Modelling the Relationships Between Quantities: Meaning in Literal Expressions

Marijana Zeljić

doi:10.12973/eurasia.2015.1362a

Full Text (PDF)

Marijana Zeljić ¹ ^*

More Detail

¹ Teacher Education Faculty^* Corresponding Author

Abstract

Background:
Algebra is often considered to be difficult and mysterious doctrine due to numerous symbols that represent mathematical notions. Results of the research on students’ interpretation of literal expressions show that only a small number of students are ready to accept that a letter can represent a variable. The aim of this research conducted with students of the fourth grade of elementary school was to examine the impact of the use of various symbols, which represent the relationships between quantities, on students’ understanding of the meanings of literal expressions.

Materials and methods:
We used an experimental method (with one group) and testing techniques (pre-test and final-test comparisons).

Results:
The research results show that the iconic representation of the structure of literal expressions, namely, giving meaning to a term through its reliance on certain schemes in the learning process, significantly affects the ability of students in modelling generic quantitative relationships, and in the development of the meaning of letters in the role of the algebraic variable.

Conclusions:
However, the results of our study show that it is possible to develop procedural as well as structural understanding of literal expressions in younger elementary school students.

Keywords

literal expression
generalization
modeling
visualization
scheme

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, Volume 11, Issue 2, 2015, 431-442

https://doi.org/10.12973/eurasia.2015.1362a

Publication date: 02 Apr 2015

Article Views: 1990

Article Downloads: 756

Open Access References How to cite this article

References

Abdullah, N., Halim, L., & Zakaria, E. (2014). VStops: A Thinking Strategy and Visual Representation Approach in Mathematical Word Problem Solving toward Enhancing STEM Literacy. Eurasia Journal of Mathematics, Science & Technology Education, 10(3), 165– 174.
Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183–198.
Andrade, A. (2011). The clock project: gears as visual-tangible representations for mathematical concepts. International Journal of Technology and Design Education, 21(1), 93–110.
Anderson, K. L., Casey, M. B., Thompson, W. L., Burrage, M. S., Perazis, E., & Kosslyn, S. M. (2008). Performance on middle school geometry problems with geometry clues matched to three different cognitive styles. Mind, Brain, and Education, 2(4), 188–197.
Asquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle school mathematics teachers‟ knowledge of students‟ understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning, 9(3), 249–272.
Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412– 446.
Brizuela, B., & Schliemann, A. (2004). Ten-year-old students solving linear equations. For the Learning of Mathematics, 24(2), 33–40.
Cai, J. (2004). Why do U.S. and Chinese students think differently in mathematical problem solving? Exploring the impact of early algebra learning and teachers‟ beliefs. Journal of Mathematical Behavior, 23(2), 135–167.
Carraher, D. W., Schliemann, A. D., & Brizuela, B. M. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87– 115.
Cooper, T.J., & Warren E. (2011). Years 2 to 6 Students‟ Ability to Generalise: Models, Representations and Theory for Teaching and Learning. In J. Cai, & E. Knuth (Eds.), Early Algebraization. A Global Dialogue from Multiple Perspectives (pp. 187–215). Berlin: Springer.
Crowley, L., Thomas, T., & Tall, D. (1994). Algebra, Simbols, and Translation of Meaning. In J. P. Ponte, & J. F. Matos (Eds.), Procedings of PME 18 (pp. 240–247). University of Lisbon, Portugal.
diSessa, A.A. (2002). Students‟ criteria for representational adequacy. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modelling and tool use in mathematics education (pp. 105–130). Dordrecht, The Netherlands: Kluwer Academic Publishers.
Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 25–41). Dordrecht: Kluwer Academic Publishers.
Drouhard, J. P., &. Teppo, A. R. (2004). Symbols and Language. In K. Stacey, H.Chick, & M. Kendal (Eds.), The Future of the Teaching and Learning of Algebra - The 12th ICMI Study (pp. 225–265). Dordrecht/Boston/London: Kluwer Academic Publishers.
Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt, & M. Santos (Eds.), Proceedings of the 21st North American PME Conference (pp. 3-26). Cuernavaca, Morelos, Mexico.
Freudenthal, H. (1962). Logical Analysis and Critical Study. In H. Freudenthal (Ed.), Report on the Relations between Arithmetic and Algebra ( pp. 20–41). Groningen, Netherlands: J.B. Wolters.
Fujii, T., & Stephens, M. (2001). Fostering an understanding of algebraic generalization through numerical expressions: The role of quasi-variables. In H. Chick, K. Stacey, & J. Vincent, (Eds.), Proceedings of the 12th study conference of the international commission on mathematical instruction: the future of the teaching and learning of algebra, Vol. 1 ( pp. 258–264). Melbourne, Australia: The University of Melbourne.
Gerofsky, S. (2009). Genre, simulacra, impossible exchange, and the real: How postmodern theory problematizes word problems. In L. Verschaffel, B. Greer, & W. V. Dooren (Eds.), Words and worlds: Modeling verbal descriptions of situations (pp. 21–38). Rotterdam: Sense Publishing.
Goldstone, R. L., & Son, J. Y. (2005). The transfer of scientific principles using concrete and idealized simulations. The Journal of the Learning Sciences, 14(1), 69– 110.
Hewitt, D. ( 2001). On learning to adopt formal algebraic notation. In H. Chick, K. Stacey, & J. Vincent (Eds.), Proceedings of the 12th study conference of the international commission on mathematical instruction: the future of the teaching and learning of algebra, Vol. 1 (pp. 305–312). Melbourne, Australia: The University of Melbourne.
Koedinger, K. R., Alibali, M. W., & Nathan, M. J. (2008). Trade-offs between grounded and abstract representations: Evidence from algebra problem solving. Cognitive Science, 32(2), 366–397.
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.
Kaput, J. J. (2000). Teaching and learning a new algebra with understanding. Dartmouth, MA: National Center for Improving Student Learning and Achievement in Mathematics and Science.
Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 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. (Edc.), ICME 8: Selected lectures (pp. 271–290). Seville, Spain, S. A. E. M. „Thales‟.
Kieran, C. (2004). Algebraic thinking in the early grades: What is it?. The Mathematics Educator, 8(1), 139–151.
Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle school students‟ understanding of core algebraic concepts: Equality and variable. International Reviews on Mathematical Education, 37(1), 1–9.
Kuchemann, D. (1981). Algebra, In K. M. Hart (Ed.), Children´s understanding mathematics: 11–16 (pp. 102–119). London: John Murray.
Lins, R.C. (2001). The Production of Meaning for Algebra: a Perspective Based on a Theoretical Model of Semantic Fields. In R. Sutherland, T. Rojano, A. Bell & R. Lins (Eds.), Perspectives on School Algebra (pp. 37– 60). Dordrecht/Boston/London: Kluwer Academic Publishers.
Mason, J., Drury, H., & Bills, E.. (2007). Explorations in the Zone of Proximal Awareness. In J. Watson, & K. Beswick (Eds.), Mathematics: Essential Research, Essential Practice, Vol. 1 (pp. 42–58). Adelaide: MERGA.
McNeil, N. M., & Weinberg, A. (2010). A Is for Apple: Mnemonic Symbols Hinder the Interpretation of Algebraic Expressions. Journal of Educational Psychology, 102 (3), 625–634.
Radford, L. (2002). On heroes and the collapse of narratives: a contribution to the study of symbolic thinking. In A. D. Cockburn, & E. Nardi (Eds.): Proceedings of the 16th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 ( pp. 81-88). Norwich, UK.
Radford, L. (2003). Gestures, speech and the sprouting of signs. Mathematical Thinking and Learning, 5(1), 37-70.
Radford, L. (2011). Students‟ Non-Symbolic Algebraic Thinking. In J. Cai, & E. Knuth (Eds.), Early Algebraization. A Global Dialogue from Multiple Perspectives (pp. 303–322). Berlin: Springer.
Radford, L., Bardini, C. & Sabena, C. (2005). Perceptual semiosis and the microgenesis of algebraic generalizations. In M. Bosch (Ed.), The Fourth Congress of the European Society in Mathematics Education (pp. 684– 696). Sant Feliu de Guixols, Spain.
Rivera, F. D. (2010). Visual templates in pattern generalization activity. Educational Studies in Mathematics, 73(3), 297–328.
Sfard, A., & Linchevski, L. (1994). The Gains and Pitfalls of Reification – The Case of Algebra. Educational Studies in Mathematics, 26(2/3), 15–39.
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. (2000). Symbolizing mathematical reality into being - or how mathematical discourse and mathematical objects create each other. In P. Cobb‚ E. Yackel‚ & K. McClain (Eds.)‚ Symbolizing and communicating in mathematics classrooms: Perspectives on discourse‚ tools‚ and instructional design (pp. 37-98). Mahwah‚ NJ: Lawrence Erlbaum.
Smith, E. (2006). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 133-160). Mahwah NJ: Lawrence Erlbaum.
Stacey, K., & Macgregor, M. (1997). Building foundations for algebra. Mathematics Teaching in the Middle School, 2(4), 252–260.
Tall, D. (2008): The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20 (2), 5–24.
Terwel, J., van Oers, B., van Dijk, I., & van den Eeden, P. (2009). Are representations to be provided or generated in primary mathematics education? Effects on transfer. Educational Research and Evaluation. 15(1), 25–44.
Tolar, T. D., Lederberg, A. R., & Fletcher, J. M. (2009). A structural model of algebra achievement: Computational fluency and spatial visualization as mediators of the effect of working memory on algebra achievement. Educational Psychology, 29, 39–266. doi: 10.1080/01443410802708903
van Oers, B. (2001). Contextualisation for abstraction. Cognitive Science Quarterly, 1(3/4), 279–306.

How to cite this article

APA

Zeljić, M. (2015). Modelling the Relationships Between Quantities: Meaning in Literal Expressions. Eurasia Journal of Mathematics, Science and Technology Education, 11(2), 431-442. https://doi.org/10.12973/eurasia.2015.1362a

Vancouver

Zeljić M. Modelling the Relationships Between Quantities: Meaning in Literal Expressions. EURASIA J Math Sci Tech Ed. 2015;11(2):431-42. https://doi.org/10.12973/eurasia.2015.1362a

AMA

Zeljić M. Modelling the Relationships Between Quantities: Meaning in Literal Expressions. EURASIA J Math Sci Tech Ed. 2015;11(2), 431-442. https://doi.org/10.12973/eurasia.2015.1362a

Chicago

Zeljić, Marijana. "Modelling the Relationships Between Quantities: Meaning in Literal Expressions". Eurasia Journal of Mathematics, Science and Technology Education 2015 11 no. 2 (2015): 431-442. https://doi.org/10.12973/eurasia.2015.1362a

Harvard

Zeljić, M. (2015). Modelling the Relationships Between Quantities: Meaning in Literal Expressions. Eurasia Journal of Mathematics, Science and Technology Education, 11(2), pp. 431-442. https://doi.org/10.12973/eurasia.2015.1362a

MLA

Zeljić, Marijana "Modelling the Relationships Between Quantities: Meaning in Literal Expressions". Eurasia Journal of Mathematics, Science and Technology Education, vol. 11, no. 2, 2015, pp. 431-442. https://doi.org/10.12973/eurasia.2015.1362a