LETTER - VARIABLES: HOW DO STUDENTS UNDERSTAND THEM AND HOW DO THEY APPEAR IN TEXTBOOKS

PDF (ελληνικά)
Published: Oct 25, 2018
DOI: https://doi.org/10.12681/enedim.18938
Keywords:
variables natural number bias textbooks algebra
Στέλλα Δημητρακοπούλου (Styliani Dimitrakopoulou)
Secondary Education
Κωνσταντίνος Χρήστου (Konstantinos Christou)
University of Western Macedonia
Abstract
This paper reports results from two empirical studies. The first study focuses on how students interpret literal symbols that stand for variables in school mathematics and more specifically students’ tendency to think that variables stand mostly for natural numbers. The results showed that students tended to interpret literal symbols as generalized numbers, that is more than one specific number, but these numbers are natural numbers in priority. The second study examined how middle school textbooks of mathematics in the Greek public junior high school present the literal symbols as variables and whether the numerical values assigned to them are mostly natural numbers. The use of variables as generalized number dominated the uses of variables and natural and non-natural numbers appeared with the same frequency as values of the variables. The results support the findings from other related studies in the literature. Educational implications are discussed.
Article Details
  • Section
  • Articles
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution licence that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g. post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.

Authors are permitted and encouraged to post their work online (preferably in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).

 

 
Downloads
Download data is not yet available.
Author Biographies
Στέλλα Δημητρακοπούλου (Styliani Dimitrakopoulou), Secondary Education
Teacher
Κωνσταντίνος Χρήστου (Konstantinos Christou), University of Western Macedonia
Assistant Professro
References
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.
Booth, L. R. (1984). Algebra: children's strategies and errors. Windsor: Berkshire: NFER-Nelson.
Carraher, D., Schliemann, A., & Brizuela, B. (2001). Can young students operate on unknowns? In M. v. d. Heuvel-Panhuizen (Ed.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (p. 130-140). Utrecht University, The Netherlands.
Christou, K. P. (2015). Natural number bias in operations with missing numbers." ZDM Mathematics Education, 47(5), 1-12.
Christou, K. P. (2017). Students’ interpretation of variables and the phenomenal sign of algebraic expressions, MENON: Journal of Educational Research, 4, 161-175.
Christou, K. P., & Vosviadou St. (2005). How Students Interpret Literal Symbols in Algebra: A Conceptual Change. In B. G. Bara, L. Barsalou, & M. Bucciarelli (Eds.). Proceedings of the XXVII Annual Conference of the Cognitive Science Society, Italy. pp.453-458.
Christou, K. P., Vosviadou St., & Vamvakoussi X. (2007). Students’ Interpretations of Literal Symbols in Algebra, Re-Framing the Conceptual Change Approach in Learning and Instruction (pp. 283-297). Elsevier Press.
Christou, K. P., & Vosniadou, St. (2012). What kinds of numbers do students assign to literal symbols? Aspects of the transition from arithmetic to algebra. Mathematical Thinking and Learning, 14(1), 1-27.
Dimitrakopoulou, St., & Christou, K. P. (2014). How students interpret literal symbols and how they appear in the Greek high school mathematics textbooks. In Mathematics in School and Everyday Life - Proceedings of the 5th National Conference of Greek Association of Researchers in Mathematics Education (ENEDIM) (pp. 1-10). Florina, Greece, EN.E.DI.M., ISSN: 1792-8494. (in Greek).
Dogbey, J. (2016). Using Variables in School Mathematics: Do School Mathematics Curricula Provide Support for Teachers; International Journal of Science and Mathematics Education. 14 (6), 1175-1196, Springer International Publishing AG.
Dogbey J., & Kersaint, Gl., (2012). Treatment of Variables in Popular Middle-Grades Mathematics Textbooks in the USA: Trends from 1957 through 2009, International Journal for Mathematics T.eaching and Learning, 2(1)1-30.
Eisner, E. W. (1987). Why the textbook influences curriculum? Curriculum Review, 26(3), 11-13.
Farmaki, V., Klaoudatos, N., & Verikios, P. (2015). Introduction of Algebraic Thinking: Connecting the Concepts of Linear Function and Linear Equation. https://www.researchgate.net/publication /238724513
Gelman, R. (2000). The epigenesis of mathematical thinking. Journal of Applied Developmental Psychology, 21, 27-37.
Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics (27), 59-78.
Johansson, M. (2005). Mathematics textbooks – the link between the intended and the implemented curriculum? Lulea: Department of Mathematics, Lulea University of Technology.
Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (p. 390–419). New York, NY: Macmillan.
Kieran, C. (2007). Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester Jr.(Εds), Second Handbook of Research on Mathematics Teaching and Learning (pp.707-762). Reston, VA: National Council of Teachers of Mathematics.
Kieran, C., & Chalouh, L. (1993). Prealgebra: The transition from arithmetic to algebra. In D.T. Owens (Ed), Research ideas for the classroom. Middle grades mathematics (p. 179-198). National Council of Teachers of Mathematics.
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. Zentralblatt fur Didaktik der Mathematik—International Reviews on Mathematical Education, 37, 1–9.
Krippendorff, K. (2004). Content Analysis, An introduction to its methodology, 2nd Edition, Thousand Oaks, CA: Sage Publications.
Küchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in School, 7(4), 23-26.
Küchemann, D. (1981). Algebra. In K. M. Hart (Ed.), Children’s understanding of mathematics: 11-16 (pp.102-119). London: John Murray.
MacGregor, M., & Stacey, K. (1997). Students' understanding of algebraic notation:11-15. Educational studies in mathematics, 33(1), 1-19.
McKnight, C. C., Crosswhite, F. J., Dossey, J. A., Kifer, E., Swafford, J. O., Travers, K. J., & Cooney, T. J. (1987). The underachieving curriculum: Assessing U.S. school mathematics from an international perspective. Champaign, IL: Stipes.
McNeil, N., Grandau, L, Knuth, E., Alibali, M., Stephens, A., Hattikudur, S., & Krill, D. (2006). Middle-school students’ understanding of the equal sign: The books they read can’t help. Cognition & Instruction, 24(3), 367-385.
McNeil, N., & Weinberg, A., (2010). A is for Apple: Mnemonic Symbols Hinder the Interpretation of Algebraic Expressions, Journal of Educational Psychology, Vol 102, No3, 625-634.
Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52.
Nicholls J., (2003). Methods in School Textbook Research, University of Oxford. [online] [cit. 18.9.2007] http://serc.carleton.edu/textbook/resources.html.
Pingel, F. (2010). UNESCO Guidebook on Textbook Research and Textbook Revision. United Nations Educational, Scientific and Cultural Organization and Georg Eckert Institute for International Textbook Research. 2nd revised and updated edition. Paris/Braunschweig.
Peacock, A., & Cleghorn, A. (2004). Missing the meaning. The Development and Use of Print and Non-Print Text Materials in Diverse School Settings. New York: Palgrave Macmillan.
Remillard, J. T. (1999). Curriculum materials in mathematics education reform: A framework for examining teachers' curriculum development. Curriculum Inquiry, 29(3), 315-342.
Rosnick, P. (1981). Some misconceptions concerning the concept of variable. Are you careful about defining your variables? Mathematics Teacher, 74(6), 418-420,450.
Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1997). A splintered vision: An investigation of U.S. science and mathematics education. Dordrecht, Netherlands: Kluwer Academic Press.
Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H., Wiley, D. E., & Cogan, L. S., (2001). Why schools matter: a cross-national comparison of curriculum and learning. San Francisco: Jossey-Bass.
Schmittau, J. (2005). The development of algebraic thinking. ZDM, 37(1),16–22.
Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice in the world of textbooks. Dordrecht, The Netherlands.
Vamvakoussi, X., & Vosniadou, St. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ understanding of rational numbers and their notation. Cognition and Instruction, 28(2), 181–209.
Van Dooren W., Christou Κ. P., & Vamvakoussi X. (2010). Greek and Flemish Students’ Interpretation of Literal Symbols as Variables, 2010. In M.M.F. Pinto & T.F. Kawasaki (Eds.). In Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 257-264. Belo Horizonte, Brazil: PME.
Vosniadou, St., Vamvakoussi, X., & Skopeliti, I. (2008). The Framework Theory Approach to the Problem of Conceptual Change. In S.
Vosniadou (Ed.) International Handbook of Research on Conceptual Change. New York, NY: Routledge, 3-34.
ΔΕΠΠΣ, (2003). Διαθεματικό Ενιαίο Πλαίσιο Προγράμματος Σπουδών, ΦΕΚ 303Β/13-03-2003; ΦΕΚ 304Β/13-03-2003.
Ματσαγγούρας, Η. (2006). Διδακτικά εγχειρίδια: Κριτική αξιολόγηση της Γνωσιακής, Διδακτικής και Μαθησιακής Λειτουργίας τους. Συγκριτική και Διεθνής Εκπαιδευτική Επιθεώρηση, 7, 60-92.
Most read articles by the same author(s)