Research Article

Solving math problems requires inferential thinking skills, improved number sense, problem-solving strategies, deductive reasoning and computational skills. Mathematically talented students often use advanced number sense strategies to get the fastest and most accurate result. Although the existence of the number sense is known, it is difficult to describe it concretely. To materialize the intangible concept of number sense, a three-tier number sense test for secondary school students was developed and validated in this study. The developed test was carried out to 499 students studying in middle school in İzmir in Turkey. The first tier of the test consists of 25 multiple-choice mathematics questions. The second tier consists of the reason tier, which includes responses to the questions in the first tier (number sense-based, rule-based, misconception and guesswork). The third tier includes the confidence question, which measures the belief in the correctness of the response given to the question. The reliability of the test was calculated as .74 with the KR-20 formula. From the results of the analysis, it can be considered that the developed test is a valid and reliable measurement tool that can be used to determine the number sense levels of the students.

- Artut, P. D., & Er, Z. (2022, February). Investigation of number sense strategies used by 5th grade gifted students in Turkey. Twelfth Congress of the European Society for Research in Mathematics Education (CERME12), Feb 2022, Bozen-Bolzano, Italy.
- Baroody, A. J., & Gatzke, M. R. (1991). The estimation of set size by potentially gifted kindergarten-age children. Journal for Research in Mathematics Education, 22(1), 59–68.
- Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38(4), 333-339.
- Büyüköztürk, Ş., Çakmak, E. K., Akgün, Ö.E., Karadeniz, Ş., & Demirel, F. (2010). Scientific research methods. Ankara: PegemA
- Caleon, I., & Subramaniam, R. (2010). Development and application of a three‐tier diagnostic test to assess secondary students’ understanding of waves. International Journal of Science Education, 32(7), 939-961.
- Çekirdekçi, S., Şengül, S., & Doğan, M. C. (2016). 4. Sınıf Öğrencilerinin Sayı Hissi İle Matematik Başarıları Arasındaki İlişkinin İncelenmesi (Examining the relationship between number sense and mathematics achievement of the 4th grade students). Qualitative Studies, 11(4), 48-66.
- Chinn, S., J. ve Ashcroft, J. R. (1993). Mathematics for dyslexics. London: WhulT Publishers Ltd.
- Dehaene, S. (2001). Précis of the number sense. Mind and Language, 16(1), 16-36.
- Faulkner, V. N., & Cain, C. (2009). The components of number sense: An instructional model for teachers. Teaching Exceptional Children, 41(5), 24-30.
- Gardner, H. (1983). Frames of mind: The theory of multiple intelligence. New York: Basic Books Inc.
- Geary, D. C. (1995). Reflections of evolution and culture in children's cognition: Implications for mathematical development and instruction. American Psychologist, 50(1), 24.
- Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. The Journal of Special Education, 33(1), 18-28.
- Harling, P., & Roberts, T. (1988). Primary mathematics schemes. London: Hodder and Stoughton.
- Howden, H. (1989). Teaching number sense. The Arithmetic Teacher, 36(6), 6-11.
- Hrich, N., Lazaar, M., & Khaldi, M. (2019). Improving cognitive decision-making into adaptive educational systems through a diagnosis tool based on the competency approach. Int. J. Emerg. Technol. Learn., 14(7), 226-235.
- Jordan, N. C., Kaplan, D., Locuniak, M. N., & Ramineni, C. (2007). Predicting first‐grade math achievement from developmental number sense trajectories. Learning Disabilities Research & Practice, 22(1), 36-46.
- Krutetskii, V. A. (1976), The psychology of mathematical abilities in schoolchildren, University of Chicago Press, Chicago.
- Markovits, Z., & Sowder, J. (1994). Developing number sense: An intervention study in grade 7th . Journal for Research in Mathematics Education, 25(1), 4-29.
- McIntosh, A., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12(3), 2-44.
- McIntosh, A., Reys, B., Reys, R., Bana, J., & Farrell, B. (1997). Number sense in school mathematics: student performance in four countries. Perth, Australia: Mathematics, Science & Technology Education Centre, Edith Cowan University.
- Milenković, D. D., Hrin, T. N., Segedinac, M. D., & Horvat, S. (2016). Development of a three-tier test as a valid diagnostic tool for identification of misconceptions related to carbohydrates. Journal of Chemical Education, 93(9), 1514-1520.
- Montague, M., & van Garderen, D. (2003). A cross-sectional study of mathematics achievement, estimation skills, and academic self-perception in students of varying ability. Journal of Learning Disabilities, 36, 437– 448.
- National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
- Peşman, H., & Eryılmaz, A. (2010). Development of a three-tier test to assess misconceptions about simple electric circuits. The Journal of Educational Research, 103(3), 208-222.
- Resnick, L. B. (1989). Defining, assesing, 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. 35-40). San Diego, CA: San Diego State University, Center for Research in Mathematics and Science Education.
- Reys, B. J. (1991). Developing Number Sense. Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 5-8. National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 22091.
- Reys, R. E. & Yang, D. (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.
- Robinson, C. S., Menchetti, B. M., & Torgesen, J. K. (2002). Toward a two‐factor theory of one type of mathematics disabilities. Learning Disabilities Research & Practice, 17(2), 81-89.
- Rotigel, J. V., & Fello, S. (2004). Mathematically gifted students: How can we meet their needs? Gifted Child Today, 27(4), 46-51.
- Salaria, N. (2012). Meaning of the term descriptive survey research method. International Journal of Transformations in Business Management, 1(6), 1-7.
- Sia, D. T., Treagust, D. F., & Chandrasegaran, A. L. (2012). Hıgh school students’profıcıency and confıdence levels in displayıng their understanding of basic electrolysis concepts. International Journal of Science and Mathematics Education, 10(6), 1325-1345.
- Singh, P. (2009). An assessment of number sense among secondary school students. International Journal for Mathematics Teaching and Learning, 155, 1-29.
- Sowder, J. T. (1992). Estimation and number sense. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 371–389). Macmillan Publishing Co, Inc.
- Stankov, L., & Crawford, J. D. (1997). Self-confidence and performance on tests of cognitive abilities. Intelligence, 25(2), 93-109.
- UNESCO. (1977). New trends in mathematics teaching (Second Impressions). Volume In. France.
- Wang, J. J., Halberda, J., & Feigenson, L. (2017). Approximate number sense correlates with math performance in gifted adolescents. Acta Psychologica, 176, 78-84.
- Whitacre, I., Henning, B., & Atabaș, Ș. (2020). Disentangling the research literature on number sense: Three constructs, One name. Review of Educational Research, 90(1), 95-134.
- Whitfield, P. (1987). Assessment and evaluation. M. Preston (Ed). Mathematics in Primary Education. London: The Falmer Press.
- Wirtz, R. W. (1974). Mathematics for everyone. Washington, DC: Curriculum Development Associates.
- Yang, D. C. & Tsai, Y. F. (2010). Promoting sixth graders' number sense and learning attitudes via technology-based environment. Journal of Educational Technology & Society, 13(4), 112-125.
- Yang, D. C. (2003). Teaching and learning number sense–an interventıon study of fıfth grade students in Taiwan. International Journal of Science and Mathematics Education, 1(1), 115-134.
- Yang, D. C. (2005). Number sense strategies used by 6th‐grade students in Taiwan. Educational Studies, 31(3), 317-333.
- Yang, D. C. (2007). Investigating the strategies used by pre‐service teachers in Taiwan when responding to number sense questions. School Science and Mathematics, 107(7), 293-301.
- Yang, D. C. (2019). Development of a three-tier number sense test for fifth-grade students. Educational Studies in Mathematics, 101(3), 405-424.
- Yang, D. C., & Lin, Y. C. (2015). Assessing 10- to 11-year-old children’s performance and misconceptions in number sense using a four-tier diagnostic test. Educational Research, 57(4), 368–388.
- Yang, D. C., Li, M. N., & Lin, C. I. (2008). A study of the performance of 5th graders in number sense and its relationship to achievement in mathematics. International Journal of Science and Mathematics Education, 6(4), 789-807.

- Artut, P. D., & Er, Z. (2022, February). Investigation of number sense strategies used by 5th grade gifted students in Turkey. Twelfth Congress of the European Society for Research in Mathematics Education (CERME12), Feb 2022, Bozen-Bolzano, Italy.
- Baroody, A. J., & Gatzke, M. R. (1991). The estimation of set size by potentially gifted kindergarten-age children. Journal for Research in Mathematics Education, 22(1), 59–68.
- Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38(4), 333-339.
- Büyüköztürk, Ş., Çakmak, E. K., Akgün, Ö.E., Karadeniz, Ş., & Demirel, F. (2010). Scientific research methods. Ankara: PegemA
- Caleon, I., & Subramaniam, R. (2010). Development and application of a three‐tier diagnostic test to assess secondary students’ understanding of waves. International Journal of Science Education, 32(7), 939-961.
- Çekirdekçi, S., Şengül, S., & Doğan, M. C. (2016). 4. Sınıf Öğrencilerinin Sayı Hissi İle Matematik Başarıları Arasındaki İlişkinin İncelenmesi (Examining the relationship between number sense and mathematics achievement of the 4th grade students). Qualitative Studies, 11(4), 48-66.
- Chinn, S., J. ve Ashcroft, J. R. (1993). Mathematics for dyslexics. London: WhulT Publishers Ltd.
- Dehaene, S. (2001). Précis of the number sense. Mind and Language, 16(1), 16-36.
- Faulkner, V. N., & Cain, C. (2009). The components of number sense: An instructional model for teachers. Teaching Exceptional Children, 41(5), 24-30.
- Gardner, H. (1983). Frames of mind: The theory of multiple intelligence. New York: Basic Books Inc.
- Geary, D. C. (1995). Reflections of evolution and culture in children's cognition: Implications for mathematical development and instruction. American Psychologist, 50(1), 24.
- Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. The Journal of Special Education, 33(1), 18-28.
- Harling, P., & Roberts, T. (1988). Primary mathematics schemes. London: Hodder and Stoughton.
- Howden, H. (1989). Teaching number sense. The Arithmetic Teacher, 36(6), 6-11.
- Hrich, N., Lazaar, M., & Khaldi, M. (2019). Improving cognitive decision-making into adaptive educational systems through a diagnosis tool based on the competency approach. Int. J. Emerg. Technol. Learn., 14(7), 226-235.
- Jordan, N. C., Kaplan, D., Locuniak, M. N., & Ramineni, C. (2007). Predicting first‐grade math achievement from developmental number sense trajectories. Learning Disabilities Research & Practice, 22(1), 36-46.
- Krutetskii, V. A. (1976), The psychology of mathematical abilities in schoolchildren, University of Chicago Press, Chicago.
- Markovits, Z., & Sowder, J. (1994). Developing number sense: An intervention study in grade 7th . Journal for Research in Mathematics Education, 25(1), 4-29.
- McIntosh, A., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12(3), 2-44.
- McIntosh, A., Reys, B., Reys, R., Bana, J., & Farrell, B. (1997). Number sense in school mathematics: student performance in four countries. Perth, Australia: Mathematics, Science & Technology Education Centre, Edith Cowan University.
- Milenković, D. D., Hrin, T. N., Segedinac, M. D., & Horvat, S. (2016). Development of a three-tier test as a valid diagnostic tool for identification of misconceptions related to carbohydrates. Journal of Chemical Education, 93(9), 1514-1520.
- Montague, M., & van Garderen, D. (2003). A cross-sectional study of mathematics achievement, estimation skills, and academic self-perception in students of varying ability. Journal of Learning Disabilities, 36, 437– 448.
- National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
- Peşman, H., & Eryılmaz, A. (2010). Development of a three-tier test to assess misconceptions about simple electric circuits. The Journal of Educational Research, 103(3), 208-222.
- Resnick, L. B. (1989). Defining, assesing, 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. 35-40). San Diego, CA: San Diego State University, Center for Research in Mathematics and Science Education.
- Reys, B. J. (1991). Developing Number Sense. Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 5-8. National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 22091.
- Reys, R. E. & Yang, D. (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.
- Robinson, C. S., Menchetti, B. M., & Torgesen, J. K. (2002). Toward a two‐factor theory of one type of mathematics disabilities. Learning Disabilities Research & Practice, 17(2), 81-89.
- Rotigel, J. V., & Fello, S. (2004). Mathematically gifted students: How can we meet their needs? Gifted Child Today, 27(4), 46-51.
- Salaria, N. (2012). Meaning of the term descriptive survey research method. International Journal of Transformations in Business Management, 1(6), 1-7.
- Sia, D. T., Treagust, D. F., & Chandrasegaran, A. L. (2012). Hıgh school students’profıcıency and confıdence levels in displayıng their understanding of basic electrolysis concepts. International Journal of Science and Mathematics Education, 10(6), 1325-1345.
- Singh, P. (2009). An assessment of number sense among secondary school students. International Journal for Mathematics Teaching and Learning, 155, 1-29.
- Sowder, J. T. (1992). Estimation and number sense. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 371–389). Macmillan Publishing Co, Inc.
- Stankov, L., & Crawford, J. D. (1997). Self-confidence and performance on tests of cognitive abilities. Intelligence, 25(2), 93-109.
- UNESCO. (1977). New trends in mathematics teaching (Second Impressions). Volume In. France.
- Wang, J. J., Halberda, J., & Feigenson, L. (2017). Approximate number sense correlates with math performance in gifted adolescents. Acta Psychologica, 176, 78-84.
- Whitacre, I., Henning, B., & Atabaș, Ș. (2020). Disentangling the research literature on number sense: Three constructs, One name. Review of Educational Research, 90(1), 95-134.
- Whitfield, P. (1987). Assessment and evaluation. M. Preston (Ed). Mathematics in Primary Education. London: The Falmer Press.
- Wirtz, R. W. (1974). Mathematics for everyone. Washington, DC: Curriculum Development Associates.
- Yang, D. C. & Tsai, Y. F. (2010). Promoting sixth graders' number sense and learning attitudes via technology-based environment. Journal of Educational Technology & Society, 13(4), 112-125.
- Yang, D. C. (2003). Teaching and learning number sense–an interventıon study of fıfth grade students in Taiwan. International Journal of Science and Mathematics Education, 1(1), 115-134.
- Yang, D. C. (2005). Number sense strategies used by 6th‐grade students in Taiwan. Educational Studies, 31(3), 317-333.
- Yang, D. C. (2007). Investigating the strategies used by pre‐service teachers in Taiwan when responding to number sense questions. School Science and Mathematics, 107(7), 293-301.
- Yang, D. C. (2019). Development of a three-tier number sense test for fifth-grade students. Educational Studies in Mathematics, 101(3), 405-424.
- Yang, D. C., & Lin, Y. C. (2015). Assessing 10- to 11-year-old children’s performance and misconceptions in number sense using a four-tier diagnostic test. Educational Research, 57(4), 368–388.
- Yang, D. C., Li, M. N., & Lin, C. I. (2008). A study of the performance of 5th graders in number sense and its relationship to achievement in mathematics. International Journal of Science and Mathematics Education, 6(4), 789-807.

Primary Language | English |
---|---|

Subjects | Education, Scientific Disciplines |

Published Date | September 2022 |

Journal Section | Identification and Test Development |

Authors | |

Publication Date | September 30, 2022 |

Published in Issue | Year 2022, Volume 9, Issue 3 |

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