Why Johnny Can’t Apply Multiplication? Revisiting the Choice of Operations with Fractions

Susanne Prediger

doi:10.29333/iejme/262

Full Text (PDF)

Susanne Prediger

More Detail

Abstract

The inadequate choice of operations and misleading interpretations of operations are major obstacles for students to mathematize and solve word problems successfully. Based on theoretical considerations on conceptual change concerning interpretations, this article reports on an empirical study conducted with 830 students. It extends existing investigations on the issue in three ways: 1. by focusing on fractions rather than on decimals, 2. by using an enriched test design, including several aspects of competence and various models for multiplication, and 3. by a deeper explorative analysis of 197 reasons for choices given in an operation choice item format with open responses. The article extends existing theoretical approaches to the problem by reconstructing four main strategies for choosing operations.

Keywords

Fractions
conceptual change
discontinuities of mental models

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

INT ELECT J MATH ED, 2011, Volume 6, Issue 2, 65-88

https://doi.org/10.29333/iejme/262

Publication date: 08 Aug 2011

Article Views: 3019

Article Downloads: 2512

Open Access References How to cite this article

References

American Educational Research Association, American Psychological Association, & National Council on Measurement in Education. (1999). Standards for educational and psychological testing. Washington, DC: American Educational Research Association.
Aksu, M. (1997). Student performance in dealing with fractions. Journal of Educational Research, 90(6), 375-380.
Anghileri, J., & Johnson, D. C. (1992). Arithmetic operations on whole numbers: multiplication and division. In T. R. Post (Ed.), Teaching mathematics in grades K-8: Research based methods (Second Ed., pp.157-200). Boston, MA: Allyn and Bacon.
Bell, A., Fischbein, E., & Greer, B. (1984). Choice of operations in verbal arithmetic problems: The effects of number size, problem structure and context. Educational Studies in Mathematics, 15, 129-147. http://dx.doi.org/10.1007/BF00305893
Bell, A., Greer, B., Grimision, L., & Mangan, C. (1989). Children‟s performance in multiplicative word problems: elements of a descriptive theory. Journal for Research in Mathematics Education, 20(5), 434-449. http://dx.doi.org/10.2307/749419
Bell, A., Swan, M., & Taylor, G. M. (1981). Choice of operation in verbal problems with decimal numbers. Educational Studies in Mathematics, 12, 399-420. http://dx.doi.org/10.1007/BF00308139
Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer.
Cohen, J. (1960). A coefficient for agreement for nominal scales. Education and Psychological Measurement, 20, 37–46. http://dx.doi.org/10.1177/001316446002000104
De Castro, S. C. (1998). Teaching probability for conceptual change. Educational Studies in Mathematics, 35(3), 233-254. http://dx.doi.org/10.1023/A:1003182219483
De Corte, E. & Verschaffel, L. (1996). An empirical test of the impact of primitive intuitive models of operations on solving problems with a multiplicative structure. Learning and Instruction, 6(3), 219-242. http://dx.doi.org/10.1016/0959-4752(96)00004-7
Fischbein, E. (1989). Tacit models and mathematical reasoning. For the Learning of Mathematics, 9(2), 9-14.
Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving problems in multiplication and division. Journal of Research in Mathematics Education, 16(1), 3-17. http://dx.doi.org/10.2307/748969
Greer, B. (1994). Extending the meaning of multiplication and division. In G. Harel & G. J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 61–85). Albany NY: SUNY Press.
Hallett, D. (2008). Effects of Fraction Situations and Individual Differences: A Review of Recent Research Regarding Children‟s Understanding of Fractions. In D. De Bock, B. D. Søndergaard, B. A. Gómez, & C. C. L. Cheng (Eds.), Proceedings of ICME-11 – Topic Study Group 10, Research and Development of Number Systems and Arithmetic (pp. 17- 25). Monterrey: ICMI.
Harel, G., Behr, M., Post, T., & Lesh, R. (1994). The impact of number type on the solution of multiplication and division problems: Further considerations. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 365-388). Albany, NY: SUNY Press.
Hartnett, P. & Gelman, R. (1998). Early understandings of number: Paths or barriers to the construction of new understandings? Learning and Instruction, 8(4), 341-374. http://dx.doi.org/10.1016/S0959-4752(97)00026-1
Hasemann, K. (1981). On difficulties with fractions. Educational studies in mathematics, 12(1), 71-87. http://dx.doi.org/10.1007/BF00386047
Johnson-Laird, P.M. (1983). Mental models: Towards a cognitive science of language, inferences, and consciousness. Cambridge: Cambridge University Press.
Kline, Morris (1973). Why Johnny can’t add: The failure of the new mathematics. New York: St. Martin‟s Press.
Lehtinen, E., Merenluoto, K., & Kasanen, E. (1997). Conceptual change in mathematics: From rational to (un)real numbers. European Journal of Psychology of Education, 12(2), 131-145. http://dx.doi.org/10.1007/BF03173081
Moser Opitz, E. (2007). Rechenschwäche/Dyskalkulie. Theoretische Klärungen und empirische Studien an betroffenen Schülerinnen und Schülern. Bern: Haupt.
Nesher, P. & Teubal. E. (1975). Verbal cues as an interfering factor in verbal problem solving. Educational Studies in Mathematics, 6(1), 41-51. http://dx.doi.org/10.1007/BF00590023
Prediger, S. (2009). “…because „of‟ is always minus…” - Students explaining their choice of operations in multiplicative word problems with fractions. In M. Tzekaki, M. Kaldrimidou, & C. Sakonidis (Eds.), Proceedings of the 33rd Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 401-408). Thessaloniki: PME.
Prediger, S. (2006). Continuities and discontinuities for fractions. A proposal for analysing in different levels. In J. Novotná et al. (Eds.), Proceedings of the 30th Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 377-384). Prague: PME.
Prediger, S. (2008a). The relevance of didactic categories for analysing obstacles in conceptual change: Revisiting the case of multiplication of fractions. Learning and Instruction, 18(1), 3-17. http://dx.doi.org/10.1016/j.learninstruc.2006.08.001
Prediger, S. (2008b). Discontinuities for mental models: A source for difficulties with the multiplication of fractions. In D. De Bock, B.D. Søndergaard, B.A. Gómez, & C.C.L. Cheng (Eds.). Proceedings of ICME-11 – Topic Study Group 10, Research and Development of Number Systems and Arithmetic (pp. 29-37). Monterrey: ICMI.
Prediger, S. (2012, in press). Discontinuities of mental models. In N.M. Seel (Ed.), Encyclopedia of the Sciences of Learning. New York: Springer.
Prediger, S. & Matull, I. (2008). Vorstellungen und Mathematisierungskompetenzen zur Multiplikation von Brüchen [Conceptions and modelling competencies for the multiplication of fractions]. Internal Research Report for the DFG-Study „Schichtung von Schülervorstellungen“. Dortmund: IEEM.
Posner, G., Strike, K., Hewson, P., & Gertzog, W. (1982). Accommodation of a scientific conception: Toward a theory of conceptual change. Science Education, 66(2), 211-227. http://dx.doi.org/10.1002/sce.3730660207
Stafylidou, S. & Vosniadou, S. (2004). The development of students‟ understanding of the numerical value of fractions. Learning and Instruction, 14(5), 503-518. http://dx.doi.org/10.1016/j.learninstruc.2004.06.015
Streefland, L. (1984). Unmasking N-distractors as a source of failures in learning fractions. In B. Southwell, R. Eyland, M. Cooper, J. Conroy. & K. Collis (Eds.), Proceedings of the eighth international conference for the psychology of mathematics education (pp. 142- 152). Sydney: PME.
Taber, S. B. (2007). Using Alice in Wonderland to teach Multiplication of Fractions. Mathematics Teaching in the Middle School, 12(5), 244-249.
Usiskin, Z. (1991). Building Mathematics Curricula with Applications and Modelling. In M. Niss, W. Blum & I. Huntley (Eds.), Teaching of Mathematical Modelling and Applications (pp. 30-45). Chichester: Horwood.
Usiskin, Z. (2008). The Arithmetic Curriculum and the Real World. In D. De Bock, B. D. Søndergaard, B. A. Gómez, & C. C. L. Cheng (Eds.), Proceedings of ICME-11 – Topic Study Group 10, Research and Development of Number Systems and Arithmetic (pp. 9- 16). Monterrey: ICMI.
Verschaffel, L., Greer, B., Van Dooren, W., & Mukhopadhyay, S. (Eds.) (2009). Words and worlds: Modelling verbal descriptions of situations. Rotterdam: Sense Publications.
Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse: Swets & Zeitlinger.
vom Hofe, R., Kleine, M., Blum, W., & Pekrun, R. (2006). The effect of mental models ("Grundvorstellungen") for the development of mathematical competencies. First results of the longitudinal study PALMA. In M. Bosch (Eds.), Proceedings of the CERME 4, 142-151, online http://ermeweb.free.fr/CERME4/CERME4_WG1.pdf.
Vosniadou, S. & Verschaffel, L. (2004) (eds.). The Conceptual Change Approach to Mathematics Learning and Teaching, Learning and Instruction, 14 (5), 445-548.
Wartha, S. (2007). Längsschnittliche Untersuchungen zur Entwicklung des Bruchzahlbegriffs. Hildesheim: Franzbecker.

How to cite this article

APA

Prediger, S. (2011). Why Johnny Can’t Apply Multiplication? Revisiting the Choice of Operations with Fractions. International Electronic Journal of Mathematics Education, 6(2), 65-88. https://doi.org/10.29333/iejme/262

Vancouver

Prediger S. Why Johnny Can’t Apply Multiplication? Revisiting the Choice of Operations with Fractions. INT ELECT J MATH ED. 2011;6(2):65-88. https://doi.org/10.29333/iejme/262

AMA

Prediger S. Why Johnny Can’t Apply Multiplication? Revisiting the Choice of Operations with Fractions. INT ELECT J MATH ED. 2011;6(2), 65-88. https://doi.org/10.29333/iejme/262

Chicago

Prediger, Susanne. "Why Johnny Can’t Apply Multiplication? Revisiting the Choice of Operations with Fractions". International Electronic Journal of Mathematics Education 2011 6 no. 2 (2011): 65-88. https://doi.org/10.29333/iejme/262

Harvard

MLA

Prediger, Susanne "Why Johnny Can’t Apply Multiplication? Revisiting the Choice of Operations with Fractions". International Electronic Journal of Mathematics Education, vol. 6, no. 2, 2011, pp. 65-88. https://doi.org/10.29333/iejme/262