Does representation matter? Teacher-provided tables and drawings as cognitive tools for solving non-routine word problems in primary school

Timo Reuter 1 * , Wolfgang Schnotz 2, Renate Rasch 3
More Detail
1 Graduate School “Teaching and Learning Processes”, University of Koblenz-Landau, Landau, Germany
2 Department of General and Educational Psychology, University of Koblenz-Landau, Landau, Germany
3 Institute for Mathematics, University of Koblenz-Landau, Landau, Germany
* Corresponding Author
EUR J SCI MATH ED, Volume 2, Issue 2A, pp. 34-43.
OPEN ACCESS   1491 Views   935 Downloads
Download Full Text (PDF)


According to the German educational standards, students should be familiar with problem-solving as a general mathematical competency by the end of grade 4. Non-routine word problems are suitable tasks for mathematical problem-solving in elementary mathematics classes. They are characterized by the fact that the problem-solver cannot simply use well-trained algorithmic calculating procedures (Rasch, 2001). As a result, many students struggle with word problems in mathematics, especially with non-routine word problems (Hohn, 2012). Representation plays a central role in the process of problem-solving. It involves representing a problem situation adequately, constructing a mental model and comparing it in a dynamic and iterative process with the information externalized in the representation (Schnotz et al, 2011). This study aims to shed light on teacher-provided representations as cognitive tools for students when working on non-routine word problems. In an experimental study, we examined a sample of 67 primary school students who worked on six non-routine word problems with provided representations. The tasks were accompanied by a table or a drawing. Furthermore, the tables and drawings differed with regard to the amount of information provided. Statistical data analysis generated, among other findings, two results: Overall solution rates were low (10 to 24%). Tables and drawings facilitated the solution process differently depending on the type of word problem. Consequences for subsequent future research are discussed.


Reuter, T., Schnotz, W., & Rasch, R. (2014). Does representation matter? Teacher-provided tables and drawings as cognitive tools for solving non-routine word problems in primary school. European Journal of Science and Mathematics Education, 2(2A), 34-43.


  • Ainsworth, S. (1999). The functions of multiple representations. Computers & Education, 33(2-3), 131-152.
  • Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. Zentralblatt für Didaktik der Mathematik 38(2), 86-95.
  • Bruder, R. and Collet, C. (2011). Problemlösen lernen im Mathematikunterricht. Berlin: Cornelsen Scriptor Praxis.
  • Chi, M. T. H., Feltovich, P. J., and Glaser, R. (1981). Categorization and Representation of Physics Problems by Experts and Novices. Cognitive Science 5(2), 121-152.
  • Cox, R. (1999). Representation construction, externalised cognition and individual differences. Learning and Instruction 9(4), 343-363.
  • De Bock, D., Verschaffel, L. and Janssens, D. (1998). The predominance of the linear model in secondary school students’ solutions of word problems involving length and area of similar plane figures. Educational Studies in Mathematics 35(1), 65-83.
  • Duncker, K. (1974). Zur Psychologie des produktiven Denkens. Berlin, Heidelberg, New York: Springer.
  • Elia, I., Van den Heuvel-Panhuizen, M., and Kolovou, A. (2009). Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics. ZDM Mathematics Education 41(5), 605-618.
  • Hohn, K. (2012). Gegeben, gesucht, Lösung? Selbstgenerierte Repräsentationen bei der Bearbeitung problemhaltiger Textaufgaben. PhD thesis. Koblenz-Landau, Landau. Psychologie.
  • Kaiser, G. (1995). Realitätsbezüge im Mathematikunterricht – Ein Überblick über die aktuelle und historische Diskussion. In G. Graumann et al (Eds.), Materialien für einen realitätsbezogenen Mathematikunterricht (pp. 66-84). Bad Salzdetfurth: Franzbecker.
  • Kintsch, W. and Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review 92(1), 109-129.
  • Mayer, R. E. (2005). Principles for Reducing Extraneous Processing in Multimedia Learning: Coherence, Signaling, Redundancy, Spatial Contiguity, and Temporal Contiguity Principles. In R. E. Mayer (Ed.), The Cambridge handbook of multimedia learning (pp. 183–200). Cambridge, U.K, New York: Cambridge University Press.
  • Mayer, R. E.; Hegarty, M. (1996). The Process of Understanding Mathematical Problems. In R. Sternberg and T. Ben-Zeev (Eds.), The Nature of Mathematical Thinking (pp. 29-53). Mahwah, NJ: L. Erlbaum Associates.
  • Nesher, P., Hershkowitz, S. and Novotna, J. (2003). Situation model, text base and what else? Factors affecting problem solving. Educational Studies in Mathematics 52(2), 151-176.
  • Ng, E. L. and Lee, K. (2009). The Model Method: Singapore Children's Tool for Representing and Solving Algebraic Word Problems. Journal for Research in Mathematics Education 40(3), 282-313.
  • Pantziara, M., Gagatsis, A. and Elia, I. (2009). Using diagrams as tools for the solution of non-routine mathematical problems. Educational Studies in Mathematics 72(1), 39-60.
  • Schnotz, W. (2002). Towards an Integrated View of Learning From Text and Visual Displays. Educational Psychology Review 14(1), 101-120.
  • Schnotz, W. (2005). An Integrated Model of Text and Picture Comprehension. In: R. E. Mayer (Ed.), The Cambridge handbook of multimedia learning (pp. 49–69). Cambridge, U.K, New York: Cambridge University Press.
  • Schnotz, W., Baadte, C., Müller, A., and Rasch, R. (2011). Kreatives Problemlösen mit bildlichen und beschreibenden Repräsentationen. In K. Sachs-Hombach and R. Totzke (Eds.), Bilder - Sehen - Denken. Zum Verhältnis von begrifflichen-philosophischen und empirisch-psychologischen Ansätzen in der bildwissenschaftlichen Forschung (pp. 204-252). Köln: Herbert von Halem Verlag.
  • Stern, E. (2005). Kognitive Entwicklungspsychologie des mathematischen Denkens. In M. von Aster and J. H. Lorenz (Ed.), Rechenstörungen bei Kindern. Neurowissenschaft, Psychologie, Pädagogik (pp. 137-149). Göttingen: Vandenhoeck & Ruprecht.
  • Van Meter, P. and Garner, J. (2005). The Promise and Practice of Learner-Generated Drawing: Literature Review and Synthesis. Educ Psychol Rev 17(4), 285-325.
  • Verschaffel, L., De Corte, E., Lasure, S., Van Vaerenbergh, G., Bogaerts, H. and Ratinckx, E. (1999). Learning to solve mathematical application problems: A design experiment with fifth graders. Mathematical Thinking and Learning 1(3), 195-229.
  • Verschaffel, L., Greer, B., and De Corte, E. (2000). Making sense of word problems. Lisse, Exton, PA: Swets & Zeitlinger Publishers.
  • Winter, H. (1992). Sachrechnen in der Grundschule. Problematik des Sachrechnens, Funktionen des Sachrechnens, Unterrichtsprojekte. Berlin: Cornelsen Scriptor