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ZDM

, Volume 50, Issue 1–2, pp 343–354 | Cite as

Conceptualization and measuring of metacognitive modelling competencies: empirical verification of theoretical assumptions

  • Katrin Vorhölter
Original Article
First Online:

Abstract

Metacognitive competencies are of great importance for developing modelling competencies. However, there are assumptions about useful metacognitive knowledge and strategies for individuals working on modelling problems as well as for whole groups, but their coherence as well as their influence on modelling processes is not evaluated satisfactorily. Furthermore, there exist different conceptualizations of metacognition. In this paper, the structure of metacognitive strategies used by 431 grade nine students is analyzed. Strategy use was measured via self-reports at individual as well as at group level. The results reveal the same structure for metacognitive strategies at individual and at group level. These metacognitive strategies can be differentiated into strategies ensuring a smooth modelling process, strategies for regulating when problems occur, and strategies for evaluating the whole modelling process.

Keywords

Modelling competencies Metacognition Strategies Social metacognition 
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References

  1. Aebli, H. (1997). Zwölf Grundformen des Lehrens: Eine allgemeine Didaktik auf psychologischer Grundlage. Stuttgart: Klett-Cotta.Google Scholar
  2. Artelt, C. (2000). Strategisches Lernen. Münster. Potsdam: Waxmann.Google Scholar
  3. Artelt, C., Schiefele, U., & Schneider, W. (2001). Predictors of reading literacy. European Journal of Psychology of Education, 16(3), 363–383.CrossRefGoogle Scholar
  4. Artzt, A. F., & Armour-Thomas, E. (1992). Development of a cognitive-metacognitive framework for protocol analysis of mathematical problem solving in small groups. Cognition and Instruction, 9(2), 137–175.CrossRefGoogle Scholar
  5. Baten, E., Praet, M., & Desoete, A. (2017). The relevance and efficacy of metacognition for instructional design in the domain of mathematics. ZDM Mathematics Education, 49(4), 613–623.CrossRefGoogle Scholar
  6. 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.), International perspectives on the teaching and learning of mathematical modelling, trends in teaching and learning of mathematical modelling (pp. 15–30). Dordrecht: Springer.CrossRefGoogle Scholar
  7. Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J. Cho (Eds.), The proceedings of the 12th International congress on mathematical education (pp. 73–96). Cham: Springer.Google Scholar
  8. Brand, S. (2014). Erwerb von Modellierungskompetenzen: Empirischer Vergleich eines holistischen und eines atomistischen Ansatzes zur Förderung von Modellierungskompetenzen. Wiesbaden: Springer Fachmedien Wiesbaden.CrossRefGoogle Scholar
  9. Brown, A. L. (1978). Knowing when, where, and how to remember: A problem of metacognition. In R. Glaser (Ed.), Advances in instructional psychology (pp. 77–165). Hillsdale: Erlbaum.Google Scholar
  10. Bühner, M. (2011). Einführung in die Test- und Fragebogenkonstruktion. München. Boston [u.a.]: Pearson Studium.Google Scholar
  11. Busse, A., & Borromeo Ferri, R. (2003). Methodological reflections on a three-step-design combining observation, stimulated recall and interview. Zentralblatt für Didaktik der Mathematik, 35(6), 257–264.CrossRefGoogle Scholar
  12. Chalmers, C. (2009). Group metacognition during mathematical problem solving. In R. K. Hunter, B. A. Bicknell, & T. A. Burgess (Eds.), Crossing divides. MERGA 32 conference proceedings (pp. 105–111). Palmerston North: MERGA.Google Scholar
  13. Cooke, N. J., Salas, E., Kiekel, P. A., & Bell, B. (2004). Advances in measuring team cognition. In E. Salas & S. M. Fiore (Eds.), Team cognition: Understanding the factors that drive process and performance (pp. 83–106). Washington, DC: American Psychological Assoc.CrossRefGoogle Scholar
  14. Flavell, J. H. (1976). Metacognitive aspects of problem-solving. In L. B. Resnick (Ed.), The nature of intelligence (pp. 231–235). Hillsdale: Erlbaum.Google Scholar
  15. Flavell, J. H. (1979). Metacognition and cognitive monitoring: A new area of cognitive-developmental inquiry. American Psychologist, 34(10), 906–911.CrossRefGoogle Scholar
  16. Flavell, J. H., Miller, P. H., & Miller, S. A. (1993). Cognitive development. Englewood Cliffs: Prentice Hall.Google Scholar
  17. Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16(3), 163.CrossRefGoogle Scholar
  18. Goos, M. (2002). Understanding metacognitive failure. Journal of Mathematical Behavior, 21(3), 283–302.CrossRefGoogle Scholar
  19. Goos, M., & Galbraith, P. (1996). Do it this way! Metacognitive strategies in collaborative mathematical problem solving. Educational Studies in Mathematics, 30(3), 229–260.CrossRefGoogle Scholar
  20. Greefrath, G., & Vorhölter, K. (2016). Teaching and learning mathematical modelling: Approaches and developments from German speaking countries. Cham: Springer.CrossRefGoogle Scholar
  21. Hasselhorn, M. (1992). Metakognition und Lernen. In G. Nold (Ed.), Lernbedingungen und Lernstrategien: Welche Rolle spielen kognitive Verstehensstrukturen? (pp. 35–63). Tübingen: Narr.Google Scholar
  22. Herget, W., Jahnke, T., & Kroll, W. (2001). Produktive Aufgaben für den Mathematikunterricht in der Sekundarstufe I. Berlin: Cornelsen.Google Scholar
  23. Kaiser, G. (2007). Modelling and modelling competencies in school. In C. Haines (Ed.), Mathematical modelling (ICTMA 12) (pp. 110–119). Chichester: Horwood Publishing.CrossRefGoogle Scholar
  24. Kaiser, G., & Brand, S. (2015). Modelling competencies: Past development and further perspectives. In G. Stillman, W. Blum & M. Salett Biembengut (Eds.), Mathematical modelling in education research and practice (pp. 129–149). Cham: Springer.CrossRefGoogle Scholar
  25. Kaiser, G., & Stender, P. (2013). Complex modelling problems in co-operative, self-directed learning environments. In G. Stillman, G. Kaiser, W. Blum & J. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 277–293). Dordrecht: Springer.CrossRefGoogle Scholar
  26. Lingel, K., Götz, L., Artelt, C., & Schneider, W. (2014). Mathematisches Strategiewissen für fünfte und sechste Klassen: MAESTRA 5–6+; Manual. Göttingen: Hogrefe.Google Scholar
  27. Maaß, K. (2006). What are modelling competencies? ZDM, 38(2), 113–142.CrossRefGoogle Scholar
  28. Maaß, K. (2007). Modelling in class: What do we students want to learn? In C. Haines, P. Galbraith, W. Blum & S. Khan (Eds.), Mathematical modelling (ICTMA 12) (pp. 63–78). Chichester: Horwood Publishing.CrossRefGoogle Scholar
  29. Ng, K. E. D. (2010). Partial metacognitive blindness in collaborative problem solving. In: L. Sparrow, B. Kissane & C. Hurst (Eds.), Shaping the future of mathematics education: MERGA 33 conference proceedings (Volume 2, pp. 446–453). Freemantle, Australia.Google Scholar
  30. Pressley, M., Borkowski, J., & 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
  31. Rakoczy, K., & Klieme, E. (2005). Dokumentation der Erhebungs- und Auswertungsinstrumente zur schweizerisch-deutschen Videostudie. “Unterrichtsqualität, Lernverhalten und mathematisches Verständnis”: 1. Befragungsinstrumente. Frankfurt am Main: GFPF [u.a.].Google Scholar
  32. Schellings, G., & van Hout-Wolters, B. (2011). Measuring strategy use with self-report instruments: theoretical and empirical considerations. Metacognition and Learning, 6(2), 83–90.CrossRefGoogle Scholar
  33. Schraw, G., & Moshman, D. (1995). Metacognitive theories. Educational Psychology Review, 7(4), 351–371.CrossRefGoogle Scholar
  34. Schukajlow, S., & Krug, A. (2013). Planning, monitoring and multiple solutions while solving modelling problems. In A. Lindmeier & A. Heinze (Eds.), Mathematics learning across the life span: Proceedings of the 37th conference of the International Group for the Psychology of Mathematics Education (pp. 177–184). Kiel: IPN Leibniz Inst. for Science and Mathematics Education.Google Scholar
  35. Schukajlow, S., & Leiss, D. (2011). Selbstberichtete Strategienutzung und mathematische Modellierungskompetenz. Journal für Mathematikdidaktik, 32, 53–77.CrossRefGoogle Scholar
  36. Scott, B. M., & Levy, M. G. (2013). Metacognition: examining the components of a fuzzy concept. Educational Research eJournal, 2(2), 120–131.CrossRefGoogle Scholar
  37. Siegel, M. A. (2012). Filling in the distance between us: Group metacognition during problem solving in a secondary education course. Journal of Science Education and Technology, 21(3), 325–341.CrossRefGoogle Scholar
  38. Sjuts, J. (2003). Metakognition per didaktisch-sozialem Vertrag. Journal für Mathematikdidaktik, 24(1), 18–40.CrossRefGoogle Scholar
  39. Stillman, G. (2004). Strategies employed by upper secondary students for overcoming or exploiting conditions affecting accessibility of applications tasks. Mathematics Education Research Journal, 16(1), 41–71.CrossRefGoogle Scholar
  40. Stillman, G. (2011). Applying metacognitive knowledge and strategies in applications and modelling tasks at secondary school. In G. Kaiser, W. Blum, R. Borromeo Ferri & G. Stillman (Eds.), International perspectives on the teaching and learning of mathematical modelling, Trends in teaching and learning of mathematical modelling (pp. 165–180). Dordrecht: Springer.CrossRefGoogle Scholar
  41. Stillman, G., & Galbraith, P. (1998). Applying mathematics with realworld connections: metacognitive characteristics of secondary students. Educational Studies in Mathematics, 36(2), 157–194.CrossRefGoogle Scholar
  42. Veenman, M. (2005). The assessment of metacognitive skills: What can be learned from multi-method designs? In C. Artelt & B. Moschner (Eds.), Lernstrategien und Metakognition: Implikationen für Forschung und Praxis (pp. 77–99). Münster: Waxmann.Google Scholar
  43. Veenman, M. (2006). The role of intellectual and metacognitive skills. In A. Desoete & M. Veenman (Eds.), Metacognition in mathematics education (pp. 35–50). New York: Nova Science Publishers.Google Scholar
  44. Veenman, M. (2011). Alternative assessment of strategy use with self-report instruments: a discussion. Metacognition and Learning, 6(2), 205–211.CrossRefGoogle Scholar
  45. Vorhölter, K. (2017). Measuring metacognitive modelling competencies. In G. Stillman, W. Blum & G. Kaiser (Eds.), Mathematical modelling and applications: Crossing and researching boundaries in mathematics education (pp. 175–185). Cham: Springer.CrossRefGoogle Scholar
  46. Vorhölter, K. (accepted). Metacognitive modelling competencies in small groups. In: T. Dooley, T. & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education. Dublin, Ireland: DCU Institute of Education and ERME.Google Scholar
  47. Vorhölter, K., Krüger, A., & Wendt, L. (submitted). Metacognition in mathematical modeling–an overview. In S. Chamberlain & B. Sriraman (Eds.), Affect and mathematical modeling.Google Scholar
  48. Weinert, F. E. (2001). Leistungsmessungen in Schulen. Weinheim: Beltz.Google Scholar

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© FIZ Karlsruhe 2018

Authors and Affiliations

  1. 1.University of HamburgHamburgGermany

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