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An Examination of Fourth-Grade Elementary School Students’ Number Sense in Context-Based and Non-Context-Based Problems

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Abstract

Research on number sense presents contradictory arguments and results regarding the effects of context-based tasks and activities on students’ number sense. The aim of this study was to examine fourth-grade elementary school students’ number sense as they solve context-based (CB) and non-context-based (NCB) items. A survey design was employed and data collected using CB and NCB tests for this purpose. The results showed that students were more successful on NCB test compared to the CB test. Students performed best on the problems related to the “using a reference point” component of number sense, while they performed worst on the “understanding of the number size” component. The majority of students used rule-based strategies (RBSs) for both CB and NCB tests. Interviews with students who solved CB items using RBSs showed that students were able to recognize and use number sense-based strategies (NSBSs) to some degree when they were asked for alternative solution methods. The variability found in students’ recognition and use of number sense seem to depend on such factors as the lack of number sense, preference to RBS, task structure (e.g. involving visual representations and easy numbers), and task requirements (e.g. asking for estimation or comparison vs. exact answer, requiring additional adjustments on reference points).

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References

  • Almedia, R., & Bruno, A. (2017). Establishing profiles on the use of number sense. REDIMAT, 6(1), 56–84.

    Google Scholar 

  • Anghileri, J. (2000). Teaching number sense. London: Cromwell Press.

    Google Scholar 

  • Antell, S. E., & Keating, D. P. (1983). Perceptions of numerical invariance in neonates. Child Development, 54, 695–701.

    Google Scholar 

  • Barwell, R. (2001). Difficulties with mathematical word problems. Proceedings of the British Society for Research in Learning Mathematics, 21(2), 63–74.

    Google Scholar 

  • Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38(4), 333–339.

    Google Scholar 

  • Can, D. (2017). İlkokul dördüncü sınıf öğrencilerinin sayı duyularının bağlam temelli ve bağlam temelli olmayan problem durumlarında incelenmesi, [Examination of fourth grade elementary school students' number sense in context-based and noncontext-based problems] (Unpublished doctoral dissertation). Hacettepe Üniversitesi, Eğitim Bilimleri Enstitüsü, Ankara.

  • Cramer, K. A., Post, T. R., & del Mas, R. C. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of 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.

    Google Scholar 

  • Cummins, D. D., Kintsch, W., Reusser, K., & Weimer, R. (1988). The role of understanding in solving word problems. Cognitive Psychology, 20, 405–438.

    Google Scholar 

  • Çekirdekci, S., Şengül, S., & Doğan, M. C. (2016). Examining the relationship between number sense and mathematics achievement of the 4th grade students. Qualitative Studies (NWSAQS), 11(4), 48–66.

    Google Scholar 

  • Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. The Journal of Special Education, 33(1), 18–28.

    Google Scholar 

  • Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293–304.

    Google Scholar 

  • Gravemeijer, K. (1997). Commentary. Solving word problems: A case of modelling? Learning and Instruction, 7, 389–397.

    Google Scholar 

  • Gülbağcı Dede, H. (2015). İlköğretim ve ortaöğretim matematik öğretmen adaylarının sayı hissinin incelenmesi [An investigation of pre-service elementary and secondary mathematics teachers’ number sense] (Unpublished doctoral dissertation). Eğitim Bilimleri Enstitüsü, Marmara Üniversitesi, İstanbul.

  • Hiebert, J. (1999). Relationships between research and the NCTM Standards. Journal for Research in Mathematics Education, 30, 3–19.

    Google Scholar 

  • Hope, J. (1989). Promoting number sense in school. Arithmetic Teacher, 36(6), 12–16.

  • Irwin, K. C. (2001). Using everyday knowledge of decimals to enhance understanding. Journal for Research in Mathematics Education, 32(4), 399–420.

    Google Scholar 

  • İymen, E. (2012). 8. sınıf öğrencilerinin üslü ifadelerle ilgili sayı duyularının sayı duyusu bileşenleri bakımından incelenmesi. [Analysis of 8 grade students’ number sense on exponents in terms of number sense components] (Unpublished master dissertation). Fen Bilimleri Enstitüsü, Pamukkale Üniversitesi, Denizli.

  • Jordan, N. C., Kaplan, D., Olah, L. N., & Locuniak, M. N. (2006). Number sense growth in kindergarten: A longitudinal investigation of children at risk for mathematics difficulties. Child Development, 77, 153–175.

    Google Scholar 

  • Kayhan Altay, M. (2010). İlköğretim ikinci kademe öğrencilerinin sayı duyularının; sınıf düzeyine, cinsiyete ve sayı duyusu bileşenlerine göre incelenmesi, [An investigation of middle grade students’ number sense in terms of grade level, gender, and components of number sense] (Unpublished doctoral dissertation). Eğitim Bilimleri Enstitüsü, Hacettepe Üniversitesi, Ankara.

  • Kelley, T. L. (1939). The selection of upper and lower groups for the validation of test items. Journal of Educational Psychology, 30(1), 17–24.

    Google Scholar 

  • Kerslake, D. (1986). Fractions: Children’s strategies and errors: A report of the strategies and errors in secondary mathematics project. England: NFER-Nelson Publishing Company.

    Google Scholar 

  • Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review, 92, 109–129.

    Google Scholar 

  • Koedinger, K. R., Alibali, M. W., & Nathan, M. J. (2008). Trade-offs between grounded and abstract representations: Evidence from algebra problem solving. Cognitive Science, 32, 366–397.

    Google Scholar 

  • Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. The Journal of the Learning Sciences, 13, 129–164.

    Google Scholar 

  • Lago, R., & DiPerna, J. (2010). Number sense in kindergarten: A factor-analytic study of the construct. School Psychology Review, 39, 164–180.

    Google Scholar 

  • Lan, Y. J., Sung, Y. T., Tan, N. C., Lin, C. P., & Chang, K. E. (2010). Mobile-device-supported problem-based computational estimation instruction for elementary school students. Educational Technology & Society, 13(3), 55–69.

    Google Scholar 

  • Li, M. N. F., & Yang, D. C. (2010). Development and validation of a computer-administered number sense scale for fifth-grade children in Taiwan. School Science and Mathematics, 110(4), 220–230.

    Google Scholar 

  • Louange, J. E. G. (2007). An examination of the relationships between teaching and learning styles, and the number sense and problem solving ability of Year 7 students (Unpublished doctoral dissertation). Perth, Australia: Edith Cowan University.

    Google Scholar 

  • Markovits, Z., & Sowder, J. (1994). Developing number sense: An intervention study in grade 7. Journal for Research in Mathematics Education, 25(1), 4–29.

    Google Scholar 

  • McIntosh, A., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12(3), 2–9.

    Google Scholar 

  • McIntosh, A., Reys, R. E., & Reys, B. J. (1997). Mental computation in the middle grades: The importance of thinking strategies. Mathematics Teaching in the Middle School, 2, 322–327.

    Google Scholar 

  • Ministry of National Education. (2009). Elementary School Curricula: Mathematics Curricula for Grades 1–5. Ankara: Milli Eğitim Basımevi.

    Google Scholar 

  • Phipps, M. C. (2008). A phenomenological investigation on eighth graders’ number sense of fractions (Unpublished doctoral dissertation). University of Northern Colorado, Greeley, CO.

  • Pollitt, A., Marriott, C., & Ahmed, A. (2000). Language, contextual and cultural constraints on examination performance. Paper presented at International Association for Educational Assessment, Jerusalem, Israel.

  • Reys, R., Reys, B., McIntosh, A., Emanuelsson, G., Johansson, B., & Yang, D. C. (1999). Assessing number sense of students in Australia, Sweden, Taiwan, and the United States. School Science and Mathematics, 99(2), 61–70.

    Google Scholar 

  • 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(2), 225–237.

    Google Scholar 

  • Tsao, Y. L. (2005). The number sense of preservice elementary teachers. College Student Journal, 39(4), 647–679.

    Google Scholar 

  • Wynn, K. (1992). Addition and subtraction by human infants. Nature, 358(27), 749–750.

    Google Scholar 

  • Yang, D. C. (1995). Number sense performance and strategies possessed by sixth and eighth grade students in Taiwan (Unpublished doctoral dissertation). University of Missouri, Columbia, MO.

  • Yang, D. C. (2003). Developing number sense through realistic settings. APMC, 8(3), 12–17.

    Google Scholar 

  • Yang, D. C. (2005). Number sense strategies used by sixth grade students in Taiwan. Educational Studies, 31(3), 317–334.

    Google Scholar 

  • Yang, D.C. (2019). Performance of fourth graders when judging the reasonableness of a computational result. International Journal of Science and Mathematics Education, 17(1), 197–215.

  • 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.

    Google Scholar 

  • Yang, D. C., & Li, M. F. (2008). An investigation of 3rd-grade Taiwanese students’ performance in number sense. Educational Studies, 34(5), 443–455.

    Google Scholar 

  • Yang, D. C., Li, M. F., & Li, W. J. (2008a). Development of a computerized number sense scale for 3rd graders: Reliability and validity analysis. International Electronic Journal of Mathematics Education, 3(2), 110–124.

    Google Scholar 

  • Yang, D. C., Li, M. N., & Lin, C. I. (2008b). 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, 789–807.

    Google Scholar 

  • Yang, D. C., & Liu, Y. F. (2013). Examining the differences on comparing fraction size for 5th graders between contextual and numerical problems. Asian Journal of Education and e-Learning, 1(2), 112–117.

    Google Scholar 

  • Yang, D. C., Reys, R. E., & Wu, L. L. (2010). Comparing how fractions were developed in textbooks used by the 5th- and 6th-graders in Singapore, Taiwan, and the U.S.A. School Science and Mathematics, 110(3), 118–127.

    Google Scholar 

  • Yang, D. C., & Wu, W. R. (2010). The study of number sense: Realistic activities integrated into third-grade math classes in Taiwan. The Journal of Educational Research, 103, 379–392.

    Google Scholar 

  • Yang, D. C., & Wu, S. S. (2012). Examining the differences of 8th graders’ estimation performance between contextual and numerical problems. US-China Educational Review, 12, 1061–1067.

    Google Scholar 

  • Yapıcı, A. (2013). 5, 6 ve 7. sınıf öğrencilerinin yüzdeler konusunda sayı duyularının incelenmesi [An investigation of 5, 6 and 7th grade students’ number sense related to percentages] (Unpublished master’s dissertation). Hacettepe University, Ankara.

  • Zanzali, N. A., & Ghazali, M. (2002). Assessment of school children’s number sense. Available at http://math.unipa.it/~grim/ENoor8.

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Authors and Affiliations

  1. Faculty of Education, Mehmet Akif Ersoy University, Burdur, Turkey

    Derya Can

  2. Faculty of Education, Hacettepe University, Ankara, Turkey

    İ. Elif Yetkin Özdemir

Authors
  1. Derya Can
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  2. İ. Elif Yetkin Özdemir
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Corresponding author

Correspondence to Derya Can.

Additional information

This paper is based on the PhD Thesis written by Derya Can under the supervision of Assoc. Prof. Dr. İ. Elif Yetkin Özdemir.

Appendix 1. Number sense test for context-based problems

Appendix 1. Number sense test for context-based problems

  1. 1.

    A factory bought 98 boxes, each of which contained 50 buttons. Which of the following is closest to the number of buttons bought by the factory?

  1. A.

    500

  2. B.

    5000

  3. C.

    50,000

  1. 2.

    Ekin wanted to check 372 − 38 = 334 using a calculator. However, she typed in 372 − 18 instead of 372 − 38. Which of the following statements best describes the result she obtained with the calculator?

    1. A.

      2 less than 334.

    2. B.

      2 more than 334.

    3. C.

      20 less than 334.

    4. D.

      20 more than 334.

  1. 3.

    Quarter of a chocolate bar

figure a

The visual representation above shows one-quarter of a chocolate bar. Draw the amount of chocolate eaten by Sevil, who ate 2 \( \frac{1}{4} \) chocolate bars.

  1. 4.

    Mert saved 150 TL to buy a bicycle costing 325 liras. Later, its price was cut in half during a sale. Does Mert have enough money to purchase the bike now?

A. Yes.

B. No.

  1. 5.

    Theater Hall A contains 9 rows, and in each row there are 18 chairs. Hall B of the theatre has 10 rows, and in each row there are again 18 chairs. Hall C of the theatre has 9 rows, and in each row there are 17 chairs. Rank the halls from the largest to smallest based on their capacity.

  1. 6.

    Ahmet and Nilay were each given 32 liras. Ahmet saved \( \frac{1}{2} \) of his money and Nilay saved \( \frac{5}{8} \) of her money. Which of the following statements is true?

A. Nilay saved more than Ahmet.

B. Ahmet saved more than Nilay.

C. Ahmet and Nilay saved the same amount.

  1. 7.
    figure b

    The height of the tree given in the visual representation is 490 cm. The height of a boy is one-fifth that of the tree. Which of the following statements is true of the boy’s height?

  1. A.

    It is 100 cm.

  2. B.

    It is higher than 100 cm.

  3. C.

    It is shorter than 100 cm.

  1. 8.

    A supermarket sells 18 packs of water, in each of which there are 12 water bottles, so it sells 216 bottles?? of water per day in total. How many bottles of water would the supermarket sell if it sold 9 packs of water with 24 bottles per day?

  2. 9.

    A bookstore joined a book fair and its four-day sales are given as follows:

Day 1: 91 liras, Day 2: 93 liras, Day 3: 97 liras, Day 4: 99 liras.

The bookstore needs to pay 380 liras to rent the bookstand. Are its sales enough to pay for the bookstand?

  1. 10.
    figure c

The expenditures of Veli at the supermarket are given above. How much did he pay for these products?

Appendix 2. Number sense test for non-context-based problems

  1. 1.

    Which of the following is closest to 50 × 98?

  1. A.

    500

  2. B.

    5000

  3. C.

    50,000

  1. 2.

    If 372 − 38 = 334, which of the following is true of 372 − 18?

  1. A.

    2 less than 334.

  2. B.

    2 more than 334.

  3. C.

    20 less than 334.

  4. D.

    20 more than 334.

  1. 3.
    figure d

In the visual representation above, the shaded part shows one-quarter of the whole. Draw a visual representation representing the fraction of 2 \( \frac{1}{4} \).

  1. 4.

    Which of the following statements is true for the result of 325 ÷ 2?

  1. A.

    It is 150.

  2. B.

    It is bigger than 150.

  3. C.

    It is smaller than 150.

  1. 5.

    Rank the results of the operations 18 × 9, 18 × 10, and 17 × 9 from largest to smallest.

  1. 6.

    Make a comparison between \( \frac{1}{2} \) of 32 and \( \frac{5}{8} \) of 32.

    1. A.

      \( \frac{1}{2} \) of 32 is smaller than \( \frac{5}{8} \) of 32.

    2. B.

      \( \frac{1}{2} \) of 32 is larger than \( \frac{5}{8} \) of 32.

    3. C.

      \( \frac{1}{2} \) of 32 is equal to \( \frac{5}{8} \) of 32.

  2. 7.

    Which of the following statements is true of 490 ÷ 5?

  1. A.

    It is 100.

  2. B.

    It is larger than 100.

  3. C.

    It is smaller than 100.

  1. 8.

    If 12 × 18 = 216, what is 24 × 9?

  1. 9.

    Which of the following statements is true of 91 + 93 + 97 + 99?

  1. A.

    It is larger than 380.

  2. B.

    It equals to 380.

  3. C.

    It is smaller than 380.

  1. 10.

    What is 39 + 23 + 52 + 48 + 61 + 77?

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Can, D., Yetkin Özdemir, İ.E. An Examination of Fourth-Grade Elementary School Students’ Number Sense in Context-Based and Non-Context-Based Problems. Int J of Sci and Math Educ 18, 1333–1354 (2020). https://doi.org/10.1007/s10763-019-10022-3

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  • DOI: https://doi.org/10.1007/s10763-019-10022-3

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