Mаријана Ж. Зељић, Универзитет у Београду , Учитељски факултет, имејл: marijana.zeljic@uf.bg.ac.rs
Светлана М. Илић, Универзитет у Београду , Учитељски факултет
Мила С. Јелић, Универзитет у Београду , Учитељски факултет
Иновације у настави, XXX, 2017/4, стр. 49–61

| PDF | | Extended summary PDF |
doi: 10.5937/inovacije1704049Z

Резиме: Ментална аритметика има важно место у настави и учењу математике. Конкретно, она развија вештине решавања проблема, пружа могућности за развијање способности процене у рачунању и доприноси дубљем разумевању појма броја и декадног система. Основне карактеристике менталне аритметике су следеће: 1) рачуна се бројевима, а не цифрама (без записивања парцијалних резултата); 2) стратешка флексибилност у смислу бирања стратегије у зависности од карактеристика бројева у задатку. Циљ ове студије је да се испита способност ученика да одузму два броја без папира и оловке (ментално), као и да се идентификују стратегије које ученици у том поступку користе. Посебно значајно питање које се поставља у раду јесте – да ли ученици флексибилно користе стратегије рачунања, то јест да ли избор стратегије зависи од структуре задатка. Узорак истраживања чини шездесет и шест ученика трећег разреда из две београдске основне школе. Коришћене су дескриптивна метода и техника интервјуисања. Наши резултати показују да ученици као доминантну стратегију приликом рачунања без папира и оловке користе алгоритам цифарског рачунања, што представља узрок грешака у рачунању. Даље, као важан резултат истраживања истичемо да ученици не поседују стратешку флексибилност приликом менталног одузимања, што може бити показатељ недовољног концептуалног разумевања структуре бројева и рачунских процедура. Резултати истраживања указују на потребу да се измене приступи аритметичким садржајима померањем фокуса са развијања вештина алгоритамског рачунања на развијање дубљег разумевања и коришћења различитих поступака рачунања.

Кључне речи: ментална аритметика, стратегије одузимања, стратешка флексибилност

Summary: Mental arithmetic plays an important role in teaching and learning mathematics. More precisely, it develops problem-solving skills, helps students to develop a skill of making estimations in calculations, and contributes to a better understanding of the concept of numbers and decade system. The basic characteristics of mental arithmetic are as follows: 1) mental calculation uses numbers, not digits (without recording partial results); 2) strategic flexibility in terms of selecting a strategy relative to the characteristics of numbers in a mathematics task. The aim of this paper is to examine pupils’ ability to subtract two numbers without using paper and pencil (mentally) and to identify the strategies used by pupils while doing the calculations. The paper also focuses on the very important question of pupils’ flexibility in using calculation strategies, or more specifically, whether the choice of a strategy depends on the structure of a mathematical task. The research was conducted on a sample of 66 third-graders from two primary schools in Belgrade. A descriptive method and interview were used in the research. The obtained results indicate that pupils predominantly use the algorithm of digital calculation in trying to do a calculation without a paper and pencil, which is the cause of many errors. Equally important, the pupils lack flexibility in doing mental calculation, which may indicate the insufficient understanding of the structure of numbers and calculation procedures. In addition, the results point to the need to change the approaches to arithmetic content by shifting the focus from developing algorithm calculation skills to developing a more in-depth understanding and use of different calculation procedures.

Кeywords: mental arithmetic, subtraction strategies, strategic flexibility.

Литература

• Baranes, R., Perry, M. & Stiegler, J. W. (1989). Activation of real-world knowledge in the solution of word problems. Cognition and Instruction. 6 (4), 287–318.
• Baroody, A. J. & Tiilikainen, S. H. (2003). Two perspectives on addition development. In: Baroody, A. J. & Dowker, A. (Eds.). The development of arithmetic concepts and skills (75–125). Mahwah, N. J.: Lawrence Erlbaum Associates.
• Blöte, A. W., Klein, A. S. & Beishuizen, M. (2000). Mental computation and conceptual understanding. Learning and Instruction. 10, 221–247.
• Blöte, A. W., Van der Burg, E. & Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: Instructional effects. Journal of Educational Psychology. 93 (3), 627–638. DOI:10.1037/0022-0663.93.3.627.
• Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E. & Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education. 29 (1), 3–20.
• Cohen, L., Manion, L. & Morrison, K. (2000). Metode istraživanja u obrazovanju. Zagreb: Naklada Slap.
• Cooper, T. & Warren, E. (2011). Years 2 to 6 students’ ability to generalise: Models. representations and theory. In: Cai, J. & Knuth, E. (Eds). Early algebraization: A global dialogue from multiple perspectives (187–214). Netherlands: Springer.
• Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I. et al. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education. 28 (2), 130–162. DOI: 10.2307/749759.
• Heirdsfield, A., Dole, S. & Beswick, K. (2007). Instruction to support mental computation development in young children of diverse ability. In: Jeffery, P. L. (еd.). Proceedings Australian Association for Research in Education Conference (26–30). November 2006. Adelaide, Australia. Retrieved January 18, 2017. from: https://eprints.qut.edu.au/6604/1/6407.pdf.
• Ilić, S. (2016). Razumevanje i primena pravila o stalnosti zbira i razlike (master rad). Beograd: Učiteljski fakultet.
• Ilić, S., Zeljić, M. (2016). Razumevanje i primena pravila aritmetike od strane učenika četvrtog razreda osnovne škole. Pedagogija. 71 (4), 419–432.
• Imbo, I. & Vandierendonck, A. (2006). The development of strategy use in elementary school children:Working memory and individual differences. Journal of Experimental Child Psychology. 96, 284–309.
• Kilpatrick, J., Swafford, J. & Findell, B. (Eds.) (2001). Adding it up. Helping children learn mathematics. Washington, DC: National Academy Press.
• Linsen, S., Verschaffel, L., Reynvoet, B. & De Smedt, B. (2015). The association between numerical magnitude processing and mental versus algorithmic multi-digit subtraction in children. Learning and Instruction. 35, 42–50. DOI: 10.1016/j.learninstruc.2014.09.003.
• Maclellan, E. (2001). Mental Calculation: its place in the development of numeracy. Westminster Studies in Education. 24 (2), 145–154. Retrieved January 18, 2017 from: https://pure.strath.ac.uk/portal/files/185113/strathprints007321.pdf.
• M., Van Den Heuvel−Panhuizen, M. & Robitzsch, A. (2012). Special education students᾿ use of indirect addition in solving subtraction problems up to 100 − A proof of the didactical potential of an ignored procedure. Educational Studies in Mathematics. 79 (3), 351–369. DOI:10.1007/s10649-011-9351-0.
• Pravilnik o nastavnom planu za prvi, drugi, treći i četvrti razred osnovnog obrazovanja i vaspitanja i nastavnom programu za treći razred osnovnog obrazovanja i vaspitanja. Prosvetni glasnik, br. 1/05, 15/06, 2/08, 7/10, 3/11, 7/11, 1/13 i 11/14.
• Pravilnik o nastavnom programu za četvrti razred osnovnog obrazovanja i vaspitanja Prosvetni glasnik, br. 3/06, 15/06, 2/08, 3/11, 7/11, 1/13 i 11/14.
• Selter, C., Prediger, S., Nührenbörger, M. & Hußmann, S. (2012). Taking away and determinig the difference – a longitudinal perspective on two models of subtraction and the inverse relation to addition. Educational Studies in Mathematics. 79 (3), 389–408. DOI:10.1007/s10649-011-9305-6.
• Star, J. R. & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology. 31 (3), 280–300. DOI: 10.1016/j.cedpsych.2005.08.001.
• Threlfall, J. (2002). Flexible Mental Calculation. Educational Studies in Mathematics. 50 (1), 29–47. DOI: 10.1023/A:1020572803437.
• Threlfall, J. (2009). Strategies and flexibility in mental calculation. ZDM Mathematics Education. 41, 541–555. DOI:10.1007/s11858-009-0195-3.
• Torbeyns, J. & Verschaffel, L. (2013). Efficient and flexible strategy use on multidigit sums: a choice/nochoice study. Research in Mathematics Education. 15 (2), 129–140. http://dx.doi.org/10.1080/14794802.2013.797745.
• Torbeyns, J., De Smedt, B., Stassens, N., Ghesquière, P. & Verschaffel, L. (2009a). Solving subtraction problems by means of indirect addition. Mathematical Thinking and Learning. 11 (1–2), 79–91. http://dx.doi.org/10.1080/10986060802583998.
• Torbeyns, J., De Smedt, B., Ghesquiere, P. & Verschaffel, L. (2009b). Acquisition and use of shortcut strategies by traditionally schooled children. Educational Studies in Mathematics. 71, 1–17. DOI: 10.1007/s10649-008-9155-z.
• Torbeyns, J., Ghesquiere, P. & Verschaffel, L. (2009c). Efficiency and flexibility of indirect addition in the domain of multi-digit subtraction. Learning and Instruction. 19 (1), 1–12.
• Torbeyns, J., Verschaffel, L. & Ghesquiere, P. (2006). The development of children’s adaptive expertise in the number domain 20 to 100. Cognition and Instruction. 24 (4), 439–465.
• Van de Walle, J. A. (2007). Elementary and Middle School Mathematics: Teaching Developmentally (6th ed.). New York: Pearson Education. Inc.
• Van den Heuvel-Panhuizen, M. (Ed.) (2001). Children learn mathematics. A teaching-learning trajectory with intermediate attainment targets for calculation with whole numbers in primary school. Utrecht, The Netherlands: Freudenthal Institute. University of Utrecht. Retrieved January 18, 2017 from: https://www.sensepublishers.com/media/161-children-learn-mathematics.pdf.
• Van Den Heuvel-Panhuizen, М. (2005). The role of contexts in assessment problems in mathematics. For the Learning of Mathematics. 25 (2), 2–23.
• Verschaffel, L., Greer, B. & De Corte, E. (2007). Whole number concepts and operations. In: Lester, F. K. (Ed.). Second handbook of research on mathematics teaching and learning (557–628). Greenwich, CT: Information Age Publishing.
• Verschaffel, L., Luwel, K., Torbeyns, J. & Van Dooren, W. (2009). Conceptualising. investigating and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education. 24 (3), 35–59.
• Verschaffel, L., Torbeyns, J., De Smedt, B., Peters, G. & Ghesquière, P. (2010). Solving subtraction problems flexibly by means of indirect addition. In: Cowan, R., Saxton, M. & Tolmie, A. (Eds.). Special issue of the British journal of educational psychology. Understanding number development and difficulties (monograph). 2 (7), 51–63.

Copyright © 2017 by the authors, licensee Teacher Education Faculty University of Belgrade, SERBIA. This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original paper is accurately cited.