АУТОР(И): Nenad S. Milinković, Sanja M. Maričić, Bojan D. Lazić
Е-АДРЕСА: milinkovic.nenad84@gmail.com
DOI: 10.46793/Uzdanica19.S.095M
САЖЕТАК:
Correct understanding of the equals sign is the key to understanding arithme- tic, and a fundamental concept important for learning other areas of mathematics. Research around the world repeatedly mentions problems with correctly understanding the equals sign, emphasising the limited view of the equals sign as a command ”to calculate“ among students. The goal of the research was to identify the development of the concept of equality in line with the operationalised levels (operational, relational, and relational in the context of real-life prob- lem solving) and determine differences in understanding between students of the second (N = 190) and the fourth (N = 210) grade of primary school. The research was carried out using the testing technique. The research results show that students do not possess sufficiently developed relational understanding of the equals sign and that operational understanding prevails. Students of the fourth grade demonstrated better understanding of the equals sign at all levels of under- standing than the second graders.
КЉУЧНЕ РЕЧИ:
equals sign, equivalence, operational understanding, relational understanding, mathematics, mathematics education
ЛИТЕРАТУРА:
- Alexandrou-Leonidou, Philippou (2011): V. Alexandrou-Leonidou, G. N. Philip- pou, Can they ’see’ the equality?, In: M. Pytlak, T. Rowland, E. Swoboda, Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (CERME 7), (410–419), Poland: University of Rzeszów.
- Alibali, Knuth, Hattikudur, McNeil, Stephens (2007): M. W. Alibali, E. J. Knuth, S. Hattikudur, N. M. McNeil, A. C. Stephens, A longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9, 221–247.
- Baroody, Ginsburg (1983): A. J. Baroody, H. P. Ginsburg, The effects of instruction on children’s understanding of the “equals” sign, The Elementary School Journal, 84(2), 199–212.
- Behr, Erlwanger, Nichols (1976): M. Behr, S. Erlwanger, E. Nichols, How Children View Equality Sentences. Project for the Mathematical Development of Children. Techni- cal Report No. 3, Florida State University. ERIC Document Reproduction Services No. ED144802
- Behr, Erlwanger, Nichols (1980): M. Behr, S. Erlwanger, E. Nichols, How children view the equals sign, Mathematics teaching, 92(1), 13–15.
- Booth (1988): L. R. Booth, Children’s difficulties in beginning algebra, The ideas of algebra, K-12, 19, 20–32.
- Capraro, Ding, Matteson, Capraro, Li (2007): M. M. Capraro, M. Ding, S. Mat- teson, R. M. Capraro, X. Li, Representational implications for understanding equivalence, School Science and Mathematics, 107(3), 86–88.
- Carpenter, Levi (2000): T. P. Carpenter, L. Levi, Developing conceptions of alge- braic reasoning in the primary grades, Madison, WI: National Center for Improving Student Learning and Achievement in Mathematics and Science.
- Carpenter, Franke, Levi (2003): T. P. Carpenter, M. L. Franke, L. Levi, Thinking mathematically: Integrating arithmetic and algebra in elementary school, Portsmouth, NH: Heinemann.
- Cobb (1987): P. Cobb, An investigation of young children’s academic arithmetic contexts, Educational Studies in Mathematics, 18(2), 109–124.
- Cockburn, Littler (2008): A. D. Cockburn, G. Littler, Mathematical Misconcep- tions, A Guide for Primary Teacher, SAGE.
- Dabić Boričić, Zeljić (2021): M. Dabić Boričić, M. Zeljić, Modelovanje ekviva- lencije matematičkih izraza u početnoj nastavi, Inovacije u nastavi ‒ časopis za savremenu nastavu, 34(1), 30–43.
- Duncan (2015): C. D. Duncan, A study of mathematical equivalence: The impor- tance of the equal sign (Master’s thesis, Southern University). LSU Digital Commons.
- Ilić, Zeljić (2017): S. Ilić, M. Zeljić, Pravila stalnosti zbira i razlike kao osnova strategija računanja, Inovacije u nastavi, 30 (1), 55–66.
- Jacobs, Franke, Carpenter, Levi, Battey (2007): V. R. Jacobs, M. L. Franke, T. P. Carpenter, L. Levi, D. Battey, Professional development focused on children’s algebraic reasoning in elementary school, Journal for Research in Mathematics Education, 38(3), 258–288.
- Jones, Inglis, Gilmore, Dowens (2012): I. Jones, M. Inglis, C. Gilmore, M. Dowens, Substitution and sameness: Two components of a relational conception of the equals sign, Journal of Experimental Child Psychology, 113(1), 166–176.
- Kieran (1981): C. Kieran, Concepts associated with the equality symbol, Educa- tional Studies in Mathematics, 12(3), 317–326.
- Kieran (1989): C. Kieran, The early learning of algebra: A structural perspective, In: S. Wagner, C. Kieran (Eds.), Research issues in the learning and teaching of algebra (33–56), Reston, VA: NCTM
- Knuth, Stephens, McNeil, Alibali (2006): E. Knuth, A. C. Stephens, N. M. McNeil, M. W. Alibali, Does understanding the equal sign matter? Evidence from solving equation, Journal for Research in Mathematics Education, 37(4), 297–312.
- Knuth, Alibali, Hattikudar, McNeil, Stephens (2008): E. J. Knuth, M. W. Alibali, S. Hattikudar, N. M. McNeil, A. S. Stephens, The Importance of the Equal Sign Under- standing in the Middle Grades, Middle Teaching in the Middle School, 13(9), 514–519.
- Milinković, Maričić Đokić (2022): N. Milinković, S. Maričić, O. Đokić, The equals sign – the problem of early algebra learning and how to solve it, Inovacije u nastavi, XXXV(3), 26–41.
- Molina, Ambrose (2008): M. Molina, R. Ambrose, From an operational to a rela- tional conception of the equal sign. Thirds graders’ developing algebraic thinking, Focus on Learning Problems in Mathematics, 30(1), 61‒80.
- Molina, Castro, Castro (2009): M. Molina, E. Castro, E. Castro, Elementary stu- dents’ understanding of the equal sign in number sentences, Education & Psychology, 17, 341–368.
- McAuliffe, Tambara, Simsek (2020): S. McAuliffe, C. Tambara, E. Simsek, Young students’ understanding of mathematical equivalence across different schools in South Af- rica, South African Journal of Childhood Education, 10(1), 1–8.
- McNeil, Grandau, Knuth, Alibali, Stephens, Hattikudur, et al. (2006): N. M. Mc- Neil, L. Grandau, E. J. Knuth, M. W. Alibali, A. S. Stephens, S. Hattikudur, et al., Middle- school students’ understanding of the equal sign: The books they read can’t help, Cognition & Instruction, 24(3), 367–385.
- McNeil (2007): N. M. McNeil, U-shaped development in math: 7-year-olds out- perform 9-year-olds on equivalence problems, Developmental Psychology, 43, 687–695.
- McNeil (2008): N. M: McNeil, Limitations to teaching children 2 + 2 = 4: Typical arithmetic problems can hinder learning of mathematical equivalence, Child Development, 79(5), 1524–1537.
- McNeil, Weinberg, Hattikudur, Stephens, Asquith, Knuth, Alibali (2010): N. M. McNeil, A. Weinberg, S. Hattikudur, A. C. Stephens, P. Asquith, E. J. Knuth, M. W. Ali- bali, A is for apple: Mnemonic symbols hinder the interpretation of algebraic expressions, Journal of Educational Psychology, 102(3), 625–634.
- Parslow-Williams, Cockburn (2008): P. Parslow-Williams, A. D. Cockburn, Equal- ity: getting the right balance, In: A. D. Cockburn, G. Littler (eds.), Mathematical Miscon- ceptions (23‒38), SAGE.
- Pravilnik (2019). Правилник о плану и програму наставе и учења за четврти разред основног образовања и васпитања, Просветни гласник, Службени гласник Ре- публике Србије, бр. 11/2019.
- Rittle-Johnson, Alibali (1999): B. Rittle-Johnson, M. W. Alibali, Conceptual and procedural knowledge of mathematics: Does one lead to the other?, Journal of Educational Psychology, 91, 175–189.
- Rittle-Johnson, Matthews, Taylor, McEldoon (2011): B. Rittle-Johnson, P. G. Mat- thews, R. S. Taylor, K. L. McEldoon, Assessing knowledge of mathematical equivalence: A construct-modeling approach, Journal of Educational Psychology, 103(1), 85.
- Seo, Ginsburg (2003): K. H. Seo, H. P. Ginsburg, “You’ve got to carefully read the math sentence…”: Classroom context and children’s interpretations of the equals sign, In: A. J. Baroody, A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (161–187), Lawrence Erlbaum Associates, Mahwah, NJ.
- Fagnant, Vlassis (2013): A. Fagnant, J. Vlassis, Schematic representations in arith- metical problem solving: Analysis of their impact on grade 4 students, Educational Studies in Mathematics, 84(1), 149–168.
- Falkner, Levi, Carpenter (1999): K. Falkner, L. Levi, T. Carpenter, Early Child- hood Corner: Children’s Understanding of Equality: A Foundation for Algebra, Teaching children mathematics, 6(4), 232–236.
- Filloy, Rojano (1989): E. Filloy, T. Rojano, Solving equations: The transition from arithmetic to algebra, For the learning of mathematics, 9(2), 19–25.
- Fyfe, Matthews, Amsel, McEldoon, McNeil (2018): E. R. Fyfe, P. G. Matthews, E. Amsel, K. L. McEldoon, N. M. McNeil, Assessing formal knowledge of math equiva- lence among algebra and pre-algebra students, Journal of Educational Psychology, 110(1), 87–101.