. 43. Nakano, A., Murakami, N., Hirashima, T., & Takeuchi, A. (2000, November). A learning environment for problem posing in simple arithmetical word problems. Paper presented at the 8 th International Conference on Com-

23 49 puters in Education, Taipei, Taiwan. 44. The National Council of Teachers of Mathematics [NCTM] (1989). Curriculum and evaluation standards for school mathematics. Retrieved September 20, 2002 from standards.nctm.org/previous/currevstds/inde x.htm. 45. The National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics. Retrieved September 20, 2002 from standards. nctm.org / document / prepost / cover.htm. 46. Özgün-Koca, S. A. (1998). Technology in mathematics education Internet resources. ERIC Digest. (ERIC Document No. EDO-SE-98-06) 47. Peterson, J. L. (1981). Petri Net Theory and the Modeling of Systems. Englewood Cliffs, N. J: Prentice-Hall, Inc. 48. The Petri Nets World (2002). Online services for the International Petri nets community. Retrieved September 20, 2002 from Pólya, G. (1971). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton, N.J.: Princeton University Press. 50. University of Minnesota (2002). The Geometry Center. Retrieved September 20, 2002 from

24 50 A CAL System for Practiceing Mathematical Problem Solving: The Underlying Design Concepts Janet Mei-Chuen Lin, Jie-Yong Juang, Ponson Sun, Chin-Yeh Hsu, Long-Hwai Huang, Szu-Ching Lu and Kevin K. Huang Department of Information and Computer Education National Taiwan Normal University Abstract This paper describes the prototype of an Internet-based computer- assisted-learning system that allows learners to practice mathematical problem solving. Petri nets are used by the system as its knowledge representation model. Math knowledge within a chosen domain is pre-analyzed to derive a set of macro functions for use in solving problems. Each macro function typically embodies a math concept or rule that may be applied to transform a math problem from a state to another. The macro functions correspond to transitions in a Petri net graph, while the conditions for triggering those functions correspond to places. With Petri net graphs, the system can store experts static problem-solving knowledge as well as learners dynamic problem-solving procedures. The graphs may then be used effectively for diagnosing learners problem-solving difficulties and for offering guidance. The system also supports synchronous and asynchronous network communication functions that may be used by learners to solicit help from other system users. Users may add new problems to the problem bank with the system-provided editor. In addition, the system can automatically detect new solution paths to any old or new problem and augment the system s knowledge base accordingly. Key words: Object-Oriented Design, Petri Nets, Computer Assisted Learning, Mathematical Problem Solving, Relational Database Systems.

2011年南臺灣教育論壇

2011年南臺灣教育論壇 2013 年南臺灣教育學術研討會十二年國教國中數學科有效教學策略探討 - 國中數學探究教學經驗分享研究者 : 高雄市梓官國中林雅雯高應大電子工程系江柏叡副教授蘭陽技術學院數位生活創意系助理教授鍾明仁發表日期 :102 年 7

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穨27.PDF