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Comparing German and Taiwanese secondary school students’ knowledge in solving mathematical modelling tasks requiring their assumptions

  • Yu-Ping Chang
  • Janina Krawitz
  • Stanislaw Schukajlow
  • Kai-Lin YangEmail author
Original Article


Mathematization is critical in providing students with challenges for solving modelling tasks. Inadequate assumptions in a modelling task lead to an inadequate situational model, and to an inadequate mathematical model for the problem situation. However, the role of assumptions in solving modelling problems has been investigated only rarely. In this study, we intentionally designed two types of assumptions in two modelling tasks, namely, one task that requires non-numerical assumptions only and another that requires both non-numerical and numerical assumptions. Moreover, conceptual knowledge and procedural knowledge are also two factors influencing students’ modelling performance. However, current studies comparing modelling performance between Western and non-Western students do not consider the differences in students’ knowledge. This gap in research intrigued us and prompted us to investigate whether Taiwanese students can still perform better than German students if students’ mathematical knowledge in solving modelling tasks is differentiated. The results of our study showed that the Taiwanese students had significantly higher mathematical knowledge than did the German students with regard to either conceptual knowledge or procedural knowledge. However, if students of both countries were on the same level of mathematical knowledge, the German students were found to have higher modelling performance compared to the Taiwanese students in solving the same modelling tasks, whether such tasks required non-numerical assumptions only, or both non-numerical and numerical assumptions. This study provides evidence that making assumptions is a strength of German students compared to Taiwanese students. Our findings imply that Western mathematics education may be more effective in improving students’ ability to solve holistic modelling problems.


Modelling tasks Making assumptions Mathematical knowledge 



This study was conducted in the framework of the TaiGer program. The meetings of the German and Taiwanese partners were funded by Deutsche Forschungsgemeinschaft (DFG, no. HE 4561/10-1) and Ministry of Science and Technology Taiwan (MOST, No. MOST 105-2911-I-003 -513), allocated to Aiso Heinze (IPN Kiel, Germany) and Kai-Lin Yang (NTNU Taipei, Taiwan) respectively. The authors thank the anonymous reviewers for their valuable comments and suggestions.


  1. Ärlebäck, J. B. (2009). On the use of realistic Fermi problems for introducing mathematical modelling in school. The Mathematics Enthusiast, 6(3), 331–364.Google Scholar
  2. Blum, W. (2011). Can modelling be taught and learnt? Some answers from empirical research. Trends in teaching and learning of mathematical modelling (pp. 15–30). Dordrecht: Springer.Google Scholar
  3. Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In J. S. Cho (Ed.), Proceedings of the 12th international congress on mathematical education (pp. 73–96). New York, NY: Springer.Google Scholar
  4. Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt? Journal of Mathematical Modelling and Application, 1(1), 45–58.Google Scholar
  5. Blum, W., & Leiß, D. (2007). How do studnets and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling: Education, engineering and economics—ICTMA12 (pp. 222–231). Chichester, UK: Horwood.Google Scholar
  6. Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects—State, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22, 37–68.Google Scholar
  7. Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. ZDM Mathematics Education, 38(2), 86–95.Google Scholar
  8. Borromeo Ferri, R. (2007). Modelling problems from a cognitive perspective. In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling: Education, engineering and economics—ICTMA12 (pp. 260–270). Chichester, UK: Horwood.Google Scholar
  9. Chang, Y.-P. (2014). Opportunities to learn mathematical proofs in geometry: Comparative analyses of textbooks from Germany and Taiwan. Riga: LAP Lampert Academic.Google Scholar
  10. Chang, Y.-P., Lin, F.-L., & Reiss, K. (2013). How do students learn mathematical proof? A comparison of geometry designs in German and Taiwanese textbooks. In C. Margolinas, et al. (Eds.), ICMI study 22: Task design in mathematics education (pp. 305–314). Oxford, UK: ICMI.Google Scholar
  11. Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Earlbaum Associates.Google Scholar
  12. Degrande, T., Van Hoof, J., Verschaffel, L., & Van Dooren, W. (2018). Open word problems: Taking the additive or the multiplicative road? ZDM Mathematics Education, 50, 91–102.Google Scholar
  13. Djepaxhija, B., Vos, P. & Fuglestad, A. B. (2015). Exploring grade 9 students’ assumption making when mathematizing. Paper presented at the Ninth Congress of European Research in Mathematics Education, Prague.Google Scholar
  14. Frejd, P. (2013). Modes of modelling assessment—A literature review. Educational Studies in Mathematics, 84, 413–438.Google Scholar
  15. Galbraith, P., & Haines, C. (2001). Conceptual and procedural demands embedded in modelling tasks. In J. F. Matos, W. Blum, K. Houston, & S. P. Carreira (Eds.), Modelling and mathematics education: ICTMA 9: Applications in science and technology (pp. 342–353). Chichester: Horwood Publishing.Google Scholar
  16. Galbraith, P. L., & Stillman, G. (2001). Assumptions and context: Pursuing their role in modelling activity. In J. Matos, W. Blum, K. Houston, & S. Carreira (Eds.), Modelling and mathematics education: ICTMA 9: Applications in science and technology (pp. 300–310). Chicester, UK: Horwood Publishing.Google Scholar
  17. Galbraith, P. L., & Stillman, G. (2006). A framework for identifying student blockages during transitions in the modelling process. ZDM Mathematics Education, 38(2), 143–162.Google Scholar
  18. Geary, D. C., Hoard, M. K., Nugent, L., & Bailey, D. H. (2013). Adolescents’ functional numeracy is predicted by their school entry number system knowledge. PLoS One, 8(1), e54651.Google Scholar
  19. Greer, B. (1997). Modelling reality in mathematics classrooms: The case of word problems. Learning and Instruction, 7(4), 293–307.Google Scholar
  20. Heinze, A., Cheng, Y.-H., & Yang, K.-L. (2004). Students’ performance in reasoning and proof in Taiwan and Germany: Results, paradoxes and open questions. ZDM Mathematics Education, 36(5), 162–171.Google Scholar
  21. Jonassen, D. H. (2000). Toward a design theory of problem solving. Educational Technology Research and Development, 48(4), 63–85.Google Scholar
  22. Joram, E., Gabriele, A., Bertheau, M., Gelman, R., & Subrahmanyam, K. (2005). Children’s use of the reference point strategy for measurement estimation. Journal for Research in Mathematics Education, 36(1), 4–23.Google Scholar
  23. Krawitz, J., Schukajlow, S., & van Dooren, W. (2018). Unrealistic responses to realistic problems with missing information: What are important barriers? Educational Psychology, 38, 1221–1238.Google Scholar
  24. Li, H.-C. (2014). A comparative analysis of British and Taiwanese students’ conceptual and procedural knowledge of fraction addition. International Journal of Mathematical Education in Science and Technology, 45(7), 968–979.Google Scholar
  25. Lin, F.-L., & Yang, K.-L. (2005). Distinctive characteristics of mathematical thinking in non-modelling friendly environment. Teaching Mathematics and Its Applications, 24(2–3), 97–106.Google Scholar
  26. Lyons, I. M., Price, G. R., Vaessen, A., Blomert, L., & Ansari, D. (2014). Numerical predictors of arithmetic success in grades 1–6. Developmental Science, 17(5), 714–726.Google Scholar
  27. Maaß, K. (2006). What are modelling competencies? ZDM Mathematics Education, 38(2), 113–142.Google Scholar
  28. Maaß, K. (2010). Classification scheme for modelling tasks. Journal für Mathematik-Didaktik, 31(2), 285–311.Google Scholar
  29. Morris, M. W., & Leung, K. (2010). Creativity East and West: Perspectives and parallels. Management and Organization Review, 6(3), 313–327.Google Scholar
  30. National Research Council. (2001). Adding it up: Helping children learn mathematics. J Kilpatrick, J. Swafford, and B. Findell (Eds.). Washington, DC: National Academy Press.Google Scholar
  31. Niss, M. (2003). Quantitative literacy and mathematical competencies. In B. L. Madison & L. A. Steen (Eds.), Quantitative literacy: Why numeracy matters for schools and colleges (pp. 215–220). Princeton, NJ: National Council on Education and the Disciplines.Google Scholar
  32. Niss, M. (2015). Mathematical competencies and PISA. In K. Stacey & R. Turner (Eds.), Assessing mathematical literacy: The PISA experience (pp. 35–55). Cham: Springer.Google Scholar
  33. Organisation for Economic Co-operation and Development (OECD). (2003). The PISA 2003 assessment framework—Mathematics, reading, science and problem solving knowledge and skills. Paris: OECD.Google Scholar
  34. Organisation for Economic Co-operation and Development (OECD). (2016). PISA 2015 results in focus. Paris: OECD.Google Scholar
  35. Organisation for Economic Co-operation and Development (OECD). (2017). PISA 2015 Assessment and Analytical framework: Science, reading, mathematics, financial literacy and collaborative problem solving (revised edition). Paris: OECD.Google Scholar
  36. Peter-Koop, A. (2004). Fermi problems in primary mathematics classrooms: Pupils’ interactive modelling processes. In I. Putt, R. Faragher, & M. McLean (Eds.), Mathematics education for the third millennium: Towards 2010 (Proceedings of the 27th annual conference of the Mathematics Education Research Group of Australasia, Townsville) (pp. 454–461). Sydney: MERGA.Google Scholar
  37. Renkl, A., Mandl, H., & Gruber, H. (1996). Inert knowledge: Analyses and remedies. Educational Psychologist, 31(2), 115–121.Google Scholar
  38. Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. In R. C. Kadosh & A. Dowker (Eds.), Oxford handbook of numerical cognition (pp. 1118–1134). New York: Oxford University Press.Google Scholar
  39. Schukajlow, S., Achmetli, K., & Rakoczy, K. (2019). Does constructing multiple solutions for real-world problems affect self-efficacy? Educational Studies in Mathematics, 100, 43–60.Google Scholar
  40. 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.Google Scholar
  41. 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.Google Scholar
  42. Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., et al. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697.Google Scholar
  43. Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.Google Scholar
  44. Stillman, G., Galbraith, P., Brown, J., & Edwards, I. (2007). A framework for success in implementing mathematical modelling in the secondary classroom. In J. Watson & K. Beswick (Eds.), The Proceedings of the 30th annual conference of the Mathematics Education Group of Australasia (pp. 688–697). MERGA: Tasmania.Google Scholar
  45. Stillman, G. A., Kaiser, G., Blum, W., & Brown, J. P. (2013). Teaching mathematical modelling: Connecting to research and practice. Dordrecht: Springer.Google Scholar
  46. Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse: Swets & Zeitlinger.Google Scholar
  47. Verschaffel, L., Luwel, K., Torbeyns, J., & van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335–359.Google Scholar
  48. Watts, T. W., Duncan, G. J., Siegler, R. S., & Davis-Kean, P. E. (2014). What’s past is prologue: Relations between early mathematics knowledge and high school achievement. Educational Researcher, 43(7), 352–360.Google Scholar

Copyright information

© FIZ Karlsruhe 2019

Authors and Affiliations

  1. 1.National Pingtung UniversityPingtung CityTaiwan
  2. 2.University of MünsterMünsterGermany
  3. 3.National Taiwan Normal UniversityTaipei CityTaiwan

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