ПУТ КА СИМБОЛИМА У АЛГЕБРИ: ОД РЕАЛИСТИЧНИХ СИТУАЦИЈА ДО СИМБОЛИЧКОГ ЈЕЗИКА
Ključne reči:
алгебра, контекстуални приступ, математика, непозната, променљива
Sažetak
У раду аутори представљају оквир учења за увођење слова као ознаке за непознату и променљиву који се заснива на ситуација из реалног контекста, њеном моделовању и превођењу на алгебарски језик. У емпиријском делу рада организовано је истраживање с циљем да испитају ефекти оваквог приступа на развијање алгебарске симболике код ученика млађих разреда основне школе. Истраживање је спроведено на узорку од 257 ученика четвртог разреда основних школа у Републици Србији, применом експерименталне методе са паралелним групама. Резултати истраживања показују позитивне ефекте примењеног приступа на боље разумевање симболичке нотације и већи степен алгебарске генерализације. Ови налази потврђују хипотезу да увођење алгебарских симбола кроз реалистичне проблеме и контекстуално повезане активности може значајно унапредити способност ученика да правилно интерпретирају и користе алгебарску нотацију.
Reference
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Akgün, L., & Özdemir, M. E. (2006). Students' understanding of the variable as general number and unknown: A case study. The teaching of mathematics, (16), 45–51. DOI: 10.5937/teacchilmath0601045A
Arcavi, A. (1994). Symbol sense: informal sense-making in formal mathematics. For the Learning of Mathematics, 25(2), 24–35. https://www.jstor.org/stable/40248497
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. https://doi.org/10.1007/s10649-016-9745-0
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Brizuela, B. M., Blanton, M., Sawrey, K., Newman-Owens, A., & Murphy Gardiner, A. (2015). Children’s Use of Variables and Variable Notation to Represent Their Algebraic Ideas. Mathematical Thinking and Learning, 17(1), 34–63. https://doi.org/10.1080/10986065.2015.981939
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 (pp. 235‒272). Mahwah, NJ: Lawrence Erlbaum Associates/Taylor & Francis Group and National Council of Teachers of Mathematics.
Christou, K., & Vosniadou, S. (2005). How students interpret literal symbols in algebra: A conceptual change approach. In B.G. Bara, L. Barsalou, & M. Bucciarelli (Eds.), Proceedings of the XXVII Annual Conference of the Cognitive Science Society (pp. 453–458). Stresa, Italy. Mahwah, NJ: Lawrence Erlbaum Associates, Inc
Dabić Boričić, M. (2019). Metodički aspekti formiranja algebarskih zakona u početnoj nastavi matematike (Doktorska disertacija, Univerzitet u Beogradu, Učiteljski fakultet).
Ferrari, P. L. (2006). From verbal texts to symbolic expressions: A semiotic approach to early algebra. In: Ј. Novotná, H. Moraová, M. Krátká & N. Stehlíková (Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 3, (pp. 73–80). Prague: PME.
Goldin, G. A. (2020). Mathematical representations. U S. Lerman (ur.), Encyclopedia of Mathematics Education (pp. 566–572). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-15789-0_103
Heffernan, N. T., & Koedinger, K. R. (2022). A developmental model for algebra symbolization: The results of a difficulty factors assessment. In: M. A. Gernsbacher, S. J. Derry (Eds.) Proceedings of the twentieth annual conference of the cognitive science society (pp. 484–489). Routledge.
Jupri, A., Drijvers, P., & van den Heuvel-Panhuizen, M. (2014). Difficulties in initial algebra learning in Indonesia. Mathematics Education Research Journal, 26(4), 683–710. https://doi.org/10.1007/s13394-013-0097-0
Kieran, C. (1981). Concepts associated with the equality symbol. Educational studies in Mathematics, 12(3), 317–326. https://doi.org/10.1007/BF00311062
MacGregor, M., & Stacey, K. (1997). Students'understanding of algebraic notation: 11–15. Educational studies in mathematics, 33(1), 1–19. https://doi.org/10.1023/A:1002970913563
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. https://psycnet.apa.org/doi/10.1037/a0019105
Milinković, N., Maričić, S. (2022). Nepoznata i promenljiva – problem učenja rane algebre, u: S. Marinković (ur.): Nauka i obrazovanje – izazovi i perspektive, 245–262. Užice: Pedagoški fakultet.
Milinković, N., Maričić, S. & Đokić, O. (2022). The equals sign–the problem of early algebra learning and how to solve it. Teaching innovations, 35(3), 26–43. https://doi.org/10.5937/inovacije2203026M
Özgeldi, M. (2013). Interpretation of unknown and variable prior to formal algebraic instruction. Kastamonu Education Journal, 21(4), 1317–1326.
Pallant, J. (2017). SPSS priručnik za preživljavanje: postupni vodič kroz analizu podataka pomoću SPSS-a, Prevod 6. izdanja. Beograd: Mikro knjiga.
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.), Proc. 26th Conf. of the Int. Group for the Psychology of Mathematics Education, Norwich (UK),. (Vol.4, pp. 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, (pp. 161–166). Norway: Bergen University College.
Radford, L. (2011). Grade 2 students’ non-symbolic algebraic thinking. In: J. Cai & E. Knuth (Eds.), Early algebraization. (pp. 303–22). Berlin: Springer.
Radford, L. (2018). The emergence of symbolic algebraic thinking in primary school. In: C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds: The global evolution of an emerging field of research and practice (pp. 3–25). New York: Springer. DOI: 10.1007/978-3-319-68351-5_1
Radford, L. (2022). Introducing equations in early algebra. ZDM–Mathematics Education, 54(6), 1151–1167. https://doi.org/10.1007/s11858-022-01422-x
Russell, S. J., Schifter, D. & Bastable, V. (2011). Developing algebraic thinking in the context of arithmetic. In: J. Cai & E. J. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives, (pp. 43–69). Berlin: Springer. DOI: 10.1007/978-3-642-17735-4_4
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. https://doi.org/10.7146/nomad.v21i1.148715
Specht, B. J. (2005). Early Algebra – Processes and Concepts of Fourth Grades Solving Algebraic Problems. In: M. Bosch, M. Perpiñán, M. Àngels Portabella, and R. Llull, (Eds.). Proceedings of the 4th Congress of the European Society for Research in Mathematics Education (pp. 706‒716). Sant Feliu de Guíxols, Spain.
Sun, S., Sun, D., & Xu, T. (2023). The Developmental Progression of Early Algebraic Thinking of Elementary School Students. Journal of Intelligence, 11(12), 222. https://doi.org/10.3390/jintelligence11120222
Sfard, A. & Linchevski, L. (1994). Between artithmetic and algebra: In the search of a missing link, the case of equations and inequalities. Rendiconti Del Seminario Matematico, 52(3), 279–307. DOI: 10.1007/BF02574811
Stacey, K., & MacGregor, M. (1999). Learning the algebraic method of solving problems. The Journal of Mathematical Behavior, 18(2), 149‒167. https://doi.org/10.1016/S0732-3123(99)00026-7
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. https://doi.org/10.5951/teacchilmath.22.2.0092
Ünal, Z. E., Ala, A. M., Kartal, G., Özel, S., & Geary, D. C. (2023). Visual and symbolic representations as components of algebraic reasoning. Journal of Numerical Cognition, 9(2), 327–345.
Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. Coxford (Ed.), The ideas of algebra, K-12 (8–19). Reston: National Council of Teachers of Mathematics.
Van Amerom, B. A. (2002). Reinvention of early algebra: Developmental research on the transition from arithmetic to algebra (Doctoral dissertation). Utrecht University.
Van Reeuwijk, M. (2001). From informal to formal, progressive formalization: An example on “solving systems of equations.” In: H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra: Proceedings of the 12th ICMI Study Conference, 2, (pp. 613–620). University of Melbourne, Australia.
Weinberg, A., Dresen, J., & Slater, T. (2016). Students’ understanding of algebraic notation: A semiotic systems perspective. The Journal of Mathematical Behavior, 43, 70‒88. https://doi.org/10.1016/j.jmathb.2016.06.001
Zeljić, M. (2014). Metodički aspekti rane algebre. Učiteljski fakultet: Univerzitet u Beogradu.
Akgün, L., & Özdemir, M. E. (2006). Students' understanding of the variable as general number and unknown: A case study. The teaching of mathematics, (16), 45–51. DOI: 10.5937/teacchilmath0601045A
Arcavi, A. (1994). Symbol sense: informal sense-making in formal mathematics. For the Learning of Mathematics, 25(2), 24–35. https://www.jstor.org/stable/40248497
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. https://doi.org/10.1007/s10649-016-9745-0
Booth, L. R. (1988). Children’s Difficulties in Beginning Algebra. In A. F. Coxford & A. P. Shulte (Eds.), The Ideas of Algebra, K–12: 1988 Yearbook (pp. 20–32). Reston, VA: National Council of Teachers of Mathematics.
Brizuela, B. M., Blanton, M., Sawrey, K., Newman-Owens, A., & Murphy Gardiner, A. (2015). Children’s Use of Variables and Variable Notation to Represent Their Algebraic Ideas. Mathematical Thinking and Learning, 17(1), 34–63. https://doi.org/10.1080/10986065.2015.981939
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 (pp. 235‒272). Mahwah, NJ: Lawrence Erlbaum Associates/Taylor & Francis Group and National Council of Teachers of Mathematics.
Christou, K., & Vosniadou, S. (2005). How students interpret literal symbols in algebra: A conceptual change approach. In B.G. Bara, L. Barsalou, & M. Bucciarelli (Eds.), Proceedings of the XXVII Annual Conference of the Cognitive Science Society (pp. 453–458). Stresa, Italy. Mahwah, NJ: Lawrence Erlbaum Associates, Inc
Dabić Boričić, M. (2019). Metodički aspekti formiranja algebarskih zakona u početnoj nastavi matematike (Doktorska disertacija, Univerzitet u Beogradu, Učiteljski fakultet).
Ferrari, P. L. (2006). From verbal texts to symbolic expressions: A semiotic approach to early algebra. In: Ј. Novotná, H. Moraová, M. Krátká & N. Stehlíková (Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 3, (pp. 73–80). Prague: PME.
Goldin, G. A. (2020). Mathematical representations. U S. Lerman (ur.), Encyclopedia of Mathematics Education (pp. 566–572). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-15789-0_103
Heffernan, N. T., & Koedinger, K. R. (2022). A developmental model for algebra symbolization: The results of a difficulty factors assessment. In: M. A. Gernsbacher, S. J. Derry (Eds.) Proceedings of the twentieth annual conference of the cognitive science society (pp. 484–489). Routledge.
Jupri, A., Drijvers, P., & van den Heuvel-Panhuizen, M. (2014). Difficulties in initial algebra learning in Indonesia. Mathematics Education Research Journal, 26(4), 683–710. https://doi.org/10.1007/s13394-013-0097-0
Kieran, C. (1981). Concepts associated with the equality symbol. Educational studies in Mathematics, 12(3), 317–326. https://doi.org/10.1007/BF00311062
MacGregor, M., & Stacey, K. (1997). Students'understanding of algebraic notation: 11–15. Educational studies in mathematics, 33(1), 1–19. https://doi.org/10.1023/A:1002970913563
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. https://psycnet.apa.org/doi/10.1037/a0019105
Milinković, N., Maričić, S. (2022). Nepoznata i promenljiva – problem učenja rane algebre, u: S. Marinković (ur.): Nauka i obrazovanje – izazovi i perspektive, 245–262. Užice: Pedagoški fakultet.
Milinković, N., Maričić, S. & Đokić, O. (2022). The equals sign–the problem of early algebra learning and how to solve it. Teaching innovations, 35(3), 26–43. https://doi.org/10.5937/inovacije2203026M
Özgeldi, M. (2013). Interpretation of unknown and variable prior to formal algebraic instruction. Kastamonu Education Journal, 21(4), 1317–1326.
Pallant, J. (2017). SPSS priručnik za preživljavanje: postupni vodič kroz analizu podataka pomoću SPSS-a, Prevod 6. izdanja. Beograd: Mikro knjiga.
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.), Proc. 26th Conf. of the Int. Group for the Psychology of Mathematics Education, Norwich (UK),. (Vol.4, pp. 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, (pp. 161–166). Norway: Bergen University College.
Radford, L. (2011). Grade 2 students’ non-symbolic algebraic thinking. In: J. Cai & E. Knuth (Eds.), Early algebraization. (pp. 303–22). Berlin: Springer.
Radford, L. (2018). The emergence of symbolic algebraic thinking in primary school. In: C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds: The global evolution of an emerging field of research and practice (pp. 3–25). New York: Springer. DOI: 10.1007/978-3-319-68351-5_1
Radford, L. (2022). Introducing equations in early algebra. ZDM–Mathematics Education, 54(6), 1151–1167. https://doi.org/10.1007/s11858-022-01422-x
Russell, S. J., Schifter, D. & Bastable, V. (2011). Developing algebraic thinking in the context of arithmetic. In: J. Cai & E. J. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives, (pp. 43–69). Berlin: Springer. DOI: 10.1007/978-3-642-17735-4_4
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. https://doi.org/10.7146/nomad.v21i1.148715
Specht, B. J. (2005). Early Algebra – Processes and Concepts of Fourth Grades Solving Algebraic Problems. In: M. Bosch, M. Perpiñán, M. Àngels Portabella, and R. Llull, (Eds.). Proceedings of the 4th Congress of the European Society for Research in Mathematics Education (pp. 706‒716). Sant Feliu de Guíxols, Spain.
Sun, S., Sun, D., & Xu, T. (2023). The Developmental Progression of Early Algebraic Thinking of Elementary School Students. Journal of Intelligence, 11(12), 222. https://doi.org/10.3390/jintelligence11120222
Sfard, A. & Linchevski, L. (1994). Between artithmetic and algebra: In the search of a missing link, the case of equations and inequalities. Rendiconti Del Seminario Matematico, 52(3), 279–307. DOI: 10.1007/BF02574811
Stacey, K., & MacGregor, M. (1999). Learning the algebraic method of solving problems. The Journal of Mathematical Behavior, 18(2), 149‒167. https://doi.org/10.1016/S0732-3123(99)00026-7
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. https://doi.org/10.5951/teacchilmath.22.2.0092
Ünal, Z. E., Ala, A. M., Kartal, G., Özel, S., & Geary, D. C. (2023). Visual and symbolic representations as components of algebraic reasoning. Journal of Numerical Cognition, 9(2), 327–345.
Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. Coxford (Ed.), The ideas of algebra, K-12 (8–19). Reston: National Council of Teachers of Mathematics.
Van Amerom, B. A. (2002). Reinvention of early algebra: Developmental research on the transition from arithmetic to algebra (Doctoral dissertation). Utrecht University.
Van Reeuwijk, M. (2001). From informal to formal, progressive formalization: An example on “solving systems of equations.” In: H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra: Proceedings of the 12th ICMI Study Conference, 2, (pp. 613–620). University of Melbourne, Australia.
Weinberg, A., Dresen, J., & Slater, T. (2016). Students’ understanding of algebraic notation: A semiotic systems perspective. The Journal of Mathematical Behavior, 43, 70‒88. https://doi.org/10.1016/j.jmathb.2016.06.001
Zeljić, M. (2014). Metodički aspekti rane algebre. Učiteljski fakultet: Univerzitet u Beogradu.