Abstract
Teaching mathematics for understanding requires teachers to start from each student’s current level of comprehension to provide effective pedagogical strategies that will meet the needs of their students and promote the building of further knowledge and skills. To do this, teachers’ diagnostic competence is needed. In particular, mathematical errors found in students’ work and statements are useful sources of information about their erroneous conceptualizations and hence a good opportunity for teachers to interpret and analyse students’ understanding and make decisions to provide targeted learning experiences.
This contribution discusses three projects, two of which aimed at fostering future primary and secondary school teachers’ diagnostic competence for dealing with students’ errors by providing focused learning opportunities in the context of university seminars. The university courses as well as pre- and post-tests were developed based on a model of diagnostic processes in error situations. Relations between facets of the diagnostic competence for dealing with students’ errors, namely cause-diagnosis and preferences in dealing with errors, and other attributes of the preservice teachers were investigated. The third project explored deeper the characteristics of in-service teachers dealing with errors in teaching situations using videotaped lessons. This project thus adds a further perspective on the strategies, which were implemented in real teaching situations. Significant results of the three studies and the connections between them are discussed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
To address the problem of multiple testing, the Bonferroni method was used in both studies.
- 2.
For the present study, two teachers were removed from the sample because technical problems prevented the conversations between teacher and learners from being heard in the case of one, and too few students’ errors occurred in the case of the other to form a differentiated picture.
- 3.
Citation rules: Emphasis is underlined; elongated words have greater character spacing; hash marks indicate simultaneous speech; parentheses with numbers or a dot indicate a pause, with numbers corresponding to seconds and a dot indicating a pause of less than one second; activities are added in italics in brackets.
References
Beck, E. (2008). Adaptive Lehrkompetenz: Analyse und Struktur, Veränderbarkeit und Wirkung handlungssteuernden Lehrerwissens. Waxmann.
Biza, I., Nardi, E., & Zachariades, T. (2018). Competences of mathematics teachers in diagnosing teaching situations and offering feedback to students: Specificity, consistency and reification of pedagogical and mathematical discourses. In T. Leuders, K. Philipp, & J. Leuders (Hrsg.), Diagnostic Competence of Mathematics Teachers (S. 55–78). Springer.
Blömeke, S., Kaiser, G., & Lehmann, R. (Hrsg.). (2010a), TEDS-M 2008 – Professionelle Kompetenz und Lerngelegenheiten angehender Primarstufenlehrkräfte im internationalen Vergleich. Waxmann.
Blömeke, S., Kaiser, G., & Lehmann, R. (Hrsg.). (2010b), TEDS-M 2008 – Professionelle Kompetenz und Lerngelegenheiten angehender Mathematiklehrkräfte für die Sekundarstufe I im internationalen Vergleich. Waxmann.
Borasi, R. (1994). Capitalizing on errors as „springboards for inquiry“: A teaching experiment. Journal for research in mathematics education, 25(2), 166–208.
Bray, W. S. (2011). A collective case study of the influence of teachers’ beliefs and knowledge on error-handling practices during class discussion of mathematics. Journal for research in mathematics education, 42(1), 2–38.
Cooper, S. (2009). Preservice teachers’ analysis of children’s work to make instructional decisions. School Science and Mathematics, 109(6), 355–362.
Cox, L. S. (1975). Systematic errors in the four vertical algorithms in normal and handicapped populations. Journal for Research in Mathematics Education, 202–220.
Heinrichs, H. (2015). Diagnostische Kompetenz von Mathematik-Lehramtsstudierenden: Messung und Förderung. Springer.
Heinrichs, H., & Kaiser, G. (2018). Diagnostic Competences for Dealing with Students’ Errors: Fostering Diagnostic Competence in Error Situations. In T. Leuders, K. Philipp, & J. Leuders (Hrsg.), Diagnostic Competence of Mathematics Teachers (S. 79–94). Springer.
Helmke, A. (2010). Unterrichtsqualität und Lehrerprofessionalität: Diagnose, Evaluation und Verbesserung des Unterrichts (3. Aufl.). Klett Kallmeyer.
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.
Kuckartz, U. (2014). Qualitative text analysis: A guide to methods, practice and using software. Sage.
Mayring, P. (2014). Qualitative content analysis: Theoretical foundation, basic procedures and software solution. Klagenfurt. URN: Http://nbn-resolving.de/urn:Nbn:De:0168-ssoar-395173.
Peterson, P., & Clark, C. M. (1978). Teachers’ reports of their cognitive processes during teaching. American Educational Research Journal, 15, 555–565.
Rach, S., Ufer, S., & Heinze, A. (2013). Learning from Errors: Effects of Teachers Training on Students’ Attitudes towards and Their Individual Use of Errors. PNA, 8(1), 21–30.
Reisman, F. K. (1982). A guide to the diagnostic of teaching of arithmetic (3. Aufl.). C. E. Merrill.
Rheinberg, F. (1978). Gefahren Pädagogischer Diagnostik. In K. J. Klauer (Hrsg.), Handbuch der pädagogischen Diagnostik (S. 27–38). Schwann.
Rittle-Johnson, B., Schneider, M., & Star, J.R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics Educational Psychology Review, 27(4), 587–597.
Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2019). Relationships among prospective secondary mathematics teachers’ skills of attending, interpreting and responding to students’ understanding. Educational Studies in Mathematics, 100(1), 83–99.
Santagata, R. (2005). Practices and beliefs in mistake-handling activities: A video study of Italian and US mathematics lessons. Teaching and Teacher Education, 21(5), 491–508.
Schoy-Lutz, M. (2005). Fehlerkultur im Mathematikunterricht. Franz-Becker.
Seifried, J., Wuttke, E., Türling, J. M., Krille, C., & Paul, O. (2015). Teachers’ strategies for handling student errors–the contribution of teacher training programs. In M. Gartmeier, H. Gruber, T. Hascher, & H. Heid (Hrsg.), Fehler: Ihre Funktionen im Kontext individueller und gesellschaftlicher Entwicklung (S. 177–188). Waxmann.
Seifried, J., & Wuttke, E. (2010). Student errors: How teachers diagnose them and how they respond to them. Empirical research in vocational education and training, 2(2), 147–162.
Shavelson, R. J., & Stern, P. (1981). Research on teachers’ pedagogical thoughts, judgements, decisions and behavior. Review of Educational Research, 51(4), 455–498.
Son, J. W. (2013). How preservice teachers interpret and respond to student errors: Ratio and proportion in similar rectangles. Educational studies in mathematics, 84(1), 49–70.
Son, J. W., & Sinclair, N. (2010). How preservice teachers interpret and respond to student geometric errors. School Science and Mathematics, 110(1), 31–46.
Tatto, M. T., Schwille, J., Senk, S., Ingvarson, L., Peck, R., & Rowley, G. (2008). Teacher Education and Development Study in Mathematics (TEDS-M): Policy, practice, and readiness to teach primary and secondary mathematics. Conceptual framework. East Lansing, MI: Teacher Education and Development International Study Center, College of Education, Michigan State University.
Türling, J. M., Seifried, J., & Wuttke, E. (2012). Teachers’ knowledge about domain specific student errors. Learning from errors at school and at work, 1, 95–110.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Der/die Autor(en), exklusiv lizenziert an Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature
About this chapter
Cite this chapter
Benecke, K., Heinrichs, H., Larrain, M. (2022). Assessing Primary and Secondary School Teachers’ Diagnostic Competence for Dealing with Students’ Errors. In: Buchholtz, N., Schwarz, B., Vorhölter, K. (eds) Initiationen mathematikdidaktischer Forschung. Springer Spektrum, Wiesbaden. https://doi.org/10.1007/978-3-658-36766-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-658-36766-4_9
Published:
Publisher Name: Springer Spektrum, Wiesbaden
Print ISBN: 978-3-658-36765-7
Online ISBN: 978-3-658-36766-4
eBook Packages: Life Science and Basic Disciplines (German Language)