Educational Studies in Mathematics

, Volume 95, Issue 1, pp 53–78 | Cite as

Make a drawing. Effects of strategic knowledge, drawing accuracy, and type of drawing on students’ mathematical modelling performance

  • Johanna RellensmannEmail author
  • Stanislaw Schukajlow
  • Claudia Leopold


Drawing strategies are widely used as a powerful tool for promoting students’ learning and problem solving. In this article, we report the results of an inferential mediation analysis that was applied to investigate the roles that strategic knowledge about drawing and the accuracy of different types of drawings play in mathematical modelling performance. Sixty-one students were asked to create a drawing of the situation described in a task (situational drawing) and a drawing of the mathematical model described in the task (mathematical drawing) before solving modelling problems. A path analysis showed that strategic knowledge about drawing was positively related to students’ modelling performance. This relation was mediated by the type and accuracy of the drawings that were generated. The accuracy of situational drawing was related only indirectly to performance. The accuracy of mathematical drawings, however, was strongly related to students’ performance. We complemented the quantitative approach with a qualitative in-depth analysis of students’ drawings in order to explain the relations found in our study. Implications for teaching practices and future research are discussed.


Visualization Drawing Strategy Representation Mathematical modelling Real-world problems 


Supplementary material

10649_2016_9736_MOESM1_ESM.docx (533 kb)
ESM 1 (DOCX 533 kb)


  1. Arcavi, A. (2003). The role of visual representations in learning mathematics. Educational Studies in Mathematics, 52, 215–241.CrossRefGoogle Scholar
  2. Blum, W. (2011). Can modelling be taught and learnt? Some answers from empirical research. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in the teaching and learning of mathematical modelling - Proceedings of ICTMA14 (pp. 15–30). New York: Springer.CrossRefGoogle Scholar
  3. Blum, W., & Leiss, D. (2007). How do students and teachers deal with mathematical modelling problems? The example sugarloaf and the DISUM project. In C. Haines, P. L. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling (ICTMA 12): Education, engineering and economics (pp. 222–231). Chichester: Horwood.CrossRefGoogle Scholar
  4. Booth, R. D., & Thomas, M. O. (1999). Visualization in mathematics learning: Arithmetic problem-solving and student difficulties. The Journal of Mathematical Behavior, 18(2), 169–190.CrossRefGoogle Scholar
  5. Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of the phases in the modelling process. ZDM–The International Journal on Mathematics Education, 38(2), 86–95.CrossRefGoogle Scholar
  6. Borromeo Ferri, R. (2007). Individual modelling routes of pupils – Analysis of modelling problems in mathematical lessons from a cognitive perspective. In C. Haines (Ed.), Mathematical modelling (ICTMA 12): Education, engineering and economics (pp. 260–270). Chichester: Horwood.CrossRefGoogle Scholar
  7. Bryant, F. B., & Satorra, A. (2012). Principles and practice of scaled difference chi-square testing. Structural Equation Modeling: A Multidisciplinary Journal, 19(3), 372–398.CrossRefGoogle Scholar
  8. Cox, R. (1999). Representation construction, externalised cognition and individual differences. Learning and Instruction, 9(4), 343–363.CrossRefGoogle Scholar
  9. 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.CrossRefGoogle Scholar
  10. De Bock, D., 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 modeling non-linear geometry problems. Learning and Instruction, 13(4), 441–463.CrossRefGoogle Scholar
  11. De Bock, D., 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.CrossRefGoogle Scholar
  12. Diezmann, C. M. (2005). Assessing primary students’ knowledge of networks, hierarchies and matrices using scenario-based tasks. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonagh, R. Pierce, et al. (Eds.), Proceedings 28th annual conference of the mathematics education research group of Australasia (pp. 289–296). Sydney: MERGA.Google Scholar
  13. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 95–123). Dordrecht: Kluwer.Google Scholar
  14. Dufour-Janvier, B., Bednarz, N., & Belanger, M. (1987). Pedagogical considerations concerning the problem of representation. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 110–120). Hillsdale, NJ: Erlbaum.Google Scholar
  15. Flavell, J. H. (1979). Metacognition and cognitive monitoring: A new area of cognitive developmental inquiry. American Psychologist, 34, 906–911.CrossRefGoogle Scholar
  16. Galbraith, P. L., & Stillman, G. (2006). A framework for identifying student blockages during transitions in the modelling process. ZDM–The International Journal on Mathematics Education, 38(2), 143–162.CrossRefGoogle Scholar
  17. Hegarty, M., & Kozhevnikov, M. (1999). Types of visual–spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684.CrossRefGoogle Scholar
  18. Hembree, R. (1992). Experiments and relational studies in problem solving: A meta-analysis. Journal for Research in Mathematics Education, 23(3), 242–273.CrossRefGoogle Scholar
  19. Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1–55.CrossRefGoogle Scholar
  20. Kline, R. B. (2005). Principles and practice of structural equation modeling. New York, NY: Guilford Press.Google Scholar
  21. Krug, A., & Schukajlow, S. (2013). Problems with and without connection to reality and students’ task-specific interest. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 209–216). Kiel, Germany: PME.Google Scholar
  22. Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11, 65–99.CrossRefGoogle Scholar
  23. Leiss, D., Schukajlow, S., Messner, R., & Pekrun, R. (2010). The role of the situation model in mathematical modelling – Task analyses, student competencies, and teacher interventions. Journal für Mathematik-Didaktik, 31(1), 119–141.CrossRefGoogle Scholar
  24. Leutner, D., Leopold, C., & Sumfleth, E. (2009). Cognitive load and science text comprehension: Effects of drawing and mentally imagining text content. Computers in Human Behavior, 25(2), 284–289.CrossRefGoogle Scholar
  25. Lingel, K., Neuenhaus, N., Artelt, C., & Schneider, W. (2014). Der Einfluss des metakognitiven Wissens auf die Entwicklung der Mathematikleistung am Beginn der Sekundarstufe I [The influence of metacognitive knowledge on the development of mathematics achievement at the beginning of secondary school]. Journal für Mathematik-Didaktik, 35(1), 49–77.CrossRefGoogle Scholar
  26. Lucangeli, D., & Cornoldi, C. (1997). Mathematics and metacognition: What is the nature of the relationship? Mathematical Cognition, 3, 121–139.CrossRefGoogle Scholar
  27. Muthén, L. K., & Muthén, B. O. (1998–2012). Mplus user’s guide (7th ed.). Los Angeles: Muthén & Muthén.Google Scholar
  28. 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. 1–32). New York: Springer.Google Scholar
  29. Preacher, K. J., & Hayes, A. F. (2008). Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behavior Research Methods, 40(3), 879–891.CrossRefGoogle Scholar
  30. Presmeg, N. C. (1986a). Visualisation and mathematical giftedness. Educational Studies in Mathematics, 17(3), 297–311.CrossRefGoogle Scholar
  31. Presmeg, N. C. (1986b). Visualisation in high school mathematics. For the Learning of Mathematics, 42–46.Google Scholar
  32. Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez, & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 205–235). Rotterdam: Sense Publishers.Google Scholar
  33. Pressley, M., Borkowski, J. G., & Schneider, W. (1989). Good information processing: What it is and how education can promote it. International Journal of Educational Research, 13(8), 857–867.CrossRefGoogle Scholar
  34. Renkl, A., & Nückles, M. (2006). Lernstrategien der externen Visualisierung [Learning strategies of externalized visualization]. In H. Mandl & H. Friedrich (Eds.), Handbuch Lernstrategien (pp. 135–147). Göttingen: Hogrefe.Google Scholar
  35. Schneider, W., & Artelt, C. (2010). Metacognition and mathematics education. ZDM–The International Journal on Mathematics Education, 42(2), 149–161.CrossRefGoogle Scholar
  36. Schukajlow, S. (2011). Mathematisches Modellieren. Schwierigkeiten und Strategien von Lernenden als Bausteine einer lernprozessorientierten Didaktik der neuen Aufgabenkultur [Mathematical modelling. Difficulties and strategies of learners as a means for a learning process-oriented didactic of problem posing]. Münster: Waxmann.Google Scholar
  37. Schukajlow, S., Krug, A., & Rakoczy, K. (2015). Effects of prompting multiple solutions for modelling problems on students’ performance. Educational Studies in Mathematics, 89(3), 393–417.CrossRefGoogle Scholar
  38. Schukajlow, S., & Leiss, D. (2011). Selbstberichtete Strategienutzung und mathematische Modellierungskompetenz [Self-reported use of strategies and mathematical modelling]. Journal für Mathematik-Didaktik, 32(1), 53–77.CrossRefGoogle Scholar
  39. Schukajlow, S., Leiss, D., Pekrun, R., Blum, W., Müller, M., & Messner, R. (2012). Teaching methods for modelling problems and students’ task-specific enjoyment, value, interest and self-efficacy expectations. Educational Studies in Mathematics, 79(2), 215–237.CrossRefGoogle Scholar
  40. Souvignier, E., & Mokhlesgerami, J. (2006). Using self-regulation as a framework for implementing strategy instruction to foster reading comprehension. Learning and Instruction, 16, 57–71.CrossRefGoogle Scholar
  41. Stender, P., & Kaiser, G. (2015). Scaffolding in complex modelling situations. ZDM Mathematics Education, 47(7), 1255–1267.CrossRefGoogle Scholar
  42. Stillman, G. A., & Galbraith, P. L. (1998). Applying mathematics with real world connections: Metacognitive characteristics of secondary students. Educational Studies in Mathematics, 36(2), 157–195.CrossRefGoogle Scholar
  43. Stylianou, D. A. (2011). An examination of middle school students’ representation practices in mathematical problem solving through the lens of expert work: Towards an organizing scheme. Educational Studies in Mathematics, 76, 265–280.CrossRefGoogle Scholar
  44. Uesaka, Y., Manalo, E., & Ichikawa, S. (2007). What kinds of perceptions and daily learning behaviours promote students’ use of diagrams in mathematics problem solving? Learning and Instruction, 17(3), 322–335.CrossRefGoogle Scholar
  45. Uesaka, Y., Manalo, E., & Ichikawa, S. (2010). The effects of perception of efficacy and diagram construction skills on students’ spontaneous use of diagrams when solving math word problems. In A. Goel, M. Jamnik, & N. H. Narayanan (Eds.), Diagrammatic representation and inference (Vol. 6170, pp. 197–211). Berlin: Springer.CrossRefGoogle Scholar
  46. Van Essen, G., & Hamaker, C. (1990). Using self-generated drawings to solve arithmetic word problems. Journal of Educational Research, 83(6), 301–312.CrossRefGoogle Scholar
  47. Van Garderen, D. (2006). Spatial visualization, visual imagery, and mathematical problem solving of students with varying abilities. Journal of Learning Disabilities, 39(6), 496–506.CrossRefGoogle Scholar
  48. Van Garderen, D., & Montague, M. (2003). Visual‐spatial representation, mathematical problem solving, and students of varying abilities. Learning Disabilities Research & Practice, 18(4), 246–254.CrossRefGoogle Scholar
  49. Van Garderen, D., Scheuermann, A., & Jackson, C. (2013). Examining how students with diverse abilities use diagrams to solve mathematics word problems. Learning Disability Quarterly, 36(3), 145–160.CrossRefGoogle Scholar
  50. Van Meter, P. (2001). Drawing construction as a strategy for learning from text. Journal of Educational Psychology, 93(1), 129.CrossRefGoogle Scholar
  51. Van Meter, P., Aleksic, M., Schwartz, A., & Garner, J. (2006). Learner-generated drawing as a strategy for learning from content area text. Contemporary Educational Psychology, 31(2), 142–166.CrossRefGoogle Scholar
  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.CrossRefGoogle Scholar
  53. Verschaffel, L., De Corte, E., Lasure, S., Vaerenbergh, G. V., 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.CrossRefGoogle Scholar
  54. Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse: Swets and Zeitlinger.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Johanna Rellensmann
    • 1
    Email author
  • Stanislaw Schukajlow
    • 1
  • Claudia Leopold
    • 2
  1. 1.Department of MathematicsUniversity of MünsterMünsterGermany
  2. 2.University of FribourgFribourgSwitzerland

Personalised recommendations