Skip to main content
SpringerLink
Log in

Improving the judgment of task difficulties: prospective teachers’ diagnostic competence in the area of functions and graphs

  • Published:
Journal of Mathematics Teacher Education Aims and scope Submit manuscript

Abstract

To teach adaptively, teachers should be able to take the students’ level of knowledge into account. Therefore, a key component of pedagogical content knowledge (PCK) is the ability to assume the students’ perspectives. However, due to the so-called expert blind spot, teachers tend to misestimate their students’ knowledge, such as when estimating the difficulty of a given task. This empirically well-documented estimation bias is predicted by Nickerson’s anchoring and adjustment model, which generally explains how people take on other people’s perspectives. In this article, we present an intervention study that aims to improve the accuracy of prospective teachers’ judgments of task difficulty in the area of functional thinking. Two types of treatments are derived from Nickerson’s model. In the first condition (PCK group), participants acquire knowledge about task characteristics and students’ misconceptions. The second condition (sensitizing group) serves to control the idea that potential improvements in the PCK group are not merely based on a pure sensitization of the expert’s estimation bias. Accordingly, these participants are only informed about the general tendency of overestimating task difficulties. The results showed that the PCK group improved both in terms of the accuracy of the estimated solution rates and the accuracy of rank order, whereas the sensitizing group only improved in regard to the former. Thus, the study shows that prospective teachers’ diagnostic judgments can be improved by teaching them relevant PCK in a short amount of time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Anders, Y., Kunter, M., Brunner, M., Krauss, S., & Baumert, J. (2010). Diagnostische Fähigkeiten von Mathematiklehrkräften und ihre Auswirkungen auf die Leistungen ihrer Schülerinnen und Schüler. [Mathematics Teachers‘Diagnostic Skills and Their Impact on Students‘Achievements]. Psychologie in Erziehung und Unterricht, 3, 175–193.

    Article  Google Scholar 

  • Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching what makes it special? Journal of Teacher Education, 59(5), 389–407.

    Article  Google Scholar 

  • Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., et al. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 47(1), 133–180.

    Article  Google Scholar 

  • Bell, A., & Janvier, C. (1981). The interpretation of graphs representing situations. For the Learning of Mathematics, 2(1), 34–42.

    Google Scholar 

  • Berliner, D. C. (1994). Expertise: The wonders of exemplary performance. In C. C. Block & J. N. Mangieri (Eds.), Creating powerful thinking in teachers and students (pp. 141–186). New York: Holt, Rinehart & Winston.

    Google Scholar 

  • Busch, J., Barzel, B., & 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), 1–12.

    Article  Google Scholar 

  • Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Heinemann, 361 Hanover Street, Portsmouth, NH 03801-3912.

  • Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499–531.

    Article  Google Scholar 

  • Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Cronbach, L. J. (1955). Processes affecting scores on “understanding of others” and “assumed similarity”. Psychological Bulletin, 52, 177–183.

    Article  Google Scholar 

  • Demaray, M. K., & Elliot, S. N. (1998). Teachers’ judgments of students’ academic functioning: A comparison of actual and predicted performances. School Psychology Quarterly, 13(1), 8–24.

    Article  Google Scholar 

  • Dünnebier, K., Gräsel, C., & Krolak-Schwerdt, S. (2009). Urteilsverzerrungen in der schulischen Leistungsbeurteilung: Eine experimentelle Studie zu Ankereffekten. [Biases in Teachers’ Assessments of Student Performance: An Experimental Study of Anchoring Effects.]. Zeitschrift für Pädagogische Psychologie, 23(34), 187–195.

    Article  Google Scholar 

  • Duval, R. (2002). Representation, vision and visualization: Cognitive functions in mathematical thinking- Basics issues for learning. In F. Hitt (Ed.), Representations and mathematics visualization. Mexico-City: Cinvestav-IPN, Departamento de Matemática Educativa.

    Google Scholar 

  • Ericsson, K. A., & Simon, H. A. (1993). Protocol analysis: Verbal reports as data (rev ed.). Cambridge, MA: MIT-press.

    Google Scholar 

  • Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 24(2), 94–116.

    Article  Google Scholar 

  • Fischhoff, B. (1975). Hindsight is not equal to foresight: The effect of outcome knowledge on judgment under uncertainty. Journal of Experimental Psychology: Human Perception and Performance, 1(3), 288.

    Google Scholar 

  • Gagatsis, A., & Monoyiou, A. (2011). The structure of primary and secondary school students’ geometrical figure apprehension. In B. Ubuz (Hrsg.). Proceedings of the 35th conference of the international group for the psychology of mathematics education (pp 2–369).

  • Hadjidemetriou, C., & Williams, J. (2002a). Children’s graphical conceptions. Research in Mathematics Education, 4(1), 69–87.

    Article  Google Scholar 

  • Hadjidemetriou, C., & Williams, J. (2002b). Teachers’ pedagogical content knowledge: Graphs, from a cognitivist to a situated perspective. In A. D. Cockburn, & E. Nardi (Eds.), Proceedings of the 26th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 57–64). Norwich: PME.

  • Harel, G., & Dubinsky, E. (1992). The process conception of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 85–106). New York: Mathematical Association of America.

    Google Scholar 

  • Helmke, A., & Schrader, F.-W. (1987). Interactional effects of instructional quality and teacher judgement accuracy on achievement. Teaching and Teacher Education, 3(2), 91–98.

    Article  Google Scholar 

  • Hinds, P. J. (1999). The curse of expertise: The effects of expertise and debiasing methods on prediction of novice performance. Journal of Experimental Psychology: Applied, 5(2), 205–221.

    Google Scholar 

  • Hoge, R. D., & Coladarci, T. (1989). Teacher-based judgments of academic achievement: A review of literature. Review of Educational Research, 59(3), 297–313.

    Article  Google Scholar 

  • Kaiser, J., Helm, F., Retelsdorf, J., Südkamp, A., & Möller, J. (2012). Zum Zusammenhang von Intelligenz und Urteilsgenauigkeit bei der Beurteilung von Schülerleistungen im Simulierten Klassenraum. [On the Relation of Intelligence and Judgment Accuracy in the Process of Assessing Student Achievement in the Simulated Classroom.] Zeitschrift für Pädagogische Psychologie, 251–261.

  • Kelley, C. M. (1999). Subjective experience as basis of “objective” judgments: Effects of past experience on judgments of difficulty. In D. Gopher & A. Koriat (Eds.), Attention and performance XVII: Cognitive regulation of performance: Interaction of theory and application (pp. 515–536). Cambridge, MA: MIT Press.

  • Klug, J., Bruder, S., Kelava, A., Spiel, C., & Schmitz, B. (2013). Diagnostic competence of teachers: A process model that accounts for diagnosing learning behavior tested by means of a case scenario. Teaching and Teacher Education, 30, 38–46.

    Article  Google Scholar 

  • Krauss, R. M., & Fussell, S. R. (1991). Perspective-taking in communication: Representations of others’ knowledge in reference. Social Cognition, 9(1), 2–24.

    Article  Google Scholar 

  • Krawitz, J., Achmetli, K., Blum, W., Vogel, S. & Besser, M. (2016). Report on the relative strengths and weaknesses of the United States in PISA 2012 mathematics. OECD Education working papers, 151. Paris: OECD Publishing.

  • Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1–64.

    Article  Google Scholar 

  • Leuders, T., Bruder, R., Kroehne, U., Naccarella, D., Nitsch, R., Henning-Kahmann, J., et al. (2017). Development, validation, and application of a competence model for mathematical problem solving by using and translating representations of functions. In D. Leutner, J. Fleischer, J. Grünkorn, & E. Klieme (Eds.), Competence assessment in education: Research. Models and Instruments: Springer.

    Google Scholar 

  • Morris, A. K., Hiebert, J., & Spitzer, S. M. (2009). Mathematical knowledge for teaching in planning and evaluating instruction: What can preservice teachers learn? Journal for Research in Mathematics Education, 49(5), 491–529.

  • Nathan, M. J., & Koedinger, K. R. (2000). An investigation of teachers’ beliefs of students’ algebra development. Cognition and Instruction, 18(2), 209–237.

    Article  Google Scholar 

  • Nathan, M. J., & Petrosino, A. (2003). Expert blind spot among preservice teachers. American Educational Research Journal, 40(4), 905–928.

    Article  Google Scholar 

  • Nickerson, R. S. (1999). How we know—And sometimes misjudge—What others know: Imputing one’s own knowledge to others. Psychological Bulletin, 125(6), 737–795.

    Article  Google Scholar 

  • Nickerson, R. S. (2001). The projective way of knowing: A useful heuristic that sometimes misleads. Current Directions in Psychological Science, 10(5), 168–172.

    Article  Google Scholar 

  • Ostermann, A., Leuders, T., Nückles, M. (2015). Wissen, was Schülerinnen und Schülern schwer fällt. Welche Faktoren beeinflussen die Schwierigkeitseinschätzung von Mathematikaufgaben? [Knowing What Students Know. Which Factors Influence Teachers’ Estimation of Task Difficulty?]. Journal für Mathematik-Didaktik, 36(1), 45–76.

    Article  Google Scholar 

  • Philipp, K., & Leuders, T. (2014). Diagnostic competences of mathematics teachers—Processes and resources. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the joint meeting of PME 38 and PME-NA 36 (Vol. 1, pp. 425–432). Vancouver: PME.

  • Schrader, F.-W. (2009). Anmerkungen zum Themenschwerpunkt Diagnostische Kompetenz von Lehrkräften. [The Diagnostic Competency of Teachers]. Zeitschrift für Pädagogische Psychologie, 23(34), 237–245.

    Article  Google Scholar 

  • Spinath, B. (2005). Akkuratheit der Einschätzung von Schülermerkmalen durch Lehrer und das Konstrukt der diagnostischen Kompetenz. [Accuracy of Teacher Judgments on Student Characteristics and the Construct of Diagnostic Competence]. Zeitschrift für Pädagogische Psychologie, 19(1), 85–95.

    Article  Google Scholar 

  • Stacey, K., Steinle, V., Price, B., & Gvozdenko, E. (in press). Specific mathematics assessments that reveal thinking: An online tool to build teachers’ diagnostic competence and support teaching. In T. Leuders, J. Leuders, K. Philipp, & T. Dörfler (Eds.), Diagnostic competence of mathematics teachers—Unpacking a complex construct in teacher education and teacher practice. New York: Springer.

  • Südkamp, A., Kaiser, J., & Möller, J. (2012). Accuracy of teachers’ judgments of students’ academic achievement: A meta-analysis. Journal of Educational Psychology, 104(3), 743–762.

    Article  Google Scholar 

  • Südkamp, A., & Möller, J. (2009). Referenzgruppeneffekte im Simulierten Klassenraum: direkte und indirekte Einschätzungen von Schülerleistungen. [Reference-Group-Effects in a Simulated Classroom: Direct and Indirect Judgments.]. Zeitschrift für Pädagogische Psychologie, 23(34), 161–174.

    Article  Google Scholar 

  • Swan, M. (1986). The language of functions and graphs. Manchester: Joint Matriculation Board, reprinted 2000, Nottingham: Shell Centre Publications.

  • Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124–1131.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematische Bildung, Pädagogische Hochschule Freiburg, Kunzenweg 21, 79117, Freiburg, Germany

    Andreas Ostermann & Timo Leuders

  2. Institut für Erziehungswissenschaft, Universität Freiburg, Rempartstrasse 11, 79098, Freiburg, Germany

    Matthias Nückles

Authors
  1. Andreas Ostermann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Timo Leuders
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthias Nückles
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Ostermann.

Appendix

Appendix

See Figs. 6, 7, 8, 9, 10, 11, 12, 13 and 14.

Fig. 6
figure 6

Anchoring example for a very easy task presented before the pretest (empirical solution rate 89.7%)

Full size image
Fig. 7
figure 7

Anchoring example for a very difficult task presented before the pretest (empirical solution rate 8.1%)

Full size image
Fig. 8
figure 8

“Hiking tour” is one of the tasks to be estimated in the pre- and posttest (empirical solution rate 44%)

Full size image
Fig. 9
figure 9

“Crash-Test” is one of the tasks to be estimated in the pre- and posttest. It is possible that the graph-as-picture error or the linearly smooth prototype might occur (empirical solution rate: 54.2%)

Full size image
Fig. 10
figure 10

“Intersection” is one of the tasks to be estimated in the pre- and posttest. Common errors are misreading the scale and assuming that the functions are restricted to the charts (empirical solution rate 20.9%)

Full size image
Fig. 11
figure 11

“Slope” is one of the tasks to be estimated in the pre- and posttest. A common error is the slope-height confusion (empirical solution rate 79.8%)

Full size image
Fig. 12
figure 12

Intervention item: Rating the item relevance of students’ misconceptions in lessons on the topic of “functions and graphs”

Full size image
Fig. 13
figure 13

Intervention item: Rating the item relevance of students’ misconceptions with respect to a given task

Full size image
Fig. 14
figure 14

Intervention item: Analyzing potential difficulties of a given task

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ostermann, A., Leuders, T. & Nückles, M. Improving the judgment of task difficulties: prospective teachers’ diagnostic competence in the area of functions and graphs. J Math Teacher Educ 21, 579–605 (2018). https://doi.org/10.1007/s10857-017-9369-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10857-017-9369-z

Keywords

Search

Navigation