Drawings and Tables as Cognitive Tools for Solving Non-Routine Word Problems in Primary School

Timo Reuter^1, , Wolfgang Schnotz² and Renate Rasch³

¹Graduate School Teaching & Learning Processes, University of Koblenz-Landau, Landau, Germany

²Faculty of Psychology, University of Koblenz-Landau, Landau, Germany

³Department of Mathematics, University of Koblenz-Landau, Landau, Germany

Cite this paper:
Timo Reuter, Wolfgang Schnotz and Renate Rasch. Drawings and Tables as Cognitive Tools for Solving Non-Routine Word Problems in Primary School. American Journal of Educational Research. 2015; 3(11):1387-1397. doi: 10.12691/education-3-11-7

Abstract

External representations play a central role in the process of word problem solving. This study aimed to shed light on teacher-provided representations as cognitive tools for primary students when working on non-routine word problems. Non-routine word problems are characterized by the fact that they cannot be solved by simply applying familiar routine calculations due to their demanding mathematical structure or complex situations described in the problem text. Since primary students often do not generate external representations, the present study examined the questions if providing students with a representation facilitates problem solving in general, and, more in detail, what type of representation (table or drawing) and what level of pre-structuring provided in the representation is most helpful. In an experimental design we studied a sample of 199 4th-graders who worked on non-routine word problems. The experimental design consisted of three tests: A pre-, a treatment-, and a transfer-test. In the pre-test, we measured participant’s prior performance with non-routine word problems. In the treatment-test, an experimental group received problems accompanied by tables and drawings with different levels of pre-structuring to measure student’s performance when external representations were provided for the problem solving process. A control group received no representations. In the transfer-test, participants worked on problems without provided representations to measure participant’s performance after they were exposed to external tools in the treatment-test. Results indicate that providing drawings or tables did not facilitate problem solving in general, which was against our hypothesis. If a representation was provided, a drawing was more helpful than a table, which was in line with our assumptions. However, the drawing effect was depending on the problem type and the level of pre-structuring. Obviously, simply providing external representations was not sufficient to facilitate problem solving. This speaks of the necessity of an early training in diagram literacy.

Keywords:
external representations non-routine word problems primary school problem solving drawings tables

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References:

[1]	Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16 (3), 183-198.
[2]	Beitzel, B. D., Staley, R. K. & DuBois, N. F. (2011). The (in)effectiveness of visual representations as an aid to solving probability word problems. Effective Education, 3 (1), 11-22.
[3]	Bell, A., Swan, M. & Taylor, G. (1981). Choice of operation in verbal problems with decimal numbers. Educational Studies in Mathematics, 12 (4), 399-420.
[4]	Bernardo, A. B. (1999). Overcoming Obstacles to Understanding and Solving Word Problems in Mathematics. Educational Psychology, 19 (2), 149-163.
[5]	Bock, D. de, Verschaffel, L., Janssens, D., van Dooren, W. & Claes, K. (2003). Do realistic contexts and graphical representations always have a beneficial impact on students’ performance? Negative evidence from a study on modelling non-linear geometry problems. Learning and Instruction, 13 (4), 441-463.
[6]	Bock, D. de, Verschaffel, L. & 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.
[7]	Bovenmeyr Lewis, A. (1989). Training students to represent arithmetic word problems. Journal of Educational Psychology, 81 (4), 521-531.
[8]	Bruder, R. & Collet, C. (2011). Problemlösen lernen im Mathematikunterricht (Scriptor Praxis - Mathematik). Berlin: Cornelsen Scriptor.
[9]	Corte, E. de, Verschaffel, L. & Win, L. de. (1985). Influence of rewording verbal problems on children's problem representations and solutions. Journal of Educational Psychology, 77 (4), 460-470.
[10]	Cox, R. (1999). Representation construction, externalised cognition and individual differences. Learning and Instruction, 9 (4), 343-363.
[11]	Csíkos, C., Szitányi, J. & Kelemen, R. (2012). The effects of using drawings in developing young children’s mathematical word problem solving: A design experiment with third-grade Hungarian students. Educational Studies in Mathematics, 81 (1), 47-65.
[12]	Davis-Dorsey, J., Ross, S. M. & Morrison, G. R. (1991). The role of rewording and context personalization in the solving of mathematical word problems. Journal of Educational Psychology, 83 (1), 61-68.
[13]	Dewolf, T., van Dooren, W., Ev Cimen, E. & Verschaffel, L. (2014). The Impact of Illustrations and Warnings on Solving Mathematical Word Problems Realistically. The Journal of Experimental Education, 82 (1), 103-120.
[14]	Diezmann, C. & English, L. D. (2001). Promoting the Use of Diagrams as Tools for Thinking. In A. Cuoco & F. R. Curcio (Hrsg.), The roles of representation in school mathematics (p. 77-89). Reston: National Council of Teachers of Mathematics.
[15]	Duncker, K. (1974). Zur Psychologie des produktiven Denkens (3. Aufl.). Berlin: Springer.
[16]	Elia, I., Bell, M. & Kolovou, A. (2009). Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics. ZDM, 41 (5), 605-618.
[17]	Elia, I., Gagatsis, A. & Demetriou, A. (2007). The effects of different modes of representation on the solution of one-step additive problems. Learning and Instruction, 17 (6), 658-672.
[18]	Fagnant, A. & Vlassis, J. (2013). Schematic representations in arithmetical problem solving: Analysis of their impact on grade 4 students. Educational Studies in Mathematics, 84 (1), 149-168.
[19]	Funke, J. (2003). Problemlösendes Denken (1. edition). Stuttgart: Kohlhammer.
[20]	Garofalo, J. & Lester, F. K. (1985). Metacognition, Cognitive Monitoring, and Mathematical Performance. Journal for Research in Mathematics Education, 16 (3), 163-176.
[21]	Greeno, J. G. & Hall, R. P. (1997). Practicing Representation: Learning with and about representational forms. Phi Delta Kappan, 78 (5), 361-367.
[22]	Groß, J. (2013). Analyse von Lösungsprozessen beim Bearbeiten problemhaltiger Textaufgaben durch Grundschulkinder. Unpublished doctoral thesis, University of Koblenz-Landau. Landau.
[23]	Haffner, J. (2005). Heidelberger Rechentest: HRT 1-4 ; Erfassung Mathematischer Basiskompetenzen im Grundschulalter ; Manual. Göttingen: Hogrefe.
[24]	Hasemann, K. & Stern, E. (2003). Textaufgaben und mathematisches Verständnis. Ergebnisse eines Unterrichtsversuchs im 2. Schuljahr. Grundschulunterricht, 3 (2), 2-5.
[25]	Hegarty, M. & Kozhevnikov, M. (1999). Types of visual-spatial representations and mathematical problem solving. Journal of Educational Psychology, 91 (4), 684-689.
[26]	Heinze, A., Star, J. R. & Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM, 41 (5), 535-540.
[27]	Hohn, K. (2012). Gegeben, gesucht, Lösung? Selbstgenerierte Repräsentationen bei der Bearbeitung problemhaltiger Textaufgaben. Unpublished doctoral thesis, University of Koblenz-Landau. Landau.
[28]	Johnson-Laird, P. N. (1983). Mental models. Towards a cognitive science of language, inference, and consciousness (Cognitive science series, No. 6). Cambridge: Harvard University Press.
[29]	Kalyuga, S., Ayres, P., Chandler, P. & Sweller, J. (2003). The Expertise Reversal Effect. Educational Psychologist, 38 (1), 23-31.
[30]	Kintsch, W. & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review, 92 (1), 109-129.
[31]	Larkin, J. & Simon, H. (1987). Why a Diagram is (Sometimes) Worth Ten Thousand Words. Cognitive Science, 11 (1), 65-100.
[32]	Lenhard, W. & Schneider, W. (2006). Ein Leseverständnistest für Erst- bis Sechstklässler. Göttingen: Hogrefe.
[33]	Liang, K. Y. & Zeger, S. L. (1986). Longitudinal Data Analysis Using Generalized Linear Models. Biometrika, 73 (1), 13-22.
[34]	Mayer, R. E. (2005). Cognitiye Theory of Multimedia Learning. In R. E. Mayer (Ed.), The Cambridge handbook of multimedia learning (pp. 31–48). Cambridge: Cambridge University Press.
[35]	Mayer, R. E. & Hegarty, M. (1996). The Process of Understanding Mathematical Problems. In R. J. Sternberg & T. Ben-Zeev (Ed.), The nature of mathematical thinking (pp. 29-53). Mahwah, NJ: L. Erlbaum Associates.
[36]	Ng, E. L. & 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.
[37]	Paivio, A. (1986). Mental representations. A dual coding approach (Bd. 9). New York: Oxford University Press; Clarendon Press.
[38]	Palmer, S. E. (1978). Fundamental Aspects of Cognitive Representation. In E. Rosch (Ed.), Cognition and categorization (pp. 259-303). Hillsdale, NJ: Erlbaum.
[39]	Pólya, G. (1949). Schule des Denkens. Vom Lösen mathematischer Probleme. Bern: A. Francke AG.
[40]	Rasch, R. (2001). Zur Arbeit mit problemhaltigen Textaufgaben im Mathematikunterricht der Grundschule. Eine Studie zu Herangehensweisen von Grundschulkindern an anspruchsvolle Textaufgaben und Schlussfolgerungen für eine Unterrichtsgestaltung, die entsprechende Lösungsfähigkeiten fördert. Hildesheim: Franzbecker.
[41]	Raven, J. C. & Schmidtke, A. (1986). CPM: Raven-Matrizen-Test; Coloured Progressive Matrices. London: Lewis.
[42]	Resnick, L. B. & Ford, W. W. (1978). The analysis of tasks for instruction: An information-processing approach. In A. C. Catania & T. A. Brigham (Ed.), Handbook of applied behavior analysis. Social and instructional processes. New York: Irvington Publishers; Distributed by Halsted Press.
[43]	Schnotz, W. (2014). Integrated Model of Text and Picture Comprehension. In R. E. Mayer (Ed.), Cambridge Handbook of Multimedia Learning (2nd edition, pp. 72-103). Cambridge: Cambridge University Press.
[44]	Schnotz, W., Baadte, C., Müller, A. & Rasch, R. (2011). Kreatives Problemlösen mit bildlichen und beschreibenden Repräsentationen. In K. Sachs-Hombach & R. Totzke (Ed.), Bilder - Sehen - Denken. Zum Verhältnis von begrifflichen-philosophischen und empirisch-psychologischen Ansätzen in der bildwissenschaftlichen Forschung (pp. 204-252), Köln: Herbert von Halem Verlag.
[45]	Schnotz, W. & Bannert, M. (2003). Construction and interference in learning from multiple representation. Learning and Instruction, 13 (2), 141-156.
[46]	Schoenfeld, A. H. (1983). The Wild, Wild, Wild, Wild, Wild World of problem Solving (A Review of Sorts). For the Learning of Mathematics, 3 (3), 40-47.
[47]	Sweller, J. (1994). Cognitive load theory, learning difficulty, and instructional design. Learning and Instruction, 4 (4), 295-312.
[48]	Sweller, J., van Merrienboer, J. J. G. & Paas, F. G. W. C. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10 (3), 251-296.
[49]	Van Dijk, T. A. & Kintsch, W. (1983). Strategies of discourse comprehension. New York: Academic Press.
[50]	Van Essen, G. & Hamaker, C. (1990). Using self-generated drawings to solve arithmetic word problems. Journal of Educational Research, 83 (6), 301-312.
[51]	Van Garderen, D. & Montague, M. (2003). Visual-Spatial Representation, Mathematical Problem Solving, and Students of Varying Abilities. Learning Disabilities Research and Practice, 18 (4), 246-254.
[52]	Van Meter, P. & Garner, J. (2005). The Promise and Practice of Learner-Generated Drawing: Literature Review and Synthesis. Educational Psychology Review, 17 (4), 285-325.
[53]	Verschaffel, L. (Hrsg.). (2010). Use of external representations in reasoning and problem solving. Analysis and improvement. Abingdon: Routledge.
[54]	Verschaffel, L., Corte, E. de, Lasure, S., van Vaerenbergh, G., Bogaerts, H. & Ratinckx, E. (1999). Learning to Solve Mathematical Application Problems: A Design Experiment With Fifth Graders. Mathematical Thinking and Learning, 1 (3), 195-229.
[55]	Verschaffel, L., Greer, B. & Corte, E. de. (2000). Making sense of word problems. Lisse [Netherlands]: Swets & Zeitlinger Publishers.
[56]	Vilenius‐Tuohimaa, P. M., Aunola, K. & Nurmi, J. (2008). The association between mathematical word problems and reading comprehension. Educational Psychology, 28 (4), 409-426.
[57]	Winter, H. (1992). Sachrechnen in der Grundschule. Problematik des Sachrechnens; Funktionen des Sachrechnens ; Unterrichtsprojekte (2nd Edition.). Frankfurt am Main: Cornelsen Scriptor.
[58]	Wolters, Miriam A. D. (1983). The part-whole schema and artithmetical problems. Educational Studies in Mathematics, 14 (2), 127-138.
[59]	Zhang, J. (1997). The Nature of External Representations in Problem Solving. Cognitive Science, 21 (2), 179-217.