Nepoznata i promenljiva – problem učenja rane algebre

Nauka i obrazovanje – izazovi i perspektive (2022) (str. 245-262)

AUTOR(I): Nenad Milinković, Sanja Maričić

E-ADRESA: milinkovic.nenad84@gmail.com

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DOI: 10.46793/NOIP.245M

SAŽETAK:

U radu autori skreću pažnju na poteškoće koje prate uvo- đenje simbola i simboličke notacije u mlađim razredima osnovne škole. U tom kontekstu posebnu pažnju skreću na pravilno razumevanje pojma nepo- znate i promenljive na ovom uzrastu i ukazuju na širok spektar različitih problema vezanih za razumevanje slova kao simbola kojim se označava nepo- znata ili promenljiva u ranoj algebri. Na primeru formiranja pojma nepo- znata i primeru formiranja pojma promenljiva pokazano je kako se može oblikovati problem realnog konteksta, na različitim nivoima apstraktnosti. U ovom procesu realno zasnovane situacije predstavljaju izvor za razvijanje matematičkih pojmova, alata i procedura u kojima učenici u kasnijoj fazi mogu primeniti matematička znanja, koja vremenom postaju opštija i formalnija, a samim tim i manje vezana za kontekst.

KLJUČNE REČI:

algebra, nepoznata, promenljiva, početna nastava matematike, realni kontekst.

LITERATURA:

  • Akgün, L. & Özdemir, M. E. (2006). Studentsʼ understanding of the variable as general number and unknown: A case study. The teaching of mathematics, 16, 45–51.
  • Aczel, J. (1998). Learning equations using a computerised balance model: a Popperian approach to learning symbolic algebra. (Doctoral dissertation). University of Oxford.
  • Blanton, M., Brizuela, B. M., Gardiner, A. M., Sawrey, K. & Newman-Owens, A. (2017). A progression in first-grade children’s thinking about variable and variable notation in functional relationships. Educational Studies in Mathematics, 95(2), 181–202.
  • Booth, L. R. (1988). Children’s difficulties in beginning algebra. The ideas of algebra, K-12, 19, 20–32.
  • Breiteig, T. & Grevholm, V. (2006). The transition from arithmetic to algebra: to reason, explain, argue, generalize and justify. In J. Novotná, H. Moraová, M. Krátká & N. Stehlíková (eds.): Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 2(1), 225–232. Prague: PME.
  • Van Amerom, B. A. (2003). Focusing on informal strategies when linking arithmetic to early algebra. Educational Studies in Mathematics, 54(1), 63–75.
  • Güçler, B. (2014). The role of symbols in mathematical communication: the case of the limit notation. Research in Mathematics Education, 16(3), 251–268.
  • Đokić, O. (2014). Realne situacije. U P. Pijanović (ur.): Leksikon obrazovnih termina (670–671). Beograd: Učiteljski fakultet.
  • Earnest, D. & Balti, A. (2008). Instructional strategies for teaching algebra in elementary school: Findings from a research-practice collaboration. Teaching Children Mathematics, 14(9), 518–522.
  • Kilhamn, C. (2014). When does a variable vary? Identifying mathematical content knowledge for teaching variables. Nordic Studies in Mathematics Education, 19(3– 4), 83–100.
  • Küchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in school, 7(4), 23–26.
  • Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A. & Stephens, A. S. (2005). Middle school students’ understanding of core algebraic concepts: Equality & variable. Zentralblatt fü r Didaktik der Mathematik [International Reviews on Mathematical Education], 37, 68–76, https://doi.org/10.1007/BF02655899.
  • Maracchia, S.  (2013).  The  importance  of  symbolism  in  the  development  of  algebra. Lettera Matematica, 1(3), 137–144.
  • Maričić, S. i Milinković, N. (2017). Udžbenik u stvaranju uslova za kontekstu- alni pristup učenju sadržaja algebre u početnoj nastavi matematike. Zbor- nik radova, 20(19), 117–130.
  • MacGregor, M. & Stacey, K. (1997). Students’understanding of algebraic notation: 11– 15. Educational studies in mathematics, 33(1), 1–19.
  • Meira, L. (1995). The microevolution of mathematical representations in childrenʼs activity. Cognition and instruction, 13(2), 269–313.
  • Milinković, N. i Maričić, S. (2019). Pristup sadržajima o zavisnosti rezultata od promene komponenata računskih operacija u udžbenicima matematike. Zbornik radova, 22(21), 193–206.
  • Milinković, N. (2021). Kontekstualni pristup nastavi algebre u mlađim razre- dima osnovne škole (doktorska disertacija). Užice: Pedagoški fakultet.
  • McNeil, N. M., Weinberg, A., Hattikudur, S., Stephens, A. C., Asquith, P., Knuth, E. J. & Alibali, M. W. (2010). A is for apple: Mnemonic symbols hinder the interpretation of algebraic expressions. Journal of Educational Psychology, 102(3), 625–634.
  • Radford, L. (2000). Signs and meanings in studentsʼ emergent algebraic thinking: A semiotic analysis. Educational studies in mathematics, 42(3), 237–268.
  • Radford, L. (2002). On heroes and the collapse of narratives: a contribution to the study of symbolic thinking. In: Cockburn, A. D. & E. Nardi (eds.), Proceedings of the 26th Conference of the International Group of Psychology of Mathematics Educa- tion, Norwich (UK), 4 (81–88). Norwich, UK: PME.
  • Radford, L. (2004). Syntax and meaning. In M. J. Høines & A. B. Fuglestad (eds.), Proceedings of the 28 Conference of the International Group for the Psychology of Mathematics Education, 1 (161–166). Norway: Bergen University College.
  • Rystedt, E., Kilhamn, C. & Helenius, O. (2016). What’s there in an n? Investigating contextual resources in small group discussions concerning an algebraic expression. Nordic Studies in Mathematics Education, 21(1), 5–26.
  • Skemp, R. R. (2012). The psychology of learning mathematics: Expanded American edition. Routledge.
  • Specht, B. J. (2005). Early Algebra – Processes and Concepts of Fourth Grades Solving Algebraic Problems. In M. Bosch, M. Perpiñán, M. Àngels Portabella & R. Llull (eds.): Proceedings of the 4th Congress of the European Society for Research in Mathematics Education (706‒716). Sant Feliu de Guíxols, Spain.
  • Stephens, A., Blanton, M., Knuth, E., Isler, I. & Gardiner, A. M. (2015). Just say yes to early algebra!. Teaching children mathematics, 22(2), 92‒101.
  • Sfard, A. & Linchevski, L. (1994). The gains and pitfalls of reification: The case of algebra. Educational Studies in Mathematics, 26(2‒3), 191‒228.
  • Tall, D. (2008). The transition to formal thinking in mathematic. Mathematics Education Research Journal, 20(2), 5–24.
  • Taylor-Cox, J. (2003). Algebra in the early years? Yes!. Young Children, 58(1), 14–21. Usiskin, Z. (1997). Doing algebra in grades K-4. Teaching children mathematics, 3(6), 346–356.
  • Ferrari, P. L. (2006). From verbal texts to symbolic expressions: A semiotic approach to early algebra. In J. Novotná, H. Moraová, M. Krátká & N. Stehlíková (eds.), Pro- ceedings 30th Conference of the International Group for the Psychology of Mathe- matics Education, 3 (73–80). Prague: PME.
  • Fujii, T. & Stephens, M. (2008). Using number sentences to introduce the idea of variable. In C. Greenes & R. Rubenstein (eds.): Algebra and algebraic thinking in school mathematics: Seventieth Yearbook (127–140). Reston, VA: National Council of Teachers of Mathematics.
  • Carpenter, T. P. & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. Madison, WI: National Center for Improving Student Learning and Achievement in Mathematics and Science.
  • Carpenter, T. & Franke, M. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In H. Chick, K. Stacey, J. Vincent & J. Vincent (eds.), Proceedings of the Twelfth ICMI Study Conference: The future of the teaching and learning of algebra, 1. Melbourne, Australia: University of Melbourne Press, 155–162.
  • Carraher, D. W., Schliemann, A. D. & Brizuela, B. M. (2000). Early algebra, early arithmetic: Treating operations as functions. In M. L. Fernandez (ed.): The 22nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Tucson, Arizona.
  • Carraher, D. W., Schliemann, A. D., Brizuela, B. M. & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics education, 87–115.
  • Carraher, D. W., Schliemann, A. D. & Schwartz, J. (2008). Early algebra is not the same as algebra early. In J. Kaput, D. Carraher & M. Blanton (eds.): Algebra in the Early Grades (235‒272). Mahwah, NJ: Lawrence Erlbaum Associates/Taylor & Francis Group and National Council of Teachers of Mathematics.
  • Cvijanović, G. (2016). Konceptualizacija pojma rana algebra i ranoalgebarsko miš ljenje . Istraživanjematematičkogobrazovanja,VIII(14), 1–8.