Examining the Conceptual and Procedural Knowledge of Decimal Numbers in Sixth-Grade Elementary School Students
In this article, we present the results of empirical research using a combination of quantitative and qualitative methodology, in which we examined th.
- Pub. date: July 15, 2024
- Online Pub. date: April 20, 2024
- Pages: 1227-1245
- 249 Downloads
- 827 Views
- 0 Citations
In this article, we present the results of empirical research using a combination of quantitative and qualitative methodology, in which we examined the achievements and difficulties of sixth-grade Slovenian primary school students in decimal numbers at the conceptual and procedural knowledge level. The achievements of the students (N = 100) showed that they statistically significantly (z = -7,53, p < .001) better mastered procedural knowledge (M = 0.60, SD = 0.22) than conceptual knowledge (M = 0.37, SD = 0.17) of decimal numbers. Difficulties are related to both procedural and conceptual knowledge, but significantly more students have difficulties at the level of conceptual knowledge. At the level of procedural knowledge, or in the execution of arithmetic operations with decimal numbers, we observed difficulties in transforming text notation into numerical expressions, difficulties in placing the decimal point in multiplication and division, and insufficient automation of mathematical operations with decimal numbers. At the level of conceptual knowledge of decimal numbers, the results indicate difficulties for students in understanding the place values of decimal numbers, in estimating the sum, product and quotient of decimals with reflection and in mathematical justification. In relation to difficulties in justification, we observed an insufficient understanding of the size relationship between decimal numbers and difficulties in expressing them in mathematical language. The results indicate that to overcome such difficulties in the learning and teaching of mathematics, more balance between procedural and conceptual knowledge is needed.
Keywords: Conceptual knowledge, decimal numbers, math learning difficulties, procedural knowledge.
References
Bell, A., Swan, M., & Taylor, G. (1981). Choice of operation in verbal problems with decimal numbers. Educational Studies in Mathematics, 12, 399-420. https://doi.org/10.1007/BF00308139
Bonotto, C. (2001). How to connect school mathematics with students’ out-of-school knowledge. ZDM, 33, 75-84. https://doi.org/10.1007/BF02655698%20
Canobi, K. H. (2009). Concept-procedure interactions in children’s addition and subtraction. Journal of Experimental Child Psychology, 102(2), 131-149. https://doi.org/10.1016/j.jecp.2008.07.008
Chappell, K. K., & Killpatrick, K. (2007). Effects of concept-based instruction on students' conceptual understanding and procedural knowledge of calculus. Primus, 13(1), 17-37. https://doi.org/10.1080/10511970308984043
D’Ambrosio, B. S., & Kastberg, S. E. (2012). Building understanding of decimal fractions. Teaching Children Mathematics, 18(9), 558-564. https://doi.org/10.5951/teacchilmath.18.9.0558
Deslis, D., & Desli, D. (2023). Does this answer make sense? Primary school students and adults judge the reasonableness of computational results in context-based and context-free mathematical tasks. International Journal of Science and Mathematical Education 21, 71-91. https://doi.org/10.1007/s10763-022-10250-0
Dijanić, Ž., Dika, A., & Debelec, T. (2015). Kategorije znanja u matematici [Categories of knowledge in mathematics]. Matemstika u Školi, XVII(81), 3-10.
Doerr, H. M. (2006). Examining the task of teaching when using students’ mathematical thinking. Educational Studies in Mathematics, 62, 3-24. https://doi.org/10.1007/s10649-006-4437-9
Forman, E. A., Larreamendy-Joerns, J., Stein, M. K., & Brown, C. A. (1998). “You’re going to want to find out which and prove it”: Collective argumentation in a mathematics classroom. Learning and Instruction, 8(6), 527-548. https://doi.org/10.1016/S0959-4752(98)00033-4
Fritz, C. O., Morris, P. E., & Richler, J. J. (2012). Effect size estimates: Current use, calculations, and interpretation. Journal of Experimental Psychology, 141(1), 2-18. https://doi.org/10.1037/a0024338
Girit, D., & Akyuz, D. (2016). Pre-service middle school mathematics teachers’ understanding of students’ knowledge: location of decimal numbers on a number line. International Journal of Education in Mathematics, Science and Technology, 4(2), 84-100. https://bit.ly/47CvuC8
Hiebert, J. (1999). Relationships between research and the NCTM standards. Journal for Research in Mathematics Education, 30(1), 3-19. https://doi.org/10.2307/749627
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Lawrence Erlbaum Associates, Inc.
Hiebert, J., & Wearne, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 199-223). Lawrence Erlbaum Associates, Inc.
Higgins, B., & Reid, H. (2017). Enhancing “conceptual teaching/learning” in a concept-based curriculum. Teaching and Learning in Nursing, 12(2), 95-102. https://doi.org/10.1016/j.teln.2016.10.005
Irwin, K. C. (2001). Using everyday knowledge of decimals to enhance understanding. Journal for Research in Mathematics Education, 32(4), 399-420. https://doi.org/10.2307/749701
Isleyen, T., & Işik, A. (2003). Conceptual and procedural learning in mathematics. Journal of the Korea Society of Mathematical Education Series D: Research in Mathematical Education, 7(2), 91-99. https://bit.ly/3S7KOkt
Japelj Pavešić, B. (2017). Poučevanje ulomkov in decimalnih števil na razredni stopnji v mednarodni perspektivi [Teaching fractions and decimals in the first five grades within an international perspective]. Razredni Pouk: Revija Zavoda RS za Šolstvo, 19(1), 11-19. https://bit.ly/3SnzBO7
Jelenc, D., & Novljan, E. (2001). Učitelj svetuje staršem 1 – MATEMATIKA [Teacher advice to parents 1 - MATHEMATICS]. Didakta.
Kallai, A. Y., & Tzelgov, J. (2014). Decimals are not processed automatically, not even as being smaller than one. Journal of Experimental Psychology: Learning, Memory, and Cognition, 40(4), 962-975. https://doi.org/10.1037/a0035782
Krutetskii, V. A. (1976). The psychology of mathematical abilities in school children. University of Chicago Press.
Lai, M. Y., & Tsang, K. W. (2009). Understanding primary children's thinking and misconceptions in Decimal Numbers. In Proceedings of the International Conference on Primary Education 2009 (pp. 1-8). The Hong Kong Institute of Education. https://bit.ly/4b7fYkS
Lawson, A. (2007). Learning mathematics vs. following “rules”: The value of student-generated methods. What Works? Research into practice. Ontario Ministry of Education.
Leikin, R. (2007). Habits of mind associated with advanced mathematical thinking and solution spaces of mathematical tasks. In D. Pitta-Pantazi & G. Philippo (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (CERME-5) (pp. 2331-2340). University of Cyprus and ERME. http://erme.site/wp-content/uploads/CERME5/WG14.pdf
Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40, 173-196. https://doi.org/10.1023/A:1003606308064
Liu, R.-D., Ding, Y., Zong, M., & Zhang, D. (2014). Concept development of decimals in Chinese elementary students: A conceptual change approach. School Science and Mathematics, 114(7), 326-338. https://doi.org/10.1111/ssm.12085
Lortie-Forgues, H., & Siegler, R. S. (2017). Conceptual knowledge of decimal arithmetic. Journal of Educational Psychology, 109(3), 374-386. https://doi.org/10.1037/edu0000148
Lortie-Forgues, H., Tian, J., & Siegler, R. S. (2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38, 201-221. https://doi.org/10.1016/j.dr.2015.07.008
McMullen, J., Hannula-Sormunen, M. M., Lehtinen, E., & Siegler, R. S. (2020). Distinguishing adaptive from routine expertise with rational number arithmetic. Learning and Instruction, 68, Article 101347. https://doi.org/10.1016/j.learninstruc.2020.101347
McNeil, N. M., & Alibali, M. W. (2005). Why won’t you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76(4), 883-899. https://doi.org/10.1111/j.1467-8624.2005.00884.x
Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: understanding the real numbers. In M. Limón, & L. Mason (Eds.), Reconsidering conceptual change: Issues in theory and practice (pp. 232-257). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47637-1_13
Moloney, K., & Stacey, K. (1997). Changes with age in students’ conceptions of decimal notation. Mathematics Education Research Journal, 9, 25-38. https://doi.org/10.1007/BF03217300
Mullis, I. V. S., Martin, M. O., Foy, P., & Hooper, M. (2016). TIMSS 2015 international results in mathematics. Boston College, TIMSS & PIRLS International Study Center. https://bit.ly/47JA2GM
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. VA NCTM. https://www.nctm.org/Standards-and-Positions/Principles-and-Standards/
Niss, M., Blum, W., & Galbraith, P. L. (2007). Introduction. In W. Blum, P. L. Garbraith, H. -W. Henn & M. Niss (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (pp. 3-32). Springer. https://doi.org/10.1007/978-0-387-29822-1_1
Organization for Economic Cooperation and Development. (2014). PISA 2012 results: What students know and can do – Student performance in mathematics, reading and science (Volume I, Revised edition, February 2014). OECD Publishing. https://bit.ly/423Cx5y
Organization for Economic Cooperation and Development. (2019). PISA 2018. Insights and Interpretations. OECD Publishing. https://bit.ly/47G6zxx
Perry, E. L., & Len-Ríos, M. E. (2019). Conceptual understanding. In M. E. Len-Rios & E. L. Perry (Eds.), Cross-cultural journalism and strategic communication: storytelling and diversity (pp. 1-17). Routledge. https://doi.org/10.4324/9780429488412-1
Protheroe, N. (2007). What does good math instruction look like? Principal, 87(1), 51-54. https://bit.ly/3TXIk9j
Purnomo, Y. W., Ningsih, Y. R., & Suryadi, D. (2014). The influence of teachers’ mathematics beliefs, curriculum structures, and teaching materials on teaching mathematics: a case study. International Journal of Instruction, 7(1), 117-132.
Republiški Izpitni Center [Slovenian National Examinations Centre]. (2018). Nacionalno preverjanje znanja: letno poročilo o izvedbi v šolskem letu 2017/18 [National Assessment of Knowledge: annual report for the school year 2017/18]. https://bit.ly/3u3vRrJ
Republiški Izpitni Center [Slovenian National Examinations Centre]. (2019). Nacionalno preverjanje znanja: letno poročilo o izvedbi v šolskem letu 2018/19 [National Knowledge Examination: annual report for the school year 2018/19]. Državni izpitni center. https://bit.ly/3Hrn1Hh
Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: an iterative process. Journal of Educational Psychology, 93(2), 346-362. https://www.doi.org/10.1037/0022-0663.93.2.346
Sadi, A. (2007). Misconceptions in numbers. UGRU Journal, 5, 1-7.
Sengul, S., & Gulbagci, H. (2012). An investigation of 5th grade Turkish students’ performance in number sense on the topic of decimal numbers. Social and Behavioural Science, 46, 2289-2293. https://doi.org/10.1016/j.sbspro.2012.05.472
Sianturi, I. A. J., Ismail, Z., & Yang, D.-C. (2023). Examining fifth graders’ conceptual understanding of numbers and operations using an online three-tier test. Mathematics Education Research Journal. Advance online publication. https://doi.org/10.1007/s13394-023-00452-2
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404-411. https://www.jstor.org/stable/30034943
Steinle, V. (2004). Changes with age in student misconceptions of decimal numbers [Doctoral dissertation, The University of Melbourne]. The University of Melbourne Library. https://bit.ly/3vTnboG
Takker, S., & Subramaniam, K. (2019). Knowledge demands in teaching decimal numbers. Journal of Mathematics Teacher Education, 22, 257-280. https://doi.org/10.1007/s10857-017-9393-z
Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ understanding of rational numbers and their notation. Cognition and Instruction, 28(2), 181-209. https://doi.org/10.1080/07370001003676603
Yang, D.-C., & Sianturi, I. A. J. (2019a). Assessing students’ conceptual understanding using an online three‐tier diagnostic test. Journal of Computer Assisted Learning, 35(5), 678-689. https://doi.org/10.1111/jcal.12368
Yang, D.-C., & Sianturi, I. A. J. (2019b). Sixth grade students’ performance, misconceptions, and confidence when judging the reasonableness of computational results. International Journal of Science and Mathematics Education, 17, 1519-1540. https://doi.org/10.1007/s10763-018-09941-4
Yuliandari, R. N., & Anggraini, D. M. (2021). Teaching for understanding mathematics in primary school. In S. Senjana, U. Hikmah, I. Rofiki, W. F. Antariksa, Z. Rofiq, D. E. Rakhmawati, M. N. Jauhari, A. Fattah, U. A. Sari, & R. I. Rosi (Eds.), Proceedings of the International Conference on Engineering, Technology and Social Science (ICONETOS 2020) (pp. 40-46). Atlantis Press. https://doi.org/10.2991/assehr.k.210421.007
Žakelj, A., Prinčič Röhler, A., Perat, Z., Lipovec, A., Vršič, V., Repovž, B., Senekovič, J., & Bregar Umek, Z. (2011). Učni načrt, Program osnovna šola, Matematika [Elementary school maths curriculum Slovenia]. Ministrstvo za šolstvo in šport; Zavod RS za šolstvo.
Zuya, H. E. (2017). Prospective teachers’ conceptual and procedural knowledge in mathematics: The case of algebra. American Journal of Educational Research, 5(3), 310-315. http://pubs.sciepub.com/education/5/3/12