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1 Chinese Journal of Science Education,, 79-9, 8(), 79-9 ( ) (WCT) %7% (NST) % 5% Beaton, Mullis, Martin, Gonzalez, Kelly & Smith,997; Stevenson, Chen & Lee, 99; Stevenson, Lee & Stigler, 986; NST =957 Stigler, Lee & Stevenson, number sense Reys Yang (998 Bethlehem

2 8 Muneer999 () () () () 999 (5) p. 8 NCTM () (The Principles and Standards for School Mathematics) p. 87 (p. 8) " " NCTM, NCTM, 989, ; AEC, , a, b Markovits & Sowder, 99; McIntosh, Reys & Reys, 99; McIntosh, Reys. Reys, Bana & Farrel, 997; Sowder, 99a, 99b; Sowder & Kelin, 99; Yang,997 Howden, 989; B. Reys, 99; NCTM, 989 Hiebert, What is number sense? 989; Greeno, 99; Resnick, , 998, b; Reys & Yang,998 NCTM, 989 Curriculum and Evaluation Standards for School Mathematics,989 NCTM, 989, ; AEC, 99; Emanuelsson & Johansson, 996

3 8 Number sense framework (Reys & Yang, 998) McIntosh, Reys Reys (99) Case & Sowder, 99; Markovits & Sowder, 99; NCTM, 989, ; McIntosh, Reys, Reys, Bana & Farrel, 997; B. Reys. et. al., 99; Reys & Yang, 998; Sowder, 99a, 99b; Yang, top% %-7% % interview technique

4 ().957 = 957 () 9.57 n = 8 () 9.57 () 9.57 (5) 8 n = 8 5 () n = 5 5. () /5 + /7 () / + /9 () /8 + / * () /7 + / () () () () () () (). / + 8/9 () () () *() () 9 (5) 5 * () () () L-L8 M-M8 8 H-H5 5

5 8 / /+8/9 /5 /8 H / 8/9 H 7/8 ( ) 957 L 9.57 M7 6 5= M (M5 & L7)9M6 (H) =/ R L7 9 /7 / R /7/ L7 H / /7 L7 - M 9

6 8. 5/7 5/9 * 5/7 5/9 n = 8 7 () () () n = 8 7 (6) () n = 5 5 () () , 6/5, 8/5, 9/8,.9 8 () 7 () 6. 5/6 / * 5/6 () / 7 () () () () H 5/ / L 7 9 Kamii (989; 997) L 7 Burns (99) 9 66/8 M R M...

7 85 M 5/7 5/6 5/9 5/6 5/ /9 6 H / /5/ /6 / 5/6 / 5/6 L.899 = 899/.9 = 9/.899 >.9 L Resnick 989 L 6 L L7:/ L / / M 5 6 Peck & Jencks, 98; Carpenter, 5/6 et al., 98 M6 5 6 M7/ 5/6 / Markovits Sowder99

86 n = 8 7. 6 7 * 6= n = 8 () () n = 5 () () 8. A. / / 7/6 * B. / () C. / D. 9.

8 86 n = * 6= n = 8 () () n = 5 () () 8. A. / / 7/6 * B. / () C. / D D x G () () () 6 () () () () () () Hiebert (98) p. 55 Markovits Sowder99 7 Markovits Sowder (99) 67 6

9 87. A B. 7 C. 7 * D. 7 n = 8 () n = 8 5 () () n = 5 5 () () * 5 (8 ) (8 ) () () () ( () () M 66 7 L / L7 L / / 8 9 / D x G M7 / M

10 D G M D G M D.G.7.8 H M n = L Sowder Wheeler (989) L M () () 8 Graeber Tirosh (99) Greer (987) 9 H

11 % 6% 6% % [%] [%] % [%] [%] 8% [6.6%] [%] [%] [%] [%] [%] 9% [%] % [.%] % Altizer-. Tuning (98) L. Sowder (988 Kamii (997) Warrington Kamii Reys Yang998 (998)

12 9 Bezuk & Crame, R.Reys (99) NCTM (p.) SowderWheeler (989). Wearne Hiebert (988) rounding a number to zero, which is It is more difficult for students to acquire the effect of ignoring it, is very difficult for many conceptual understanding once they have learned students to do p. 7 Reys, rote procedures Rybolt, Bestgen & Wyatt, 98; Rubenstein, 985; Threadgill-Sowder, 98.. Southwell985 p. 77 Behr, Wachsmuth, Post Lesh98 Kerslake

13 Capenter, T. P., Corbbit, M. K., Kepner, H. S. Jr., - Lindquist, M. M., & Reys, R. E. (98). An. a interpretation of the results of the Second NAEP 89 Mathematics assessment. In S. Hill (Ed.), Education in the 8 s: Mathematics (pp. -5). 5. b Washington, DC: National Education 89 Association.. Case, R., & Sowder, J. T. (99). The 6. Kamii, C. 997development of computational estimation: 997 A neo-piagetian analysis. Cognition and Instruction, 7, Alitzer-Tuning, Carol (98). One point of view: Crisis in Arithmetic teaching: The future is here. Arithmetic Teacher,,. 8. Australian Education Council (99). A national statement on mathematics for Australian schools. Melbourne: Curriculum Corporation. 9. Beaton, A. E., Mullis, I., Martin, M. O., Gonzalez, E. J., Kelly, D. L., & Smith, T. A. (997). Mathematics achievement in the middle school years, IEA s third international mathematics and science study, Chestnut Hill, MA: Boston College, Center for the Study of Testing, Evaluation and Educational Policy.. Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (98). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 5, -.. Bezuk, N., & Cramer, K. (989). Teaching about fractions: What, when, and how? In Paul R. Trafton and Albert B. Shult (Eds.), New directions for elementary school Mathematics (pp ), 989 Yearbook of NCTM, Reston, VA: NCTM.. Burns, M. (99). Arithmetic: The last holdout. Phi Delta Kappen, (Feb.), Emanuelsson, G., & Johansson, B. (996). Kommentarer till kursplan I matematik, Lpo 9. (Nn-statutory guidance to the ne Swedish curriculum of mathematics). 6. Graeber, A. O., & Tirosh, D. (99). Insights fourthand fifth graders bring to multiplication and division with decimals. Educational Studies in Mathematics,, Greeno, J. G. (99). Number sense as situated knowing in a conceptual domain. Journal for Research in Mathematics Education,, Greer, B. (987). Nonconservation of multiplication and division involving Decimals. Journal for Research in Mathematics Education, 8, Hiebert, J. (98). Children s mathematics learning: The struggle to link form and Understanding. The Elementary School Journal, 8, Hiebert, J. (989). Reflections after the conference on number sense. In J. T. Sowder & B. P. Schappelle (Eds.), Establishing foundations for research on number sense and related topics: Report of a conference (pp. 8-8). San Diego: San Diego University,

14 9 Center for Research in Mathematics and Science Education.. Howden, H. (989). Teaching number sense. Arithmetic Teacher, 6(6), 6-.. Kamii, C. (989). Young children continue to reinvent Arithmetic, second grade: Implications of Piaget s theory. New York: Teachers College Press.. Kerslake, D. (986). Fractions: Children s strategies and errors: A report of the strategies and errors in Secondary Mathematics Project. Windsor, England: NFER-Nelson.. Markovits, Z., & Sowder J. T. (99). Students understanding of the relationship between fractions and decimals. Focus on Learning Problems in Mathematics, (), Markovits, Z., & Sowder J. T. (99). Developing number sense: An intervention study in grade 7. Journal for Research in Mathematics Education, 5(), McIntosh, A., Reys, B. J., & Reys, R. E. (99). A proposed framework for examining basic number sense. For the Learning of Mathematics, (), McIntosh, A., Reys, B. J., Reys, R. E., Bana, J., & Farrel, B. (997). Number sense in school Mathematics: Student performance in four countries. Mathematics, Science, & Technology Education Centre, Edith Cowan University. 8. Muneer, Jebree (999). The relationship between number sense and computational performance in grade-ninth of Bethlehem area schools. Unpublished Master Thesis of University of Birzeit in Israel. 9. National Council of Teachers of Mathematics. (989). Curriculum and evaluation standards for school Mathematics. Reston, VA: NCTM.. National Council of Teachers of Mathematics. (). The principles and standards for school Mathematics. Reston, VA: NCTM.. Peck, D. M., & Jencks, S. M. (98). Conceptual issues in the teaching and learning of fractions. Journal for Research in Mathematics Education,, Resnick, L. B. (989). Defining, assessing, and teaching number sense. In J. T. Sowder & B. P. Schappelle (Eds.), Establishing foundations for research on number sense and related topics: Report of a conference (pp. 8-8). San Diego: San Diego Univ., Center for Research in Mathematics and Science Education.. Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. (989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics,, Reys, B. J. (Ed.). (99). Developing number sense in the middle grades. Reston, VA: NCTM. 5. Reys, B. J. (99). Promoting number sense in middle grades. Teaching Mathematics in the Middle School, (), Reys, R. E., & Yang, D. C. (998). Relationship between computational performance and number sense among sixth-and eighth-grade students in Taiwan. Journal for Research in Mathematics Education, 9, Reys, R. E., Reys, B. J., Nohda, Ishida, Yoshikawa, & Shimizu (99). Computational estimation performance and strategies used by fifth-and eighth-grade Japaneses students, Journal for Research in Mathematics Education,, Reys, R. E., Rybolt, J. F., Bestgen, B. J., &

15 9 Wyatt, J. W. (98). Processes used by good computational estimators. Journal for Research in Mathematics Education, (), Rubenstein, R. N. (985). Computational stimation and related mathematical skills. Journal for Research in Mathematics Education, 6, Southwell, B. (985). The development of rational number concepts in Papua, New Guinea. In A. Bell, B. Low, & J. Kilpatrick (Eds.),Theory, research, and practice in mathematical education. Nottingham, England: Shell Centrefor Mathematical Education, Universityof Nottingham.. Sowder, J. T. (99a). Estimation and number sense. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 7-89). New York: Macmillan.. Sowder, J. T. (99b). Making sense of numbers in school mathematics. In G. Leinhardt, & R. Hattrup (Eds.), Analysis of Arithmetic for Mathematics Teaching (pp. -5). Hillsdale, NJ: Erlbaum.. Sowder, J. T. & Kelin, J. (99). Number sense and related topics. In D. T. Owens (Ed.), Research ideas for the classroom: Middle grades mathematics (pp. -57). New York: Macmillan.. Sowder, J. T., & Wheeler, M. M. (989). The development of concepts and strategies used in computational estimation. Journal for Research in Mathematics Education,, Sowder, L. (988). Children s solution of story problems. Journal of Mathematical Behavior, 7, Stevenson, H. W., Chen, C., & Lee, S. Y. (99). Mathematical achievement of Chinese, Japanese, & American children: Ten years later. Science, 59, Stevenson, H. W., Lee, S. Y., & Stigler, J. W. (986). Mathematical achievement of Chinese, Japanese, and American children. Science,, Stigler, J. W., Lee, S. Y., & Stevenson, H. W. (99). Mathematical knowledge of Japanese, Chinese, and American elementary school children, Reston,VA: NCTM. 9. Threadgill-Sowder (98). Computational stimation procedures of school Children. Journal for Research in Mathematics Education, 77, Warrington, M. A. & Kamii, C. (998). Multiplication with Fractions: A Piagetian, Constructivist Approach, Mathematics Teaching in the Middle School, (5), Wearne, D. & Hiebert J. (988). Constructing and using meaning for mathematical symbols: The case of decimal fractions. In J. Hieber & M. Behr ( Eds.), Number concepts and operations in the middle grades (pp. -5). Reston, VA: NCTM. 5. Yang, D. C. (997). Number sense performance and strategies possessed by sixth and eighth grade students in Taiwan. Doctoral dissertation, University of Missouri Columbia, 995, Dissertation Abstracts International, 57, 865A.

16 9 The Methods Used by Sixth Grade Students in Taiwan When Responded Number Sense Questions Der-Ching Yang Department of Applied Mathematics National Chiayi University Abstract sixth grade students from four different public elementary schools participated this study. The purpose of this study was to explore the methods used by sixth grade students in Taiwan when solving specific number sense problems. The interview results indicated that most of students (including low, middle, and high level) revealed little number sense strategies when were interviewed. These students heavily relied upon paper-and-pencil computational skills when solving number sense related questions. Key words : Estimation, Benchmark, Number magnitude, Number sense.

穨e235.PDF

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穨E379.PDF