A Study of the Performance of 5th Graders in Number Sense and its Relationship to Achievement in Mathematics

  • Der-Ching YangEmail author
  • Mao-neng Li
  • Chih-I Lin


In order to investigate the performance of number sense and its relationships with mathematics achievement of Taiwanese students who had just completed the 5th-grade mathematics curriculum, a computerized number sense scale has been developed. This number sense scale includes four factors which are recognizing relative number size, using multiple representations of numbers and operations, judging the reasonableness of estimates of computed results and recognizing the relative effect of operations on numbers. A total of 1,212 students in Taiwan participated in this study. The main findings of this study are summarized as follows. First, the students perform best on “recognizing relative number size” and perform worst on “judging the reasonableness of estimates of computed results”. This finding is consistent with previous studies. It shows that students in Taiwan seem quite poor on judging the reasonableness of estimates of computed results. Second, female students, on average, have higher scores on recognizing the relative number size than male students, even though only a small effect size is found. And, third, the achievements of the students in mathematics are significantly correlated with their number sense, as measured by the average grade for the academic year of 5th-grade students.

Key words

computerized 5th graders mathematics achievement number sense scale 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Australian Education Council (1991). A national statement on mathematics for Australian schools. Melbourne: Curriculum Corporation.Google Scholar
  2. 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.CrossRefGoogle Scholar
  3. Cockcroft, W. H. (1982). Math counts. London: Her Majesty’s Sationery office.Google Scholar
  4. Cramer, K. A., Post, T. R. & delMas, R. C. (2002). Initial fraction learning by forth-and fifith-grade students: A comparison for the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33(2), 111–144.CrossRefGoogle Scholar
  5. Emanuelsson, G., Johansson, B., Reys, B. J. & Reys, R. E. (1996). Using a journal to engage teachers in developmental work. Nordic Studies in Mathematics Education, 4, 61–73.Google Scholar
  6. Graeber, A. O. & Tirosh, D. (1990). Insights fourth and fifth graders bring to multiplications and division with decimal. Educational Studies in Mathematics, 21, 565–588.CrossRefGoogle Scholar
  7. Greer, B. (1987). Nonconservation of multiplication and division involving decimals. Journal for Research in Mathematics Education, 178, 37–45.CrossRefGoogle Scholar
  8. Hart, K. (1988). Ratio and proportion. In J. Hibert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 198–279). Reston, VA: NCTM.Google Scholar
  9. Hiebert, E. H. (1999). Text matters in learning to read. The Reading Teacher, 52, 552–566Google Scholar
  10. Hsu, C. Y., Yang, D. C. & Li, M. N. (2001). The design of “the fifth and sixth grade number sense rating scale.” Chinese Journal of Science Education, 9(4), 351–374.Google Scholar
  11. Kerslake, D. (1986). Fractions: Children’s strategies and errors: A report of the strategies and errors in secondary mathematics project. Windsor, England: NFER-Nelso.Google Scholar
  12. Li, W. J. (2004). Technology integrated into the mathematics assessment∼an example of computerized number sense scale development. Unpublished master thesis, National Chiayi University, Taiwan.Google Scholar
  13. Lin, C. I. (2005). Development of a computerized number sense scale for the 5th graders and their performance on number sense. Unpublished master thesis, National Chiayi University, Taiwan.Google Scholar
  14. Markovits, Z. & Sowder, J. T. (1994). Developing number sense: An intervention study in grade 7. Journal for Research in Mathematics Education, 25(1), 4–29.CrossRefGoogle Scholar
  15. McIntosh, A., Reys, B. J. & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12, 2–8.Google Scholar
  16. McIntosh, A., Reys, B. J., Reys, R. E., Bana, J. & Farrel, B. (1997). Number sense in school mathematics: Student performance in four countries. MASTEC: Mathematics, Science & Technology Education Centre.Google Scholar
  17. Ministry of Education in Taiwan (2000). 9-year-integrated mathematics curriculum standards for national schools from grade 1 to 9 in Taiwan (pp. 19–86). Taiwan: Author. (In Chinese).Google Scholar
  18. Mullis, I. V. S., Martin, M. O., Gonzalez, E. J. & Chrostowski, S. J. (2004). TIMSS 2003 international mathematics report: Findings from IEA’s Trends in International Mathematics and Science Study at the fourth and eight grades. Chestnut Hill, MA: Boston College, TMISS & PIRLS International Study Center.Google Scholar
  19. National Council of Teachers of Mathematics (2000). The principles and standards for school mathematics. Reston, VA: NCTM.Google Scholar
  20. National Research Council (2002). Adding it up. Helping children learn mathematics. Washington, D. C.: National Academy Press.Google Scholar
  21. Organisation for Economic Co-operation and Development (2004). Learning for tomorrow’s world: First results from PISA 2003. Paris: Author.Google Scholar
  22. Post, T. R., Cramer, K., Behr, M. J., Lesh, R. & Harel, G. (1993). Curricula implications of research on the teaching and learning of rational numbers concepts. In T. Carpenter, T. E. Fennema & T. Romberg (Eds.), Research on the teaching, learning, and assessing of rational number concepts (pp. 327–362). Hillsdale NJ: Lawrence Erlbaum Associates.Google Scholar
  23. Reys, R. E. & Yang, D. C. (1998). Relationship between computational performance and number sense among sixth- and eighth-grade students in Taiwan. Journal for Research in Mathematics Education, 29, 225–237.CrossRefGoogle Scholar
  24. Sowder, L. (1988). Children’s solution of story problems. Journal of Mathematical Behavior, 7, 227–238.Google Scholar
  25. Sowder, J. (1992a). Estimation and number sense. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 371–389). New York: Macmillan.Google Scholar
  26. Sowder, J. (1992b). Making sense of numbers in school mathematics. In G. Leinhardt & R. Hatterp (Eds.), Analysis of arithmetic for mathematics teaching (pp. 1–51). Hillsdale, NJ: Erlbaum.Google Scholar
  27. Treffers, A. (1991). Meeting innumeracy at primary school. Educational Studies in Mathematics, 26(1), 333–352.CrossRefGoogle Scholar
  28. Yang, D. C. (2003). Teaching and learning number sense-an intervention study of fifth grade students in Taiwan. International Journal of Science and Mathematics Education, 1(1), 115–134.CrossRefGoogle Scholar
  29. Yang, D. C. & Huang, F. Y. (2004). Relationships among computational performance, pictorial representation, symbolic representation, and number sense of sixth grade students in Taiwan. Educational Studies, 30(4), 373–389.CrossRefGoogle Scholar
  30. Yang, D. C. & Reys, R. E. (2002). Fractional Number Sense StrategiesPossessed by Sixth Grade Students in Taiwan. Hiroshima Journal of Mathematics Education, 10, 53–70Google Scholar
  31. Yang, D. C., Hsu, C. J. & Huang, M. C. (2004). A study of teaching and learning number sense for sixth grade students in taiwan. International Journal of Science and Mathematics Education, 2(3), 407–430.CrossRefGoogle Scholar

Copyright information

© National Science Council, Taiwan 2007

Authors and Affiliations

  1. 1.Graduate Institute of Mathematics EducationNational Chiayi UniversityMing-Hsiung ChiayiRepublic of China
  2. 2.Graduate Institute of Compulsory EducationNational Chiayi UniversityMing-Hsiung ChiayiRepublic of China

Personalised recommendations