Educational Studies in Mathematics

, Volume 100, Issue 1, pp 43–60 | Cite as

Does constructing multiple solutions for real-world problems affect self-efficacy?

  • Stanislaw SchukajlowEmail author
  • Kay Achmetli
  • Katrin Rakoczy


The development of multiple solutions for a given problem is important for learning mathematics. In the present intervention study, we analyzed whether prompting students to construct multiple solutions (more precisely: prompting them to apply multiple mathematical procedures to real-world problems) and prior self-efficacy influenced students’ self-efficacy directly as well as indirectly via perceived competence. Students’ self-efficacy (N = 304) was measured before and after a 4-lesson teaching unit, and students’ perceived competence was measured during the unit. Results of the path model showed that although prompting multiple solutions did not positively affect self-efficacy, indirect effects of teaching method on self-efficacy were found. Students who were asked to develop multiple solutions perceived higher competence and reported higher self-efficacy than students who were required to provide one solution. These indirect effects were significant for students with low prior self-efficacy and nonsignificant for students with high prior self-efficacy, indicating the moderating effect of prior self-efficacy. This finding indicates that students with unfavorable learning prerequisites such as low self-efficacy might benefit from teaching methods that require them to construct multiple solutions. Further, students with low prior self-efficacy reported lower competence during the lessons regardless of whether they were asked to develop one or multiple solutions; they also reported lower self-efficacy at posttest prior self-efficacy was controlled for. Our findings therefore indicate that disadvantages for students with low prior self-efficacy for the further development of self-efficacy during learning might be balanced by teaching students to construct multiple solutions.


Multiple solutions Self-efficacy Mathematical modeling Real-world problems Word problems Teaching methods 


  1. Achmetli, K., Schukajlow, S., & Krug, A. (2014). Effects of prompting students to use multiple solution methods while solving real-world problems on students’ self-regulation. In C. Nicol, S. Oesterle, P. Liljedahl, & D. Allan (Eds.), Proceedings of the joint meeting of PME 38 and PME-NA 36 (Vol. 2, pp. 1–8). Vancouver: PME.Google Scholar
  2. Bandura, A. (1993). Perceived self-efficacy in cognitive development and functioning. Educational Psychologist, 28(2), 117–148.CrossRefGoogle Scholar
  3. Bandura, A. (2003). Self-efficacy: The exercise of control (6th ed.). New York: Freeman.Google Scholar
  4. Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J. Cho (Ed.), The proceedings of the 12th international congress on mathematical education (pp. 73–96). Cham: Springer.Google Scholar
  5. Blum, W., & Leiss, D. (2006). “Filling up” – the problem of independence-preserving teacher interventions in lessons with demanding modelling tasks. In M. Bosch (Ed.), CERME-4 – Proceedings of the fourth conference of the european eociety for research in mathematics education (pp. 1623–1633). Sant Feliu de Guíxols: Universitat Ramon Llull.Google Scholar
  6. Boaler, J., & Selling, S. K. (2017). Psychological imprisonment or intellectual freedom? A longitudinal study of contrasting school mathematics approaches and their impact on adults’ lives. Journal for Research in Mathematics Education, 48(1), 78–105.CrossRefGoogle Scholar
  7. Bonney, C. R., Kempler, T. M., Zusho, A., Coppola, B. P., & Pintrich, P. R. (2005). Student learning in science classrooms: What role does motivation play? In S. Alsop (Ed.), Beyond cartesian dualism encountering affect in the teaching and learning of science (pp. 83–97). Dordrecht: Springer.CrossRefGoogle Scholar
  8. Butz, A. R., & Usher, E. L. (2015). Salient sources of early adolescents’ self-efficacy in two domains. Contemporary Educational Psychology, 42, 49–61.CrossRefGoogle Scholar
  9. Deci, E. L., & Ryan, R. M. (2000). The “What” and “Why” of goal pursuits: Human needs and the self-determination of behavior. Psychological Inquiry, 11(4), 227–268.CrossRefGoogle Scholar
  10. Gravemeijer, K., Bruin-Muurling, G., Kraemer, J.-M., & van Stiphout, I. (2016). Shortcomings of mathematics education reform in The Netherlands: A paradigm case? Mathematical Thinking and Learning, 18(1), 25–44.CrossRefGoogle Scholar
  11. Hannula, M. S., Bofah, E., Tuohilampi, L., & Metsämuuronen, J. (2014). A longitudinal analysis of the relationship between mathematics-related affect and achievement in Finland. In C. Nicol, S. Oesterle, P. Liljedahl, & D. Allan (Eds.), Proceedings of the joint meeting of PME 38 and PME-NA 36 (Vol. 3, pp. 249–256). Vancouver: PME.Google Scholar
  12. Hänze, M., & Berger, R. (2007). Cooperative learning, motivational effects, and student characteristics: An experimental study comparing cooperative learning and direct instruction in 12th grade physics classes. Learning and Instruction, 17(1), 29–41.CrossRefGoogle Scholar
  13. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., … Stigler, J. (2003). Teaching mathematics in seven countries. In Results from the TIMSS 1999 video study. Washington, DC: NCES.Google Scholar
  14. 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
  15. Kline, R. B. (2005). Principles and practice of structural equation modeling. New York, NY: Guilford Press.Google Scholar
  16. Levav-Waynberg, A., & Leikin, R. (2012). The role of multiple solution tasks in developing knowledge and creativity in geometry. Journal of Mathematical Behavior, 31, 73–90.CrossRefGoogle Scholar
  17. MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7(1), 83–104.CrossRefGoogle Scholar
  18. MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39(1), 99–128.CrossRefGoogle Scholar
  19. Marsh, H. W., Pekrun, R., Parker, P. D., Murayama, K., Guo, J., Dicke, T., & Lichtenfeld, S. (2017). Long-term positive effects of repeating a year in school: Six-year longitudinal study of self-beliefs, anxiety, social relations, school grades, and test scores. Journal of Educational Psychology, 109(3), 425–438.CrossRefGoogle Scholar
  20. Marsh, H. W., Pekrun, R., Parker, P. D., Murayama, K., Guo, J., Dicke, T., & Arens, A. K. (2018). The murky distinction between self-concept and self-efficacy: Beware of lurking jingle-jangle fallacies. Journal of Educational Psychology.
  21. Muthén, B., Muthén, L., & Asparouhov, T. (2016). Regression and mediation analysis using Mplus. Los Angeles, CA: Muthén & Muthén.Google Scholar
  22. Muthén, L. K., & Muthén, B. O. (1998–2016). Mplus user’s guide (5th ed.). Los Angeles, CA: Muthén & Muthén.Google Scholar
  23. Niss, M. (1996). Goals of mathematics teaching. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 11–47). Dordrecht: Springer.Google Scholar
  24. 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
  25. Pajares, F., & Miller, M. D. (1994). Role of self-efficacy and self-concept beliefs in mathematical problem solving: A path analysis. Journal of Educational Psychology, 86(2), 193–203.CrossRefGoogle Scholar
  26. Pantziara, M. (2016). Student self-efficacy beliefs. In G. A. Goldin, M. S. Hannula, E. Heyd-Metzuyanim, A. Jansen, R. Kaasila, S. Lutovac, P. Di Martino, F. Morselli, J. A. Middleton, M. Pantziara, & Q. Zhang (Eds.), Attitudes, beliefs, motivation, and identity in mathematics education (pp. 7–11). Heidelberg: Springer.Google Scholar
  27. Pantziara, M., & Philippou, G. N. (2015). Students’ motivation in the mathematics classroom. Revealing causes and consequences. International Journal of Science and Mathematics Education, 13(2), 385–411.CrossRefGoogle Scholar
  28. Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36, 36–48.CrossRefGoogle Scholar
  29. Peugh, J. L., & Enders, C. K. (2004). Missing data in educational research: A review of reporting practices and suggestions for improvement. Review of Educational Research, 74(4), 525–556.CrossRefGoogle Scholar
  30. Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574.CrossRefGoogle Scholar
  31. Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836–852.CrossRefGoogle Scholar
  32. Schafer, J. L., & Graham, J. W. (2002). Missing data: Our view of the state of the art. Psychological Methods, 7(2), 147–177.CrossRefGoogle Scholar
  33. Schukajlow, S., & Krug, A. (2014). Do multiple solutions matter? Prompting multiple solutions, interest, competence, and autonomy. Journal for Research in Mathematics Education, 45(4), 497–533.CrossRefGoogle Scholar
  34. 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
  35. 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
  36. Schukajlow, S., & Rakoczy, K. (2016). The power of emotions: Can enjoyment and boredom explain the impact of individual preconditions and teaching methods on interest and performance in mathematics? Learning and Instruction, 44, 117–127.CrossRefGoogle Scholar
  37. Schukajlow, S., Rakoczy, K., & Pekrun, R. (2017). Emotions and motivation in mathematics education: Theoretical considerations and empirical contributions. ZDM Mathematics Education, 49(3), 307–322.CrossRefGoogle Scholar
  38. Schütze, B., Rakoczy, K., Klieme, E., Besser, M., & Leiss, D. (2017). Training effects on teachers' feedback practice: The mediating function of feedbackknowledge and the moderating role of self-efficacy. ZDM Mathematics Education, 49, 475–489.CrossRefGoogle Scholar
  39. Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C. Y., & Font Strawhun, B. T. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24(3–4), 287–301.CrossRefGoogle Scholar
  40. Usher, E. L., & Pajares, F. (2008). Sources of self-efficacy in school: Critical review of the literature and future directions. Review of Educational Research, 78(4), 751–796.CrossRefGoogle Scholar
  41. Usher, E. L., & Pajares, F. (2009). Sources of self-efficacy in mathematics: A validation study. Contemporary Educational Psychology, 34, 89–101.CrossRefGoogle Scholar
  42. Van Den Heuvel-Panhuizen, M. (2010). Reform under attack–forty years of working on better mathematics education thrown on the scrapheap? No way! In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the future of mathematics Education: Proceedings of the 33rd annual conference of the Mathematics Education Research Group of Australasia (pp. 1–25). Fremantle, Australia: MERGA, Merga.Google Scholar
  43. Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse, the Netherlands: Swets and Zeitlinger.Google Scholar
  44. Verschaffel, L., Van Dooren, W., Greer, B., & Mukhopadhyay, S. (2010). Reconceptualising word problems as exercises in mathematical modelling. Journal für Mathematikdidaktik, 31, 9–29.CrossRefGoogle Scholar
  45. Zan, R., Brown, L., Evans, J., & Hannula, M. S. (2006). Affect in mathematics education: An introduction. Educational Studies in Mathematics, 63(2), 113–122.CrossRefGoogle Scholar
  46. Zuffianò, A., Alessandri, G., Gerbino, M., Kanacri, B. P. L., Di Giunta, L., Milioni, M., & Caprara, G. V. (2013). Academic achievement: The unique contribution of self-efficacy beliefs in self-regulated learning beyond intelligence, personality traits, and self-esteem. Learning and Individual Differences, 23, 158–162.CrossRefGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MünsterMünsterGermany
  2. 2.Center for Research on Educational Quality and Evaluation, German Institute for International Educational ResearchFrankfurt am MainGermany

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