Abstract
During the last 20 years, many scholars have argued in various ways that the (traditional) practice of word problems in school mathematics does not foster in students, indeed inhibits, a genuine disposition towards mathematical modelling and applied problem solving. In this article we give a brief review and discussion of this research, including a summary of earlier work culminating in the book by Verschaffel et al. (2000) and with special attention to the more recent empirical work. We begin with presenting the ascertaining studies documenting and illuminating the phenomenon of “suspension of sense-making” when doing school arithmetic word problems. Then we move to studies that have contributed to the explanation of the observed effects. This explanation is followed by a review of some recent design experiments wherein the modelling perspective has been implemented and tested. Afterwards we discuss some recent studies on the difficulties encountered by teachers who try to implement this new perspective into their daily classroom practices. Finally, we discuss a number of educational implications of the research done so far and some challenges for the future of teaching mathematical modelling.
Zusammenfassung
Im Verlauf der letzten 20 Jahre haben zahlreiche Wissenschaftler in verschiedenen Studien gezeigt, dass die traditionelle Art und Weise der schulischen Behandlung mathematischer Textaufgaben die Lernenden nicht hinsichtlich ihrer Modellierungskompetenzen fördert, sondern eher sogar behindert, dass sie angemessene Voraussetzungen für das Lösen angewandter Probleme erwerben. In diesem Artikel geben wir zunächst einen kurzen Überblick über die Forschung zu Textaufgaben und beziehen uns – mit einem Fokus auf neuere empirische Untersuchungen – u.a. auch auf diesbezügliche frühere Arbeiten, die im Buch von Verschaffel et al. (2000) veröffentlicht sind. Dabei beginnen wir mit der Beschreibung von Studien, die sich mit der Beschreibung und Erklärung des Phänomens des “suspension of sense-making” im Rahmen der schulischen Bearbeitung von arithmetischen Textaufgaben auseinandersetzen. Hierauf folgt die Analyse einiger experimenteller Untersuchungen, die sich mit einer verstärkten Implementation und Evaluation von Unterrichtsexperimenten befassen, in denen Modellierungsaspekten im Vordergrund stehen. Im Anschluss daran werden verschiedene Studien hinsichtlich der Schwierigkeiten analysiert, die sich bei der Implementation einer veränderten Herangehensweise in das tägliche Unterrichten ergeben.
Abschließend werden einige sich aus den Forschungsergebnissen ergebende didaktische Implikationen sowie sich daraus abzuleitende Herausforderungen noch zu leistender Forschung zur Vermittlung von Modellierungskompetenz diskutiert.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
As pointed out by one of our reviewers, a similar response happened when Piaget’s conservation tasks were first replicated.
This point was drawn to our attention by one of our reviewers.
References
Baruk, S. (1985). L’âge du capitaine. De l’erreur en mathématiques. Berlin: Springer.
Blum, W., & Leiß, D. (2007). How do students and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling: Education, engineering, and economics (pp. 222–231). Chichester: Horwood.
Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects—state, trends, and issues in mathematics education. Educational Studies in Mathematics, 22, 37–68.
Blum, W., Galbraith, P. L., Henn, H.-W., & Niss, M. (Eds.), (2007). Modelling and applications in mathematics education (ICMI Study 14). New York: Springer.
Boaler, J. (1994). When do girls prefer football to fashion? An analysis of female underachievement in relation to “realistic” mathematical contexts. British Educational Research Journal, 20, 551–564.
Bonotto, C. (2009). Working towards teaching realistic mathematical modelling and problem posing in Italian classrooms. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 297–314). Rotterdam: Sense.
Bonotto, C., & Wilczewski, E. (2007). I problemi di matematica nella scuola primaria: Sull’attivazione o meno di conoscenze di tipo realistico. In C. Bonotto (Ed.), Quotidianizzare la matematica (pp. 101–134). Lecce: La Biblioteca Pensa Multimedia.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer. Edited and translated by N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield.
Burkhardt, H. (1994). Mathematical applications in school curriculum. In T. Husén, & T. N. Postlethwaite (Eds.), The international encyclopedia of education (2nd ed., pp. 3621–3624). Oxford: Pergamon Press.
Caldwell, L. (1995). Contextual considerations in the solution of children’s multiplication and division word problems. Unpublished Master’s thesis, Queen’s University, Belfast, Northern Ireland.
Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational Studies in Mathematics, 62, 211–230.
Cognition and Technology Group at Vanderbilt (1997). The Jasper project: Lessons in curriculum, instruction, assessment, and professional development. Mahwah: Lawrence Erlbaum.
Cooper, B., & Dunne, M. (1998). Anyone for tennis? Social class differences in children’s responses to National Curriculum mathematics testing. Sociological Review, 46, 115–148.
Cooper, B., & Harries, T. (2009). Realistic contexts, mathematics assessment, and social class: Lessons for policy from an English research programme. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 93–110). Rotterdam: Sense.
Csíkos, C. (2003). How many buses are needed? Hungarian students’ achievement on ‘problematic’ word problems. Paper presented at the 10th European Conference for Research on Learning and Instruction, Padova, Italy, August 2003.
DeFranco, T. C., & Curcio, F. R. (1997). A division problem with a remainder embedded across two contexts: Children’s solutions in restrictive versus real-world settings. Focus on Learning Problems in Mathematics, 19(2), 58–72.
Depaepe, F., De Corte, E., & Verschaffel, L. (2009a). Analysis of the realistic nature of word problems in upper elementary mathematics education in Flanders. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 245–264). Rotterdam: Sense.
Depaepe, F., De Corte, E., & Verschaffel, L. (2009b). Teachers’ approaches towards word problem solving: Elaborating or restricting the problem context. Teaching and Teacher Education. doi:10.1016/j.tate.2009.03.016
Frankenstein, M. (2009). Develo** a critical mathematical numeracy through real real-world word problems. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 111–130). Rotterdam: Sense.
Gainsburg, J. (2009). How and why secondary mathematics teachers make (or don’t make) real-world connections in teaching. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 265–282). Rotterdam: Sense.
Gerofsky, S. (1997). An exchange about word problems. For the Learning of Mathematics, 17(2), 22–23.
Gerofsky, S. (2009). Genre, simulacra, impossible exchange, and the real: How postmodern theory problematizes word problems. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 21–38). Rotterdam: Sense.
Greer, B. (1993). The modelling perspective on wor(l)d problems. Journal of Mathematical Behavior, 12, 239–250.
Greer, B. (1997). Modelling reality in mathematics classrooms: The case of word problems. Learning and Instruction, 7, 293–307.
Greer, B., & Verschaffel, L. (2007). Modelling competencies—overview. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education (ICMI Study 14) (pp. 219–224). New York: Springer.
Hatano, G. (1997). Commentary: Cost and benefit of modelling activity. Learning and Instruction, 7, 383–387.
Hidalgo, M. C. (1997). L’activation des connaissances à propos du monde réel dans la résolution de problèmes verbaux en arithmétique. Unpublished doctoral dissertation, Université Laval, Québec, Canada.
Inoue, N. (2001). The role of personal interpretation in mathematical problem solving: Enhancing the relevance of mathematical learning to everyday experience (Internal report). Teachers College, Columbia University.
Inoue, N. (2005). The realistic reasons behind unrealistic solutions: The role of interpretive activity in word problem solving. Learning and Instruction, 15, 69–83.
Jablonka, E., & Gellert, U. (2007). Mathematisation–Demathematisation. In U. Gellert, & E. Jablonka (Eds.), Mathematisation and demathematisation: Social, philosophical and educational ramifications (pp. 1–18). Rotterdam: Sense Publishers.
Kaiser, G., & Maaß, K. (2007). Modelling in lower secondary classrooms—Problems and chances. In W. Blum, P. Galbraith, H. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 99–108). New York: Springer.
Lave, J. (1992). Word problems: A microcosm of theories of learning. In P. Light, & G. Butterworth (Eds.), Context and cognition: Ways of learning and knowing (pp. 74–92). New York: Harvester Wheatsheaf.
Lehrer, R., & Schauble, L. (2000). Modelling in mathematics and science. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 5.) Mahwah: Lawrence Erlbaum.
Lesh, R., & Caylor, B. (2009). Differing conceptions of problem solving in mathematics education, science education, and professional schools. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 333–350). Rotterdam: Sense.
Lesh, R., & Doerr, H.M. (Eds.) (2003). Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning, and teaching. Erlbaum: Mahwah.
Maaß, K. (2004). Mathematisches Modellieren im Unterricht—Ergebnisse einer Empirischen Studie. Hildesheim: Franzbecker.
Mason, L., & Scrivani, L. (2004). Enhancing students’ mathematical beliefs: An intervention study. Learning and Instruction, 14, 153–176.
Mukhopadhyay, S., & Greer, B. (2001). Modelling with purpose: Mathematics as a critical tool. In B. Atweh, H. Forgasz, & B. Nebres (Eds.), Socio-cultural aspects in mathematics education (pp. 295–311). Mahwah: Lawrence Erlbaum.
Niss, M. (2001). Issues and problems of research on the teaching and learning of applications and modelling. In J. F. Matos, W. Blum, S. K. Houston, & S. P. Carreira (Eds.), Modelling and mathematics education. ICTMA 9: Applications in science and technology (pp. 72–89). Chichester: Horwood.
Niss, M., Blum, W., & Galbraith, P. L. (2007). Introduction. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (pp. 3–32). New York: Springer.
Palm, T. (2002). The realism of mathematical school tasks. Features and consequences. Unpublished doctoral dissertation, University of Umea, Sweden.
Prediger, S. (2009). Zur Bedeutung vielfältiger Theorien und wissenschaftlicher Praktiken in der Mathematikdidaktik am Beispiel von Schwierigkeiten mit Textaufgaben. In M. Neubrand (Ed.), Beiträge zum Mathematikunterricht (pp. 49–56) Münster: WTM. http://www.mathematik.uni-dortmund.de/~prediger/veroeff/09-BzMU-Theorien.pdf
Radatz, H. (1983). Untersuchungen zum Lösen eingekleideter Aufgaben. Journal für Mathematik-Didaktik, 4(3), 205–217.
Renkl, A. (1999). The gap between school and everyday knowledge in mathematics. Paper presented at the Eighth European Conference for Research on Learning and Instruction, Göteborg, Sweden, August 1999.
Reusser, K., & Stebler, R. (1997a). Every word problem has a solution: The suspension of reality and sense-making in the culture of school mathematics. Learning and Instruction, 7, 309–328.
Reusser, K., & Stebler, R. (1997b). Realistic mathematical modelling through the solving of performance tasks. Paper presented at the Seventh European Conference on Learning and Instruction, Athens, Greece, August 1997.
Roth, W.-M. (2009). On the problematic of word problems—language and the world we inhabit. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 55–69). Rotterdam: Sense.
Säljö, R., Riesbeck, E., & Wyndhamn, J. (2009). Learning to model: Coordinating natural language and mathematical operations when solving word problems. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 177–194). Rotterdam: Sense.
Schoenfeld, A. H. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In J. F. Voss, D. N. Perkins, & J. W. Segal (Eds.), Informal reasoning and education (pp. 311–343). Hillsdale: Erlbaum.
Seeger, F., Voigt, J., & Waschescio, U. (Eds.), (1998). The culture of the mathematics classroom. Cambridge: Cambridge University Press.
Selter, Ch. (1994). How old is the captain? Strategies, 5(1), 34–37.
Selter, Ch. (2001). “1/2 Busse heißt: ein halbvoller Bus!”—Zu Vorgehensweisen von Grundschülern bei einer Textaufgabe mit Rest [“1/2 bus means a bus half-full!”—Solution strategies of elementary school children for DWR problems]. In C. Selter, & G. Walther (Eds.), Mathematik lernen und gesunder Menschenverstand (pp. 162–173). Leipzig: Klett.
Skovsmose, O. (2005). Travelling through education: Uncertainty, mathematics, responsibility. Rotterdam: Sense.
Swetz, F. J. (1987). Capitalism and arithmetic: The new math of the 15th century. La Salle: Open Court.
Swetz, F. J. (2009). Word problems: Footprints from the history of mathematics. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 73–92). Rotterdam: Sense.
Thorndike, E. L. (1922). The psychology of arithmetic. New York: Macmillan.
Toom, A. (1999). Word problems: Applications or mental manipulatives. For the Learning of Mathematics, 19(1), 36–38.
Usiskin, Z. (2007). The arithmetic operations as mathematical models. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Applications and modelling in mathematics education (The 14th ICMI Study 14) (pp. 257–264). New York: Springer.
Verschaffel, L., & De Corte, E. (1997). Teaching realistic mathematical modelling and problem solving in the elementary school: A teaching experiment with fifth graders. Journal for Research in Mathematics Education, 28, 577–601.
Verschaffel, L., De Corte, E., & Lasure, S. (1994). Realistic considerations in mathematical modelling of school arithmetic word problems. Learning and Instruction, 4, 273–294.
Verschaffel, L., De Corte, E., & Borghart, I. (1997). Pre-service teachers’ conceptions and beliefs about the role of real-world knowledge in mathematical modelling of school word problems. Learning and Instruction, 4, 339–359.
Verschaffel, L., De Corte, E., Lasure, S., Van Vaerenbergh, G., Bogaerts, H., & Ratinckx, E. (1999). Design and evaluation of a learning environment for mathematical modelling and problem solving in upper elementary school children. Mathematical Thinking and Learning, 1, 195–229.
Verschaffel, L., De Corte, B., & Greer, E. (2000). Making sense of word problems. Lisse: Swets & Zeitlinger.
Verschaffel, L., Greer, B., Van Dooren, W., & Mukhopadhyay, S. (Eds.), (2009). Words and worlds. Modelling verbal descriptions of situations. Rotterdam: Sense.
Vinner, S. (1997). The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. Educational Studies in Mathematics, 34, 97–129.
Wyndhamn, J., & Säljö, R. (1997). Word problems and mathematical reasoning: A study of children’s mastery of reference and meaning in textual realities. Learning and Instruction, 7, 361–382.
**n, Z. (2009). Realistic problem solving in China. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (pp. 161–176). Rotterdam: Sense.
**n, Z., & Zhang, L. (2009). Cognitive holding power, fluid intelligence, and mathematical achievement as predictors of children’s realistic problem solving. Learning and Individual Differences, 19(1), 124–129.
**n, Z., Lin, C., Zhang, L., & Yan, R. (2007). The performance of Chinese primary school students on realistic arithmetic word problems. Educational Psychology in Practice, 23, 145–159.
Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477.
Ye**, L., & Silver, E. (2000). Can younger students succeed where older students fail? An examination of third graders’ solutions of a division-with-remainder problem. Journal of Mathematical Behavior, 19, 233–246.
Yoshida, H., Verschaffel, L., & De Corte, E. (1997). Realistic considerations in solving problematic word problems: Do Japanese and Belgian children have the same difficulties? Learning and Instruction, 7, 329–338.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Verschaffel, L., Van Dooren, W., Greer, B. et al. Reconceptualising Word Problems as Exercises in Mathematical Modelling. J Math Didakt 31, 9–29 (2010). https://doi.org/10.1007/s13138-010-0007-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13138-010-0007-x
Keywords
- Word problems
- Mathematical sense-making
- Didactical contract
- Socio-mathematical norms
- Classroom practices