Deepening prospective mathematics teachers’ diagnostic judgments: Interplay of videos, focus questions, and didactic categories

Susanne Prediger 1 * , Carina Zindel 1
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1 TU Dortmund University, Dortmund, Germany
* Corresponding Author
EUR J SCI MATH ED, Volume 5, Issue 3, pp. 222-242. https://doi.org/10.30935/scimath/9508
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ABSTRACT

This article combines different conceptualizations of teachers' diagnostic competence in listening to students' mathematical thinking processes on the levels of general perspectives, noticed aspects and activated didactic categories. An empirical study of 159 prospective mathematics teachers' diagnostic judgments investigated how these levels are related and how they can be enhanced by focus questions and an intervention introducing relevant categories. For this purpose, 2×159 written diagnostic judgments, before and after a short intervention, were compared with respect to general perspectives, activated categories and aspects noticed in the video. As a means of comparing, a coding scheme was developed with respect to a sound conceptualization of diagnostic competence and to interrater reliability. The comparison shows that video alone cannot enhance prospective teachers' capacities of noticing, hence the activation and reflection of didactic categories is crucial. Possible theoretical consequences for a multi-facetted conceptualization of diagnostic competence and practical consequences for teacher education courses are discussed.

CITATION

Prediger, S., & Zindel, C. (2017). Deepening prospective mathematics teachers’ diagnostic judgments: Interplay of videos, focus questions, and didactic categories. European Journal of Science and Mathematics Education, 5(3), 222-242. https://doi.org/10.30935/scimath/9508

REFERENCES

  • Arcavi, A., and Schoenfeld, A. (2008). Using the unfamiliar to problematize the familiar: The case of mathematics teacher in-service education. Canadian Journal of Science, Mathematics and Technology Education, 8(3), 280-295.doi 10.1080/ 14926150802315122.
  • Ball, D. L., and Cohen, D. K. (1999). Developing practice, developing practitioners: Toward a practice-based theory of professional education. In G. Sykes, and L. Darling-Hammond (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 3-32). San Francisco: Jossey-Bass.
  • Blomberg, G., Renkl, A., Sherin, M. G., Borko, H., and Seidel, T. (2013). Five research-based heuristics for using video in pre-service teacher education. Journal for Education Research Online, 5(1), 90-114.
  • Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. ZDM, 38(2), 86-95.doi 10.1007/BF02655883.
  • Brophy, J. (Ed.)(2004). Using video in teacher education. Amsterdam: Elsevier.
  • Brunner, M., Anders, Y., Hachfeld, A., and Krauss, S. (2013). The Diagnostic Skills of Mathematics Teachers. In M. Kunter, J. Baumert, W. Blum, U. Klusmann, S. Krauss, and M. Neubrand (Eds.), Cognitive Activation in the Mathematics Classroom and Professional Competence of Teachers (pp. 229-248). New York: Springer.
  • Busch, J., Barzel, B., and Leuders, T. (2015). Promoting secondary teachers’ diagnostic competence with respect to functions: development of a scalable unit in Continuous Professional Development. ZDM Mathematics Education, 47(1), 53–64.doi10.1007/s11858-014-0647-2.
  • Clarke, D., Clarke, B., and Roche, A. (2011). Building teachers’ expertise in understanding, assessing and developing children’s mathematical thinking: the power of task-based, one-to-one interviews. ZDM Mathematics Education, 43(6), 901-913.doi 10.1007/s11858-011-0345-2.
  • Davis, B., andSimmt, E. (2006). Mathematics-for-teaching: an ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, 61(3), 293–319. doi 10.1007/s10649-006-2372-4.
  • Dreher, A., and Kuntze, S. (2015). Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics, 88(1), 89-114. doi 10.1007/s10649-014-9577-8.
  • Empson, S. B., and Jacobs, V. J. (2008). Learning to Listen to Children’s Mathematics. In T. Wood, and P. Sullivan (Eds.), International handbook of mathematics teacher education (Vol. 1, pp. 257-281). Rotterdam: Sense.
  • Even, R., and Tirosh, D. (2002). Teacher knowledge and understanding of students' mathematical learning. In L. English et al. (Eds.). Handbook of International Research in Mathematics Education (pp.219–240). Mahwah, NJ: Lawrence Erlbaum.
  • Franke, M. L., Carpenter, T. P., Levi, L., and Fennema, E. (2001). Capturing Teachers’ Generative Change: A Follow-Up Study of Professional Development in Mathematics. American Educational Research Journal, 38(3), 653-689. doi: 10.3102/ 00028312038003653.
  • Galbraith, P., andStillman, G. (2006). A framework for identifying student blockages during transitions in the modelling process. ZDM, 38(2), 143-162.doi 10.1007/BF02655886.
  • Girulat, A., Nührenbörger, M., andWember, F. (2013). Fachdidaktischfundierte Reflexion von Diagnose und individuelle Förderungim Unterrichtskontext – am Beispiel des Faches Mathematikunter Beachtungsonderpädagogischer Förderung. In S. Hußmann, and C. Selter (Eds.), Diagnose und individuelle Förderung in der MINT-Lehrerbildung (pp. 150-166). Münster: Waxmann.
  • Heinrichs, H. (2015). Diagnostische Kompetenz von Mathematik-Lehramtsstudierenden. Messung und Förderung. Wiesbaden: Springer Spektrum.
  • Helmke, A., Hosenfeld, I., and Schrader, F.-W. (2003). Diagnosekompetenz in Ausbildung und Berufentwickeln. Karlsruher Pädagogische Beiträge, 55, 15-34.
  • Jungwirth, H., Steinbring, H., Voigt, J., andWollring, B. (2001). Interpretative classroom research in teacher education. In H.-G. Weigand et al. (eds.). Developments in mathematics education in Germany. Selected Papers from the Annual Conference on Didactics of Mathematics. (pp. 46–56). Hildesheim: Franzbecker.
  • Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317–326.doi10.1007/BF00311062.
  • Krauss, S., and Brunner, M. (2011). Schnelles Beurteilen von Schülerantworten: Ein Reaktionszeittes tfür Mathematiklehrer/innen. Journal für Mathematik-Didaktik, 32(2), 233-251.doi 10.1007/s13138-011-0029-z.
  • Morris, A. K., Hiebert, J., and Spitzer, S. M. (2009). Mathematical knowledge for teaching in planning and evaluating instruction: What can pre-service teachers learn? Journal for Research in Mathematics Education, 40(5), 491-529.doi www.jstor.org/stable/40539354.
  • Nunes, T., Schliemann, A.D., and Carraher, D.W. (1993). Street mathematics and school mathematics. Cam­bridge: Cam­bridge University Press.
  • Philipp, K., and Leuders. T. (2014). Diagnostic competences of mathematics teachers – processes and resources. In P. Liljedahl, S. Oesterle, C. Nicol, and D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 4, pp. 426-433). Vancouver: PME.
  • Pollak, H.O. (1979). The interaction between mathematics and other school subjects. In UNESCO (Eds.), New trends in mathematics teaching IV (pp. 232-248), Paris: UNESCO.
  • Prediger, S. (2008). The relevance of didactic categories for analysing obstacles in conceptual change: Revis­iting the case of multiplication of fractions. Learning and Instruction, 18(1), 3-17.doi10.1016/j.learninstruc.2006.08.001.
  • Prediger, S. (2010a). How to Develop Mathematics for Teaching and for Understanding. The Case of Meanings of the Equal Sign. Journal of Mathematics Teacher Education, 13(1), 73-93. doi 10.1007/s10857-009-9119-y.
  • Prediger, S. (2010b). „Aber wie sag ichesmathematisch?“ – Empirische Befunde und Konse­­quen­­zenzum Lernen von Mathematikals Mittelzur Beschreibung von Welt. In D. Höttecke (Ed.), EntwicklungnaturwissenschaftlichenDenkenszwischenPhänomen und Systematik (pp. 6-20). Berlin: LIT-Verlag.
  • Prediger, S. (2010c). Über das Verhältnis von Theorien und wissenschaftlichenPraktiken – am Beispiel von SchwierigkeitenmitTextaufgaben. Journal fürMathematik-Didaktik, 31(2), 167-195.doi10.1007/s13138-010-0011-1.
  • Renk, N. (2009). Operationsverständnis von GrundschulkindernbeimMathematisieren von Textaufgaben. Eineempirische Untersuchungzu Strategien des intermodalen Transfers, Master thesis, supervised by S. Prediger, TU Dortmund.
  • Reusser K. (1990). From Text to Situation to Equation: Cognitive Simulation of Understanding and Solving Mathematical Word Problems. In H. Mandl, E. De Corte, N. Bennet, and H.F. Friedrich (Eds.), Learning and Instruction. European Research in an International Context (Vol. II, pp. 477–498). New York: Pergamon.
  • Scherer, P., andSteinbring, H. (2006). Noticing children’s learning processes – teachers jointly reflect on their own classroom interaction for improving mathematics teaching. Journal of Mathematics Teacher Education, 9(2), 157–185. doi 10.1007/s10857-006-0004-7.
  • Schoenfeld, A. H. (2011). Noticing matters. A lot. Now what? In M. G. Sherin, V. R. Jacobs, and R. A. Philipp (Eds.), Mathematics teacher noticing (pp. 223–238). New York: Routledge.
  • Schrader, F.-W., and Helmke, A. (1987). Diagnostische Kompetenz von Lehrern: Komponenten und Wirkungen. EmpirischePädagogik, 1(1), 27–52.
  • Schwarz, B., Wissmach, B., and Kaiser, G. (2008). “Last curves not quite correct”: diagnostic competences of future teachers with regard to modelling and graphical representations. ZDM, 40(5), 777-790.doi 10.1007/s11858-008-0158-0.
  • Seago, N. M. (2000). Using video of classroom practice as a tool to study and improve teaching. In E. Silver (Ed.), Mathematics education in the middle grades. (pp. 63–74). Washington, DC: National Academy Press.
  • Selter, C. (2001). Understanding - The underlying goal of teacher education. In M. van den Heuvel-Panhuizen (Ed.),Proceedings of the 25th annual conference of PME(Vol. 1, pp. 198–202).Utrecht: Freudenthal Institute.
  • Selter, C., Prediger, S., Nührenbörger, M., and Hußmann, S. (2012). Taking away and determining the difference—a longitudinal perspective on two models of subtraction and the inverse relation to addition. Educational Studies in Mathematics, 79(3), 389-408. doi 10.1007/s10649-011-9305-6.
  • Sherin, M. G. (2004). New perspectives on the role of video in teacher education. In J. Brophy (Ed.), Using video in teacher education (pp. 1–28). Oxford: Elsevier.
  • Sherin, M. G., and van Es, E. (2009). Effects of video club participation on teachers’ professional vision. Journal of Teacher Education, 60, 20–37. doi: 10.1177/0022487108328155.
  • Sherin, M. G., Jacobs, V.R., and Philipp, R. A. (Eds.) (2011). Mathematics Teacher Noticing. New York: Routledge.
  • Smith, J. P., diSessa, A. A., and Roschelle, J. (1993). Misconceptions Reconceived: A Constructivist Analysis of Knowledge in Transition. Journal of the Learning Sciences, 3(2), 115-163. doi10.1207/s15327809jls0302_1.
  • Star, J. R., and Strickland, S. K. (2008). Learning to observe: using video to improve preservice mathematics teachers’ ability to notice. Journal of Mathematics Teacher Education, 11(2), 107-125. doi 10.1007/s10857-007-9063-7.
  • Usiskin, Z. (1991). Building mathematics curricula with applications and modelling. In M. Niss, W. Blum, and I. Huntley (Eds.), Teaching of mathematical modelling and applications (pp. 30-45). Chichester: Horwood.
  • van den Heuvel-Panhuizen, M. (2005). The role of contexts in assessment problems in mathematics. For the Learning of Mathematics, 25(2), 2-9.
  • Verschaffel, L., Greer, B., and De Corte, E. (2000). Making sense of word problems. Lisse: Swets & Zeitlinger.
  • vomHofe, R., Kleine, M., Blum, W., and Pekrun, R. (2006). The effect of mental models ("Grundvorstellun­gen") for the development of mathematical competencies. First results of the longitudinal study PALMA. In M. Bosch (Eds.), Proceedings of the CERME 4 (pp. 142-151). Barcelona: Fundemi IQS - Universitat/ERME.
  • Walkoe, J. (2014). Exploring teacher noticing of student algebraic thinking in a video club. Journal of Mathematics Teacher Education, 18(6), 523-550. doi:10.1007/s10857-014-9289-0.
  • Wilson, S., and Berne, J (1999). Teacher learning and the acquisition of professional knowledge: An examination of research on contemporary professional development. Review of research in education, 24(1), 173-209.doi 10.3102/0091732X024001173.