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ZDM

, Volume 49, Issue 6, pp 923–935 | Cite as

Hybrid task design: connecting learning opportunities related to critical thinking and statistical thinking

  • Sebastian Kuntze
  • Einav Aizikovitsh-Udi
  • David Clarke
Original Article
First Online:

Abstract

Stimulating thinking related to mathematical content is the focus of many tasks in the mathematics classroom. Beyond such content-related thinking, promoting forms of higher order thinking is among the goals of mathematics instruction as well. So-called hybrid tasks focus on combining both goals: they aim at fostering mathematical thinking and higher order thinking through the same mathematical activities—an aim which requires empirical examination. For empirically valid hybrid task design, evidence is required about the nature of the interrelatedness of content-related and higher order thinking. In this article, we choose the example of statistical thinking and critical thinking for exploring the interrelatedness of these different modes of thinking when solving hybrid tasks. Even if theories about statistical thinking and critical thinking have so far followed almost separate strands, there are commonalities at the theoretical level, which facilitate hybrid task design. We report an empirical study, in which a bottom–up analysis of thinking-aloud interviews with adult learners around solution processes of hybrid tasks affords insight into how these modes of thinking may interact when solving such tasks. The results show that the tasks did evoke both statistical thinking and critical thinking: Both modes of thinking entered in an interplay in which we observed instances of mutual support of both modes of thinking, but also cases in which a strong focus on statistical thinking or critical thinking appeared to impede the other mode of thinking, respectively. The findings can inform the further development of hybrid tasks: Based on the observations, the task format can be enriched with specific reflective stimuli intended to support a fruitful interplay of statistical thinking and critical thinking.

Keywords

Hybrid tasks Critical thinking Statistical thinking Scientific reasoning Task design 
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References

  1. Ainley, J., & Margolinas, C. (2015). Accounting for student perspectives in task design. In A. Watson & M. Ohtani (Eds.), Task design in mathematics education (pp. 115–142). New York: Springer.CrossRefGoogle Scholar
  2. Aizikovitsh-Udi, E. (2012). Developing critical thinking through probability models, intuitive judgments and decision-making under uncertainty. Doctoral dissertation. Saarbrucken: Lambert Academic Publishing.Google Scholar
  3. Aizikovitsh-Udi, E., Kuntze, S., & Clarke, D. (2012). Connections between statistical thinking and critical thinking – a case study. In D. Ben-Zvi & K. Makar (Eds.), Teaching and learning of statistics. Proceedings of Topic Study Group 12, 12th International Congress on Mathematical Education (ICME 12). July 8–15, 2012, Seoul, Korea.Google Scholar
  4. Aizikovitsh-Udi, E., & Amit, M. (2008). Developing critical thinking in probability lesson. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano & A. Sepúlveda (Eds.), Proceedings of the 32nd Conf. of the IGPME (Vol. 2, pp. 9–13). Morelia: Universidad Michoacana de San Nicolás de Hidalgo.Google Scholar
  5. Aizikovitsh-Udi, E., Clarke, D. J., & Kuntze, S. (2013). Hybrid tasks: Promoting statistical thinking and critical thinking through the same mathematical activities. In A. Watson, et al. (Eds.), Proceedings of ICMI Study 22 Task Design in Mathematics Education (pp. 457–467). Oxford: ICMI.Google Scholar
  6. Aizikovitsh-Udi, E., & Kuntze, S. (2014). Critical thinking as an impact factor on statistical literacy. In K. Makar, B. de Sousa & R. Gould (Eds.), Proceedings of ICOTS9. Voorburg: ISI.Google Scholar
  7. Blomhøj, M., & Jensen, T. H. (2003). Developing mathematical modelling competence: Conceptual clarification and educational planning. Teaching Mathematics and its Applications, 22(3), 123–139.CrossRefGoogle Scholar
  8. Blum, W. (2007). Mathematisches Modellieren – zu schwer für Schüler und Lehrer? Beiträge zum Mathematikunterricht 2007, 3–12.Google Scholar
  9. Blum, W., Galbraith, P., Niss, M., & Henn, H.-W. (Eds.). (2007). Modelling and applications in mathematics education. New ICMI studies series no. 10. New York: Springer.Google Scholar
  10. Blum, W., & Leiss, D. (2005). Modellieren im Unterricht mit der “Tanken”-Aufgabe. Mathematik Lehren, 128, 18–21.Google Scholar
  11. Bullock, M., & Ziegler, A. (1994). Scientific thinking. In F. Weinert & W. Schneider (Eds.), The Munich longitudinal study on the genesis of Ind. competencies. Munich: MPI.Google Scholar
  12. Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the educative process. Boston: Heath.Google Scholar
  13. Ennis, R. H. (1987). A taxonomy of critical thinking dispositions and abilities. In J. B. Baron & R. J. Sternberg (Eds.), Teaching thinking skills: Theory and practice (pp. 9–26). New York: Freeman.Google Scholar
  14. Ennis, R. H. (2002). Goals for a critical thinking curriculum and its assessment. In A. L. Costa (Ed.), Developing minds (3rd edn., pp. 44–46). Alexandria: Association for Supervision and Curriculum Development.Google Scholar
  15. Ennis, R. H., & Millman, J. (2005). Cornell critical thinking test, level Z (5th edn.). Seaside: The Critical Thinking Company.Google Scholar
  16. Ennis, R. R. (1989). Critical thinking and subject specificity: Clarification and needed research. Educational Researcher, 18, 4–10.CrossRefGoogle Scholar
  17. Gal, I. (2004). Statistical literacy, meanings, components, responsibilities. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 47–78). Dordrecht: Kluwer.CrossRefGoogle Scholar
  18. Innabi, H., & Sheikh, O. (2007). The change in mathematics teachers’ perceptions of critical thinking after 15 years of educational reform in Jordan. Educational Studies in Mathematics, 64(1), 45–68.CrossRefGoogle Scholar
  19. Klahr, D., & Dunbar, K. (1989). Developmental differences in scientific discovery processes. In D. Klahr & K. Kotovsky. (Eds.), Complex information processing (pp. 109–143). Hillsdale: Erlbaum.Google Scholar
  20. Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33(4), 259–289.CrossRefGoogle Scholar
  21. Kröpfl, B., Peschek, W., & Schneider, E. (2000). Stochastik in der Schule: Globale Ideen, lokale Bedeutungen, zentrale Tätigkeiten. Mathematica Didactica, 23, 25–57.Google Scholar
  22. Kuhn, D. (1989). Children and adults as intuitive scientists. Psychological Review, 96, 674–689.CrossRefGoogle Scholar
  23. Kuhn, D., Amsel, E., & O’Loughlin, M. (1988). The development of scientific thinking skills. San Diego: Academic Press.Google Scholar
  24. Kuntze, S. (2004). Wissenschaftliches Denken von Schülerinnen und Schülern bei der Beurteilung gegebener Beweisbeispiele aus der Geometrie. Journal für Mathematik-Didaktik, 25(3/4), 245–268.CrossRefGoogle Scholar
  25. Kuntze, S. (2010). Zur Beschreibung von Kompetenzen des mathematischen Modellierens konkretisiert an inhaltlichen Leitideen (Describing competencies of mathematical modeling with respect of core ideas in specific content domains). MU, 56(4), 4–19.Google Scholar
  26. Kuntze, S. (2013). Modellieren beim Nutzen von Darstellungen in statistischen Kontexten. In R. Borromeo Ferri, G. Greefrath & G. Kaiser (Eds.), Mathematisches Modellieren in Schule und Hochschule (pp. 71–94). Wiesbaden: Springer Spektrum.CrossRefGoogle Scholar
  27. Kuntze, S., Aizikovitsh-Udi, E., & Clarke, D. (2013). Strategies for evaluating claims—an aspect that links critical thinking and statistical thinking. In A. Lindmeier & A. Heinze. (Eds.), Proceedings of the 37th Conf. of the IGPME (Vol. 5, p. 235). Kiel: PME.Google Scholar
  28. Kuntze, S., Lindmeier, A. & Reiss, K. (2008). “Using models and representations in statistical contexts” as a sub-competency of statistical literacy—results from three empirical studies. Proceedings of ICME 11. http://tsg.icme11.org/document/get/474. Accessed June 26 2017.
  29. Kuntze, S., Martignon, L., Vargas, F., & Engel, J. (2015). Competencies in understanding statistical information in primary and secondary school levels. AIEM Avances de Investigación en Educación Matemática, 7, 5–25.Google Scholar
  30. Lesh, R., Galbraith, P., Haines, C., & Hurford, A. (Eds.). (2010). Modeling students’ mathematical modeling competencies. New York: Springer.Google Scholar
  31. Lipman, M. (1991). A functional definition of critical thinking. Thinking in Education, 6, 114–125.Google Scholar
  32. Maaß, K. (2006). What are modelling competencies? ZDM, 38(2), 115–118.CrossRefGoogle Scholar
  33. Mayring, P. (2015). Qualitative Inhaltsanalyse. [Qualitative content analysis]. Weinheim: Beltz.Google Scholar
  34. McPeck, J. (1981). The meaning of critical thinking. Critical Thinking and Education, 1, 1–21.Google Scholar
  35. Neubrand, J. (2002). Eine Klassifikation mathematischer Aufgaben zur Analyse von Unterrichtssituationen. Hildesheim: Franzbecker.Google Scholar
  36. OECD. (2003). The PISA 2003 assessment framework. Retrieved January 20, 2007, from http://www.pisa.oecd.org/dataoecd/46/14/33694881.pdf.
  37. OECD. (2012), Education at a glance 2012: OECD indicators. OECD Publishing. Retrieved October 12, 2016, from doi: 10.1787/eag-2012-en.
  38. Reiss, K., & Thomas, J. (2000). Wissenschaftliches Denken beim Beweisen in der Geometrie. Mathematica Didactica, 23, 96–112.Google Scholar
  39. Royalty, J. (1995). The generalizability of critical thinking: Paranormal beliefs versus statistical reasoning. The Journal of Genetic Psychology, 156(4), 477–488.CrossRefGoogle Scholar
  40. Schreier, M. (2012). Qualitative content analysis in practice. London: Sage.Google Scholar
  41. Shaughnessy, J. M. (2007). Research on statistics learning and reasoning. In F. K. Lester (Ed.), The second handbook of research on mathematics teaching and learning (pp. 957–1010). Charlotte: Information Age Publishing.Google Scholar
  42. Siegel, H. (1988). Educating reason: Rationality, critical thinking and education. New York: Routledge.Google Scholar
  43. Thomas, J. (1997). Wissenschaftliches Denken im Jugendalter. Mainz: Universität.Google Scholar
  44. Tschirigi, J. (1980). Sensible reasoning: A hypothesis about hypotheses. Child Development, 51, 1–10.CrossRefGoogle Scholar
  45. Watson, J., & Callingham, R. (2003). Statistical literacy: A complex hierarchical construct. Statistics Education Research Journal, 2(2), 3–46.Google Scholar
  46. Watson, J. M. (1997). Assessing statistical thinking using the media, In I. Gal & J. Gar-field (Eds.), The assessment challenge in statistics education (pp. 107–121). IOS Press.Google Scholar
  47. Weinert, F. (1996). Lerntheorien und Instruktionsmodelle. In F. Weinert (Ed.), Enzyklopädie der Psychologie. Vol. 2 (pp. 1–48). Göttingen: Hogrefe.Google Scholar
  48. Weinert, F. (2001). Vergleichende Leistungsmessung in Schulen. In F. Weinert (Ed.), Leistungsmessungen in Schulen (pp. 17–31). Weinheim: Beltz.Google Scholar
  49. Wild, C., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 3, 223–266.CrossRefGoogle Scholar
  50. Willingham, D. (2007). Critical Thinking—Why is it so hard to teach? American Educator, 2007(3), 8–19.Google Scholar

Copyright information

© FIZ Karlsruhe 2017

Authors and Affiliations

  1. 1.Ludwigsburg University of EducationLudwigsburgGermany
  2. 2.Davidson InstituteWeizmann Institute of ScienceRehovotIsrael
  3. 3.International Centre for Classroom ResearchUniversity of MelbourneCarltonAustralia

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