Моделовање еквиваленције математичких израза у почетној настави

Милана М. Дабић Боричић, Универзитет у Београду, Учитељски факултет, Република Србија, имејл: marijana.zeljic@uf.bg.ac.rs
Маријана Ж. Зељић, Универзитет у Београду, Учитељски факултет, Република Србија
Иновације у настави, XXXIV, 2021/1, стр. 30–43

| PDF | | Extended summary PDF |
doi: 10.5937/inovacije2101030D

 

Резиме: Један од појмова који је у литератури препознат као кључан за разумевање алгебарских идеја је појам еквивалентности израза. За разумевање наведеног појма важан је контекст који се користи као основа за развијање значења, као и језик којим се исказују генерализације. Циљ рада је двојак: а) испитати да ли контекст текстуалног задатка и активности моделовања утичу на разумевање трансформације израза у еквивалентне форме; б) утврдити да ли на разумевање еквивалентности израза утиче ниво апстрактности израза (алгебарски или аритметички) који се користe. Истраживање је квазиексперименталног дизајна са две експерименталне и контролном групом. Узорак чини 148 ученика четвртог разреда. Постојање статистички значајних разлика између ученика експерименталних и контролне групе упућује да процес моделовања утиче на развијање појма еквивалентности израза. У овом истраживању нису се показале разлике у резултатима ученика који су били подучавани помоћу алгебарских, односно аритметичких израза. Ово имплицира да разумевање еквивалентности које је развијано кроз процес моделовања није у вези са нивоом апстрактности математичког језика који се користи, већ да на основу разумевања значења појма ученици са подједнаком успешношћу могу трансформисати и аритметичке и алгебарске изразе.

Кључне речи: еквиваленција математичких израза, моделовање, математички симболизам, алгебра.

 

Summary: The notion of expression equivalence is one of the terms that has been recognized in the literature as key to understanding algebraic ideas. To understand this term, the context used as a basis for developing meaning is important, as well as the language in which generalizations are expressed. The aim of this paper is twofold: a) to examine whether the context of a textual task and modeling activities influence the understanding of the transformation of expressions into equivalent forms; b) determine whether the understanding of the equivalence of the expression is affected by the level of abstractness of the expression (algebraic or arithmetic). The research is of a quasi-experimental design with two experimental groups and one control group. The sample consists of 148 fourth-graders. The existence of statistically significant differences between the students of the experimental groups and the control group suggests that the modeling process influences the development of the notion of expression equivalence. This research did not show any differences in the results of the students who were taught using algebraic or arithmetic expressions. This implies that the understanding of equivalence developed through the modeling process is not related to the level of abstractness of the mathematical language used, but that, based on understanding the meaning of the term, students can transform arithmetic and algebraic expressions with equal success.

Keywords: equivalence of mathematical expressions, modeling, mathematical symbolism, algebra.

Литература

  • Banerjee, R., Subramaniam, K. & Naik, S. (2008). Bridging Arithmetic and Algebra: Evolution of a Teaching Sequence. In: Fogueras, O., Cortina, J. L., Alatorre, S., Rojano T. & Sepulveda, A. (Eds.). Proceedings of the Joint Meeting of PME 32, Vol. 2 (121–128). Morelia, México: Cinvestav-UMSNH.
  • Blanton, M. & Kaput, J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36 (5), 412–446.
  • Blum, W. (1994). Mathematical modeling in mathematics education and instruction. In: Breiteig, T., Huntley, I. & Kaiser-Messmer, G. (Eds.). Teaching and learning mathematics in context (3–14). Chichester, England: Ellis Horwood Limited.
  • Blum, W. & Leiss, D. (2007). How do students and teachers deal with modelling problems. In: Haines, C., Galbraith, P., Blum, W. & Khan, S. (Eds.). Mathematical modeling: Education, engineering, and economics (222–231). Chichester: Horwood.
  • Booth, L. (1988). Children’s difficulties in beginning algebra. In: Coxford, A. F. (Ed.). The Ideas of Algebra. K-12 (20–32). Reston, VA: National Council of Teachers of Mathematics.
  • Cai, J. (2014). Searching for evidence of curricular effect on the teaching and learning of mathematics: Some insights from the LieCal project. Mathematics Education Research Journal, 26, 811–831.
  • Cerulli, M. & Mariotti, M. A. (2001). Arithmetic & Algebra, Continuity or Cognitive Break? The Case of Francesca. In: Van-den Heuvel Pannhueizen, M. (Ed.). Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (225–232). Utrecht, Netherlands: PME.
  • Chaiklin, S. & Lesgold, S. B. (1984). Prealgebra Students’ Knowledge of Algebraic Tasks with Arithmetic Expressions. Retrieved May 20, 2020. from www: https://apps.dtic.mil/dtic/tr/fulltext/u2/a144672.pdf
  • Chazan, D. (2000). Beyond formulas in mathematics and teaching: Dynamics of the high school algebra classroom. New York, NY: Teachers College Press.•• Crowley, L., Thomas, T. & Tall, D. (1994). Algebra, Symbols, and Translation of Meaning. In: Ponte, J. P. & Matos, J. F. (Eds.). Proceedings of PME 18 (240–247). University of Lisbon, Portugal.
  • Ding, M. & Li, X. (2014). Transition from concrete to abstract representation: the distributive property in a Chinese textbook series. Educational Studies in Mathematics, 87, 103–121.
  • Dreyfus, T. & Eisenberg, T. (1982). Intuitive functional concepts: A baseline study on intuitions. Journal for Research in Mathematics Education, 13 (5), 360–380.
  • Duval, R. (1999). Representation, vision & visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In: Hitt, F. & Santos, M. (Eds.), Proceedings of the 21st North American PME Conference (3–26). Cuernavaca, Morelos, Mexico.
  • Fagnant, A. & Vlassis, J. (2013). Schematic representations in arithmetical problem solving: Analysis of their impact on grade 4 students. Educational Studies in Mathematics, 84, 149–168.
  • Fuji, T. & Stephens, M. (2001). Fostering an understanding of algebraic generalization trough numerical expressions: the role of quasi-variables. In: Chick, H., Stacey, K. & Vincent, J. (Eds.). Proceedings of the 12th ICMI Study Conference: The Future of the Teaching & Learning of Algebra, Vol. 1 (258–264). Melbourne: The University of Melbourne.
  • Gerofsky, S. (2009). Genre, simulacra, impossible exchange, & the real: How postmodern theory problematizes word problems. In: Verschaffel, L., Greer, B. & Dooren, W. V. (Eds.). Words and worlds: Modeling verbal descriptions of situations (21–38). Rotterdam: Sense Publishing.
  • Herscovics, N. & Linchevski, L. (1994). A Cognitive Gap between Arithmetic and Algebra. Educational Studies in Mathematics, 27, 59–78.
  • Ilić, S., Zeljić, M. (2017). Pravila stalnosti zbira i razlike kao osnova strategija računanja. Inovacije u nastavi, 30 (1), 55–66.
  • Kabaca, T. (2013). Using Dynamic Mathematics Software to Teach One-Variable Inequalities by the View of Semiotic Registers. Eurasia Journal of Mathematics, Science & Technology Education, 9 (1), 73–81.
  • Kieran, C. (1992). The learning and teaching of school algebra. In: Grouws, D. A. (Ed.). Handbook of research on mathematics teaching & learning (390–419). New York: Macmillan.
  • Kieran, C. (1996). The changing face of school algebra. In: Alsina, C., Alvares, J., Hodgson, B., Laborde, C. & Pérez, A. (Eds.). ICME 8: Selected lectures (271–290). Seville, Spain: S. A. E. M. „Thales“.
  • Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathematics Educator (Singapore), 8 (1), 139–151.
  • Kieran, C., Boileau, A., Tanguay, D. & Drijvers, P. (2013). Design researchers’ documentational genesis in a study on equivalence of algebraic expressions. ZDM Mathematics Education, 45, 1045–1056.
  • Lee, L. & Wheeler, D. (1989). The arithmetic connection. Educational Studies Mathematics, 20, 41–54.
  • Liebenberg, R. E., Linchevski, L., Sasman, M. C. & Olivier, A. (1999). Focusing on the structural aspects of numerical expressions. In: Kuiper, J. (Ed.). Proceedings of the Seventh Annual Conference of the Southern African Association for Research in Mathematics & Science Education (249–256). Harare, Zimbabwe.
  • Linchevski, L. & Livneh, D. (1999). Structure sense: The Relationship between Algebraic and Numerical Contexts. Educational Studies in Mathematics, 40, 173–196.
  • Lins, R. & Kaput, J. (2001). The Early Development of Algebraic Reasoning: The Current State of the Field. In: Chick, H., Stacey, K., Vincent, J. & Vincent, J. (Eds.). The Future of the Teaching & Learning of Algebra, Proceedings of the 12th ICMI Study Conference (47–70). Melbourne, Australia: The University of Melbourne.
  • Livneh, D. & Linchevski, L. (2007). Algebrification of Arithmetic: Developing Algebraic Structure Sense in the Context of Arithmetic. In: Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 3 (217–224). Seoul: PME.
  • Malara, N. & Navarra, G. (2001). „Brioshi“ & other mediation tools employed ina teaching of arithmetic from a relational point of view with the aim of approaching algebra as a language. In: Chick, H., Stacey, K., Vincent, J. & Vincent, J. (Eds.). The Future of the Teaching & learning of Algebra, Proceedings of the 12th ICMI Study Conference (412–419). Melbourne: University of Melbourne.
  • Malara, N. & Iaderosa, R. (1999). The interweaving of arithmetic and algebra: Some questions about syntactic and structural aspects and their teaching and learning. In: Schwank, I. (Ed.). Proceedings of the First Conference of the European Society for Research in Mathematics Education, Vol. 2 (159–171). Osnabrueck: Forschungsinstitut fuer Mathematikdidaktik.
  • Ni, Y., Zhou, D. R., Cai, J., Li, X., Li, Q. & Sun, I. X. (2018). Improving cognitive and affective learning outcomes of students through mathematics instructional tasks of high cognitive dem. The Journal of Educational Research, 111 (6), 704–719.
  • Panasuk, R. M. & Beyranev, M. L. (2010). Algebra students’ ability to recognize multiple representations and achievement. International Journal for Mathematics Teaching and Learning, 22, 1–22.
  • Rivera, F. D. (2010). Visual templates in pattern generalization activity. Educational Studies in Mathematics, 73 (3), 297–328.
  • Sfard, A. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, 22 (1), 1–36.
  • Sfard, A. & Linchevski, L. (1994). The Gains and Pitfalls of Reification – The Case of Algebra. Educational Studies in Mathematics, 26 (2/3), 15–39.
  • Stacey, K. & MacGregor, M. (1999). Learning the Algebraic Method of Solving Problems. Journal of Mathematical Behavior, 18 (2), 149–167.
  • Steele, D. & Johanning, D. J. (2004) A Schematic-theoretic View of Problem Solving and Development of Algebraic Thinking. Educational Studies in Mathematics, 57, 65–90.
  • Stylianou, D. A. (2011). An examination of middle school students’ representation practices in mathematical problem solving through the lens of expert work: Toward an organizing scheme. Educational Studies in Mathematics, 76 (3), 265–280.
  • Subramaniam, K. & Banerjee, R. (2004). Teaching arithmetic and algebraic expressions. In: Johnsen Hoines, M. & Berit Fuglestad, A. (Eds.). Proceedings of the 28th International Conference of the International Group for the Psychology of Mathematics Education, Vol. 3 (121–128). Bergen: PME.
  • Subramaniam, K. & Banerjee, R. (2011). The Arighmetic-Algebra Connection: A Historical-Pedagogical Perspective. In: Cai, E. & Knuth, J. (Eds.). Early Algebraization (87–107). Berlin-Heidelberg: Springer.
  • Verschaffel, L., Greer, B. & De Corte, E. (2000). Making sense of word problems. Lisse, The Netherlands: Swets and Zeitlinger.
  • Zeljić, M. (2015). Modelling the Relationships between Quantities: Meaning in Literal Expressions. Eurasia Journal of Mathematics, Science & Technology Education, 11 (2), 431–442.
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