The Strategy the Use of False Assumption and Word Problem Solving

  • Petr Eisenmann Univerzita J. E. Purkyně v Ústí nad Labem, Přírodovědecká fakulta
  • Jiří Přibyl Univerzita J. E. Purkyně v Ústí nad Labem, Přírodovědecká fakulta
  • Jarmila Novotná Univerzita Karlova v Praze, Pedagogická fakulta M.D. Rettigové 4 116 39 Praha 1
Keywords: problem solving, false assumption, word problems, false position method, phylogenesis and ontogenesis theory

Abstract

The paper describes one problem solving strategy – the Use of false assumption. The objective of the paper is to show, in accordance with Phylogenesis and Ontogenesis Theory, that it is worthwhile to reiterate the process of development of the concept of a variable and thus provide to pupils one of the ways helping them to eliminate usual difficulties when solving word problems using linear equations, namely construction of the equations. The paper presents the outcomes of a study conducted on three lower secondary schools in the Czech Republic with 147 14–15-year-old pupils. Pupils from the experimental group were, unlike pupils from the control group, taught the strategy the Use of false assumption before being taught the topic Solving word problems. The tool for the study was a test of four problems that was sat by all the involved pupils three weeks after finishing the topic “Solving word problems” and whose results were evaluated statistically. The experiment confirmed the research hypothesis that the introduction of the strategy the Use of false assumption into 8th grade mathematics lessons (14–15-year-old pupils) helps pupils construct equations more successfully when solving word problems.

Author Biographies

Petr Eisenmann, Univerzita J. E. Purkyně v Ústí nad Labem, Přírodovědecká fakulta

Katedra matematiky

Jiří Přibyl, Univerzita J. E. Purkyně v Ústí nad Labem, Přírodovědecká fakulta

Katedra matematiky

References


  • Artigue, M., Haspekian, M. and Corblin-Lenfant, A. (2014) ‘Introduction to the theory of didactical situations (TDS)’, In A. Bikner-Ahsbahs and S. Prediger (eds.), Networking of theories as a research practice in mathematics education: Authored by the networking theories group, Cham, Heidelberg, New York (NY), Dordrecht, London: Springer.

  • Bednarz, N. and Dufour-Janvier, B. (1994) ‘The emergence and development of algebra in a problem solving context: A problem analysis’, In J.P. da Ponte and J.F. Matos (eds.), Proceedings of the International Conference for the Psychology of Mathematics Education: Volume II, pp. 64–71, Lisbon: IG for PME.

  • Blažek, R. and Příhodová, S. (2016) Mezinárodní šetření PISA 2015: Národní zpráva [PISA 2015 results: Country overview], Prague: Czech School Inspectorate.

  • Boman, E.C. (2009) ‘False position, double false position and Cramer’s rule’, The College Mathematics Journal, Vol. 40, No. 4, pp. 279–283. https://doi.org/10.4169/193113409x458732

  • Boonen, A.J.H., van der Schoot, M., van Wesel, F., de Vries, M.H. and Jolles, J. (2013) ‘What underlies successful word problem solving? A path analysis in sixth grade students’, Contemporary Educational Psychology, Vol. 38, No. 3, pp. 271–279. https://doi.org/10.1016/j.cedpsych.2013.05.001

  • Boonen, A.J.H., de Koning, B.B., Jolles, J. and van der Schoot, M. (2016) ‘Word problem solving in contemporary math education: A plea for reading comprehension skills training’, Frontiers in Psychology, Vol. 7, Article 191, pp. 1–10. https://doi.org/10.3389%2Ffpsyg.2016.00191

  • Brousseau, G. (1997) Theory of didactical situations in mathematics, Dordrecht: Kluwer Academic Publisher.

  • Brown, T. and Heywood, D. (2010) ‘Geometry, subjectivity and the seduction of language: the regulation of spatial perception’, Educational Studies in Mathematics, Vol. 77, No. 2–3, pp. 351–367. https://doi.org/10.1007/s10649-010-9281-2

  • Bruder, R. and Weiskirch, W. (2013) CAliMERO-computer-algebra im mathematikunterricht: Entdecken, rechnen, organisieren: Lineare algebra / analytische geometrie: Schülermaterialien, Münster: Schroedel Verlag.

  • Břehovský, J., Eisenmann, P., Ondrušová, J., Přibyl, J. and Novotná, J. (2013) ‘Heuristic strategies in problem solving of 11–12-year-old pupils’, In J. Novotná and H. Moraová (eds.), Proceedings of SEMT ’13, pp. 75–82, Prague: Charles University in Prague.

  • Bunt, L.N.H., Jones, P.S. and Bedient, J.D. (1978) The historical roots of elementary mathematics, New York (NY): Dover Publications, Inc.

  • Bush, S.B. and Karp, K.S. (2013) ‘Prerequisite algebra skills and associated misconceptions of middle grade students: A review’, The Journal of Mathematical Behavior, Vol. 32, No. 3, pp. 613–632. https://doi.org/10.1016/j.jmathb.2013.07.002

  • Cajori, F. (1890) The teaching and history of mathematics in the United States, Washington: Goverment Printing Office.

  • Chabert, J.-L. (ed.) (1999) A history of algorithms: From the pebble to the microchip, Berlin, Heidelberg: Springer-Verlag.

  • Chace, A.B., Bull, L. and Manning, H.P. (1929) The Rhind mathematical papyrus: British museum 10057 and 10058: Photographic facsimile, hieroglyphic transcription, transliteration, literal translation, free translation, mathematical commentary, and bibliography in two volumes, Oberlin: Mathematical Association of Mathematica.

  • De Corte, E., Greer, B. and Verschaffel, L. (2000) Making sense of word problems, Lisse: Swets & Zeitlinger.

  • Descartes, R. (1954) The Geometry of René Descartes: With a Facsimile of the First Edition, translated from French and Latin by D.E. Smith and M.L. Latham, New York: Dover Publication, Inc.

  • Dubinsky, E., Weller, K., McDonald, M.A. and Brown, A. (2005) ‘Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis, Part 1’, Educational Studies in Mathematics, Vol. 58, No. 3, pp. 335–359. https://doi.org/10.1007/s10649-005-2531-z

  • Eisenmann, P., Novotná, J., Přibyl, J. and Břehovský, J. (2015) ‘The development of a culture of problem solving with secondary students through heuristic strategies’, Mathematics Education Research Journal, Vol. 27, No. 4, pp. 535–562. http://dx.doi.org/10.1007/s13394-015-0150-2

  • Eisenmann, P., Přibyl, J., Novotná, J., Břehovský, J. and Cihlář, J. (2017) ‘Volba řešitelských strategií v závislosti na věku’ [Choice of heuristic strategies with respect to age], Scientia in educatione, Vol. 8, No. 2, pp. 1–19.

  • Filloy, E. and Rojano, T. (1989) ‘Solving equations: the transition from arithmetic to algebra’, For the Learning of Mathematics, Vol. 9, No. 2, pp. 19–25.

  • Fischer, A. (2009) ‘Zwischen bestimmten und unbestimmten Zahlen — Zahl- und Variablenauffassungen von Fünftklaesslern’, Journal für Mathematik-Didaktik, Vol. 30, No. 1, pp. 3–29. https://doi.org/10.1007/bf03339071

  • Furinghetti, F. and Radford, L. (2002) ‘Historical conceptual developments and the teaching of mathematics: from phylogenesis and ontogenesis theory to classroom practice’, In L. English (ed.), Handbook of international research in mathematics education (pp. 631–654), New Jersey: Lawrence Erlbaum

  • Gick, M.L. (1986). ‘Problem-solving strategies’, Educational Psychologist, Vol. 21, No. 1–2, pp. 99–120. https://doi.org/10.1080/00461520.1986.9653026

  • Goodson-Epsy, T. (1998) ‘The role of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra’, Educational Studies in Mathematics, Vol. 36, No. 3, pp. 219–245. https://doi.org/10.1023/a:1003473509628

  • Høyrup, J. (2002) Lenghts, widths, surfaces: A portrait of old Babylonian algebra and its kin, Berlin, New York (NY): Springer.

  • Jacobs, V.R., Franke, L.M., Carpenter, T.P., Levi, L., and Battey, D. (2007) ‘Professional development focused on children’s algebraic reasoning in elementary school’, Journal for Research in Mathematics Education, Vol. 38, No. 3, pp. 258–288. https://doi.org/10.2307/30034868

  • Jeřábek, J., Lisnerová, R., Smejkalová, A. and Tupý, J. (eds.) (2013) Rámcový vzdělávací program pro základní vzdělávání [The framework educational programme for basic (Primary and Lower Secondary) education], Prague: The Ministry of Education, Youth and Sports.

  • Jupri, A. and Drijvers, P.H.M. (2016) ‘Student difficulties in mathematizing word problems in algebra’, EURASIA Journal of Mathematics, Science & Technology Education, Vol. 12, No. 10, pp. 2481–2502. https://doi.org/10.12973/eurasia.2016.1299a

  • Katz, V., Dorier, J.L., Bekken, O. and Sierpinska, A. (2000) ‘The role of historical analysis in predicting and interpreting students’ difficulties in mathematics’, In J. Fauvel and J.V. Maanen (eds.), History in mathematics education (pp. 149–161), Dordrecht: Kluwer Academic Publisher.

  • Lewis, A.B. (1989) ‘Training students to represent arithmetic word problems’, Journal of Educational Psychology, Vol. 81, No. 4, pp. 521–531. https://doi.org/10.1037/0022-0663.81.4.521

  • Lewis, A.B. and Mayer, R.E. (1987) ‘Students’ miscomprehension of relational statements in arithmetic word problems’, Journal of Educational Psychology, Vol. 79, No. 4, pp. 363–371. http://doi.org/10.1037//0022-0663.79.4.363

  • Linchevski, L. and Herscovics, N. (1996) ‘Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations’, Educational Studies in Mathematics, Vol. 30, No. 1, pp. 39–65. https://doi.org/10.1007/bf00163752

  • Lumpkin, B. (1996) ‘From Egypt to Benjamin Banneker: Africans origins of false position solutions’, In R. Calinger (ed.), Vita mathematica: Historical research and integration with teaching (pp. 279–303), Washington, DC: The mathematical association of America.

  • Morin, L.L., Silvana M.R.W., Hester, P. and Raver, S. (2017) ‘The use of a bar model drawing to teach word problem solving to students with mathematics difficulties’, Learning Disability Quarterly, Vol. 40, No. 2, pp. 91–104. https://doi.org/10.1177/0731948717690116

  • National Council of Teachers of Mathematics (2000) Principles and standards for school mathematics, Reston, VA: The National Council of Teachers of Mathematics.

  • Novotná, J. (2000a) Analýza řešení slovních úloh [Analysis of word problems solution], Prague: Charles University in Prague.

  • Novotná, J. (2000b) ‘Students’ levels of understanding of word problems’, Regular lecture, ICME 9, Tokyo/Makuhari, Japan. In H. Fujita (ed.), ICME 9: Abstracts of plenary lectures and regular lectures, pp. 96–97, Tokyo/Makuhari, Japan.

  • Novotná, J., Eisenmann, P. and Přibyl, J. (2014) ‘Impact of heuristic strategies on pupils’ attitudes to problem solving’, In M. Houška, I. Krejčí and M. Flégl, (eds.), Proceedings of Efficiency and Responsibility in Education 2014, pp. 514–520, Prague: Czech University of Life Sciences.

  • Novotná, J., Eisenmann, P., Přibyl, J., Ondrušová, J. and Břehovský, J. (2014) ‘Problem solving in school mathematics based on heuristic strategies’, Journal on Efficiency and Responsibility in Education and Science, Vol. 7, No. 1, pp. 1–6. https://doi.org/10.7160/eriesj.2014.070101

  • Ofir, R. and Arcavi, A. (1992) ‘Word problems and equations: An historical activity for the algebra classroom’, The Mathematical Gazette, Vol. 76, No. 475, pp. 69–84. https://doi.org/10.2307/3620379

  • Palečková, J., Tomášek, V. and Blažek, R. (2014) Mezinárodní šetření PISA 2012: Národní zpráva [PISA 2012 results: Country overview], Prague: Czech School Inspectorate.

  • Pólya, G. (2004) How to solve it: A new aspect of mathematical method, Princeton: Princeton University Press.

  • Přibyl, J., Eisenmann, P. and Gunčaga, J. (2018) ‘The phenomenon of false assumption in historical and educational texts’, Science & Education, Vol. 27, No. 7–8, pp. 737–767. https://doi.org/10.1007/s11191-018-0005-9

  • Smith, D.E. (1929) A source book in mathematics: I & II volume, New York, NY: McGraw Hill Book Company.

  • Stacey, K. and MacGregor, M. (2000) ‘Learning the algebraic method of solving problems’, Journal of Mathematical Behavior, Vol. 18, No. 2, pp. 149–167. https://doi.org/10.1016/s0732-3123(99)00026-7

  • StatSoft, Inc. (2013). Statistica (data analysis software system), version 12. http://statistica.io

  • Verschaffel, L., De Corte, E. and Pauwels A. (1992) ‘Solving compare problems: an eye movement test of Lewis and Mayer’s consistency hypothesis’, Journal of Educational Psychology, Vol. 84, No. 1, pp. 85–94. http://doi.org/10.1037/0022-0663.84.1.85

  • Winicki, G. (2000) ‘The analysis of regula falsi as an instance for professional development of elementary school teachers’, In V. Katz (ed.), Using history to teach mathematics: An international perspective (pp. 129–133), Washington, DC: Mathematical Association of America.

Published
2019-07-01
How to Cite
Eisenmann, P., Přibyl, J. and Novotná, J. (2019) ’The Strategy the Use of False Assumption and Word Problem Solving’, Journal on Efficiency and Responsibility in Education and Science, vol. 12, no. 2, pp. 51-65. https://doi.org/10.7160/eriesj.2019.120203
Section
Research Paper