Title: Dealing with student mistakes in mathematics at primary school
Source document: Studia paedagogica. 2024, vol. 29, iss. 2, pp. [66]-89
Extent
[66]-89
-
ISSN2336-4521 (online)
Persistent identifier (DOI): https://doi.org/10.5817/SP2024-2-3
Stable URL (handle): https://hdl.handle.net/11222.digilib/digilib.80624
Type: Article
Language
License: Not specified license
Rights access
open access
Notice: These citations are automatically created and might not follow citation rules properly.
Abstract(s)
The article focuses on how teachers deal with student mistakes during mathematics instruction at primary school. The theoretical framework encompasses a constructivist culture of teaching and learning and dialogic teaching. Two general strategies are distinguished: avoiding mistakes and using mistakes as opportunities to learn. The presented qualitative study is based on 24 audio recordings of mathematics instruction, supplemented by interviews with six teachers. Through axial coding, we arrived at seven categories that describe specific ways of dealing with mistakes: not responding; making students aware of a mistake; encouraging students to present proofs; leading students to a level where they do not fail; expanding the student network of mathematical knowledge; creating opportunities for presenting content to clarify mistakes; and providing opportunities for students to find the cause of their mistakes. The findings are discussed in the context of the constructivist approach to instruction.
Note
This study was supported by a project to develop a research organization called "Transdisciplinarity in curriculum and teaching in pre-primary and primary education." The ethics committee of Tomas Bata University in Zlín approved the project No. 3/1 of May 19, 2023.
References
[1] Borasi, R. (1994). Capitalizing on errors as "Springboards for Inquiry": A teaching experiment. Journal for Research in Mathematics Education, 25(2), 166–208. https://doi.org/10.2307/749507
[2] Brodie, K. (2014). Learning about learner errors in professional learning communities. Educational Studies in Mathematics, 85(2), 221–239. https://doi.org/10.1007/s10649-013-9507-1
[3] Drageset, O. G. (2019). How teachers use interactions to craft different types of student participation during whole–class mathematical work. In U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the 11th Congress of the European Society for Research in Mathematics Education (pp. 3622–3629). Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.
[4] Gagatsis, A., & Kyriakides, L. (2000). Teachers attitudes towards their pupils mathematical errors. Educational Research and Evaluation, 6(1), 24–58. https://doi.org/10.1076/1380-3611(200003)6:1;1-I;FT024
[5] Grassinger, R., Scheunpflug, A., Zeinz, H., & Dresel, M. (2018). Smart is who makes lots of errors? The relevance of adaptive reactions to errors and a positive error climate for academic achievement. High Ability Studies, 29(1), 37–49. https://doi.org/10.1080/13598139.2018.1459294
[6] Heinze, A. (2005, July 10th). Mistake–handling activities in the mathematics classroom [Paper presentation]. Conference of the International Group for the Psychology of Mathematics Education, Melbourne, Australia.
[7] Heinze, A., & Reiss, K. (2007). Mistake–handling activities in the mathematics classroom: Effects of an inservice teacher training on students' performance in geometry. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (pp. 9–16). PME.
[8] Hejný, M. (2012). Exploring the cognitive dimension of teaching mathematics through schemeoriented approach to education. Orbis scholae, 6(2), 41−55. https://doi.org/10.14712/23363177.2015.39
[9] Ingram, J., Pitt, A., & Baldry, F. (2015). Handling errors as they arise in whole–class interactions. Research in Mathematics Education, 17(3), 183–197. https://doi.org/10.1080/14794802.2015.1098562
[10] Janík, T. (2013). Od reformy kurikula k produktivní kultuře vyučování a učení [From curricular reform towards productive culture of teaching and learning]. Pedagogická orientace, 23(5), 634–663. https://doi.org/10.5817/PedOr2013-5-634
[11] Janík, T., Slavík, J., Najvar, P., & Jirotková, D. (2020). The same and the different: On semantization and instrumentalization practices in the (maths) classroom. SAGE Open, 10(3), 1–12. https://doi.org/10.1177/2158244020950380
[12] Janík, T., Slavík, J., Najvar, P., & Češková, T. (2024, in print). On the theory of content transformation in education: The 3A methodology for analysing and improving teaching and learning. Routledge.
[13] Käfer, J., Kuger, S., Klieme, E., & Kunter, M. (2019). The significance of dealing with mistakes for student achievement and motivation: Results of doubly latent multilevel analyses. European Journal of Psychology of Education, 34, 731–753. https://doi.org/10.1007/s10212-018-0408-7
[14] Köpfer, P. (2022). Teachers' perspectives on dealing with students' errors. Frontiers in Education, 7, 1-17. https://doi.org/10.3389/feduc.2022.868729
[15] Kuřina, F. (2016). Matematika jako pedagogický problém: Mé didaktické krédo. Gaudeamus.
[16] Majcík, M. (2022). Produktivní práce učitele s žákovskou chybou při komunikaci s celou třídou. Studia paedagogica, 27(1), 67–91. https://doi.org/10.5817/SP2022-1-3
[17] Matteucci, C. M., Corazza, M., & Santagata, R. (2015). Learning from errors, or not. An analysis of teachers' beliefs about errors and error-handling strategies through questionnaire and video. In R. V. Nata (Ed.), Progress in education (pp. 1–22). Nova Science.
[18] Meyer, L., Seidel, T., & Prenzel, M. (2006). When learning situations turn into achievement situations: Investigation of mistake culture in a video study. Swiss Journal of Educational Research, 28(1), 21–42. https://doi.org/10.24452/sjer.28.1.4717
[19] Myhill, D., & Warren, P. (2005). Scaffolds or straitjackets? Critical moments in classroom discourse. Educational Review, 57(1), 55−69. https://doi.org/10.1080/0013191042000274187
[20] Oser, F., & Spychiger, M. (2005). Lernen ist schmerzhaft: Zur Theorie des negativen Wissens und zur Praxis der Fehler Kultur. Beltz.
[21] Pan, S. C., Sana, F., Samani, J., Cooke, J., & Kim, J. A. (2020). Learning from errors: Students' and instructors' practices, attitudes, and beliefs. Memory, 28(9), 1105–1122. https://doi.org/10.1080/09658211.2020.1815790
[22] Pavelková, M. (2023). Podoby žákovských otázek ve výuce matematiky v primárním vzdělávání. [Doctoral dissertation]. Stag.
[23] Pinzón, A., Gómez, P., & González, M. (2022). Mathematics teachers' feedback responses to students' errors and unexpected strategies. Australian Journal of Teacher Education, 47(3), 19–34. https://doi.org/10.14221/ajte.2022v47n3.2
[24] 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. https://doi.org/10.30827/pna.v8i1.6122
[25] Rakoczy, H. (2008). Taking fiction seriously: Young children understand the normative structure of joint pretence games. Developmental Psychology, 44(4), 1195–1201. https://doi.org/10.1037/0012-1649.44.4.1195
[26] Reusser, K. (2001). Co-constructivism in educational theory and practice. In N. J. Smelser & B. Baltes (Eds.), International encyclopedia of the social and behavioral sciences (pp. 2058–2062). Elsevier.
[27] Reusser, K. (2006). Konstruktivismus – vom epistemologischen Leitbegriff zur Erneuerung der didaktischen Kultur. In M. Baer, M. Fuchs, P. Füglister, K. Reusser, & H. Wyss (Eds.), Didaktik auf psychologischer Grundlage: Von Hans Aeblis kognitionspsychologischer Didaktik zur modernen Lehr- und Lernforschung (pp. 151–168). Hep.
[28] Rybowiak, V., Garst, H., Frese, M., & Batinic, B. (1999). Error orientation questionnaire (EOQ): Reliability, validity, and different language equivalence. Journal of Organizational Behavior, 20(4), 527–547. https://doi.org/10.1002/(SICI)1099-1379(199907)20:4<527::AID-JOB886>3.0.CO;2-G
[29] Santagata, R. (2004). "Are you joking or are you sleeping?" Cultural beliefs and practices in Italian and U. S. teachers' mistake–handling strategies. Linguistics and Education, 15(1–2), 141–164. https://doi.org/10.1016/j.linged.2004.12.002
[30] 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. https://doi.org/10.1016/j.tate.2005.03.004
[31] Schleppenbach, M., Flevares, L., Sims, L., & Perry, M. (2007). Teachers' responses to student mistakes in Chinese and U.S. mathematics classrooms. The Elementary School Journal, 108(2), 131–147. https://doi.org/10.1086/525551
[32] Steuer, G., & Dresel, M. (2015). A constructive error climate as an element of effective learning environments. Psychological Test and Assessment Modeling, 57(2), 262–275.
[33] Swan, K. (2002). Building learning communities in online courses: The importance of interaction. Education, Communication & Information, 2(1), 23–49. http://dx.doi.org/10.1080/1463631022000005016
[34] Šeďová, K., Šalamounová, Z., Švaříček R., & Sedláček, M. (2020). Getting dialogic teaching into classrooms. Making change possible. Springer. https://doi.org/10.1007/978-981-15-9243-0
[35] Tulis, M. (2013). Error management behavior in classrooms: Teachers' responses to student mistakes. Teaching and Teacher Education, 33, 56–68. https://doi.org/10.1016/j.tate.2013.02.003
[36] Tulis, M., Steuer, G., & Dresel, M. (2016). Learning from errors: A model of individual processes. Frontline Learning Research, 4(2), 12–26. https://doi.org/10.14786/flr.v4i2.168
[37] Tulis, M., Steuer, G., & Dresel, M. (2017). Positive beliefs about errors as an important element of adaptive individual dealing with errors during academic learning. Educational Psychology, 38(2), 139–158. https://doi.org/10.1080/01443410.2017.1384536
[38] Voss, T., Kleickmann, T., Kunter, M., & Hachfeld, A. (2013). Mathematics teachers' beliefs. In M. Kunter, J. Baumert, W. Blum, U. Klusmann, S. Krauss, & M. Neubrand (Eds.), Cognitive activation in the mathematics classroom and professional competence of teachers. Mathematics teacher education (pp. 249–271). Springer. https://doi.org/10.1007/978-1-4614-5149-5_12
[39] Weinert, F. E. (1997). Lernkultur im Wandel. In E. Beck, T. Guldimann, & M. Zutavern (Eds.), Lernkultur im Wandel. Tagungsband drei Schweizerischen Gesellschaft für Lehrerinnen und Lehrerbildung und der Schweizerischen Gesellschaft für Bildungsforschung (pp. 11–29). UVK.
[40] Wiater, W. (2005). Die neue Lernkultur im Widerstreit der Meinungen. In E. M. Lanthaler & R. Meraner (Eds.), Neue Lernkultur im Kindergarten und Schule (pp. 46–62). Pädagogisches Institut.
[41] Wuttke, E., & Seifried, J. (2017). Competence, teacher competence and professional error competence: An introduction. In E. Wuttke & J. Seifried (Eds.), Professional error competence of preservice teachers (pp. 1–14). Springer.
[42] Zander, L., Kreutzmann, M., & Wolter, I. (2014). Constructive handling of mistakes in the classroom: The conjoint power of collaborative networks and self-efficacy beliefs. Zeitschrift für Erziehungswissenschaft, 17(5), 205–223. https://doi.org/10.1007/s11618-014-0558-6
[2] Brodie, K. (2014). Learning about learner errors in professional learning communities. Educational Studies in Mathematics, 85(2), 221–239. https://doi.org/10.1007/s10649-013-9507-1
[3] Drageset, O. G. (2019). How teachers use interactions to craft different types of student participation during whole–class mathematical work. In U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the 11th Congress of the European Society for Research in Mathematics Education (pp. 3622–3629). Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.
[4] Gagatsis, A., & Kyriakides, L. (2000). Teachers attitudes towards their pupils mathematical errors. Educational Research and Evaluation, 6(1), 24–58. https://doi.org/10.1076/1380-3611(200003)6:1;1-I;FT024
[5] Grassinger, R., Scheunpflug, A., Zeinz, H., & Dresel, M. (2018). Smart is who makes lots of errors? The relevance of adaptive reactions to errors and a positive error climate for academic achievement. High Ability Studies, 29(1), 37–49. https://doi.org/10.1080/13598139.2018.1459294
[6] Heinze, A. (2005, July 10th). Mistake–handling activities in the mathematics classroom [Paper presentation]. Conference of the International Group for the Psychology of Mathematics Education, Melbourne, Australia.
[7] Heinze, A., & Reiss, K. (2007). Mistake–handling activities in the mathematics classroom: Effects of an inservice teacher training on students' performance in geometry. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (pp. 9–16). PME.
[8] Hejný, M. (2012). Exploring the cognitive dimension of teaching mathematics through schemeoriented approach to education. Orbis scholae, 6(2), 41−55. https://doi.org/10.14712/23363177.2015.39
[9] Ingram, J., Pitt, A., & Baldry, F. (2015). Handling errors as they arise in whole–class interactions. Research in Mathematics Education, 17(3), 183–197. https://doi.org/10.1080/14794802.2015.1098562
[10] Janík, T. (2013). Od reformy kurikula k produktivní kultuře vyučování a učení [From curricular reform towards productive culture of teaching and learning]. Pedagogická orientace, 23(5), 634–663. https://doi.org/10.5817/PedOr2013-5-634
[11] Janík, T., Slavík, J., Najvar, P., & Jirotková, D. (2020). The same and the different: On semantization and instrumentalization practices in the (maths) classroom. SAGE Open, 10(3), 1–12. https://doi.org/10.1177/2158244020950380
[12] Janík, T., Slavík, J., Najvar, P., & Češková, T. (2024, in print). On the theory of content transformation in education: The 3A methodology for analysing and improving teaching and learning. Routledge.
[13] Käfer, J., Kuger, S., Klieme, E., & Kunter, M. (2019). The significance of dealing with mistakes for student achievement and motivation: Results of doubly latent multilevel analyses. European Journal of Psychology of Education, 34, 731–753. https://doi.org/10.1007/s10212-018-0408-7
[14] Köpfer, P. (2022). Teachers' perspectives on dealing with students' errors. Frontiers in Education, 7, 1-17. https://doi.org/10.3389/feduc.2022.868729
[15] Kuřina, F. (2016). Matematika jako pedagogický problém: Mé didaktické krédo. Gaudeamus.
[16] Majcík, M. (2022). Produktivní práce učitele s žákovskou chybou při komunikaci s celou třídou. Studia paedagogica, 27(1), 67–91. https://doi.org/10.5817/SP2022-1-3
[17] Matteucci, C. M., Corazza, M., & Santagata, R. (2015). Learning from errors, or not. An analysis of teachers' beliefs about errors and error-handling strategies through questionnaire and video. In R. V. Nata (Ed.), Progress in education (pp. 1–22). Nova Science.
[18] Meyer, L., Seidel, T., & Prenzel, M. (2006). When learning situations turn into achievement situations: Investigation of mistake culture in a video study. Swiss Journal of Educational Research, 28(1), 21–42. https://doi.org/10.24452/sjer.28.1.4717
[19] Myhill, D., & Warren, P. (2005). Scaffolds or straitjackets? Critical moments in classroom discourse. Educational Review, 57(1), 55−69. https://doi.org/10.1080/0013191042000274187
[20] Oser, F., & Spychiger, M. (2005). Lernen ist schmerzhaft: Zur Theorie des negativen Wissens und zur Praxis der Fehler Kultur. Beltz.
[21] Pan, S. C., Sana, F., Samani, J., Cooke, J., & Kim, J. A. (2020). Learning from errors: Students' and instructors' practices, attitudes, and beliefs. Memory, 28(9), 1105–1122. https://doi.org/10.1080/09658211.2020.1815790
[22] Pavelková, M. (2023). Podoby žákovských otázek ve výuce matematiky v primárním vzdělávání. [Doctoral dissertation]. Stag.
[23] Pinzón, A., Gómez, P., & González, M. (2022). Mathematics teachers' feedback responses to students' errors and unexpected strategies. Australian Journal of Teacher Education, 47(3), 19–34. https://doi.org/10.14221/ajte.2022v47n3.2
[24] 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. https://doi.org/10.30827/pna.v8i1.6122
[25] Rakoczy, H. (2008). Taking fiction seriously: Young children understand the normative structure of joint pretence games. Developmental Psychology, 44(4), 1195–1201. https://doi.org/10.1037/0012-1649.44.4.1195
[26] Reusser, K. (2001). Co-constructivism in educational theory and practice. In N. J. Smelser & B. Baltes (Eds.), International encyclopedia of the social and behavioral sciences (pp. 2058–2062). Elsevier.
[27] Reusser, K. (2006). Konstruktivismus – vom epistemologischen Leitbegriff zur Erneuerung der didaktischen Kultur. In M. Baer, M. Fuchs, P. Füglister, K. Reusser, & H. Wyss (Eds.), Didaktik auf psychologischer Grundlage: Von Hans Aeblis kognitionspsychologischer Didaktik zur modernen Lehr- und Lernforschung (pp. 151–168). Hep.
[28] Rybowiak, V., Garst, H., Frese, M., & Batinic, B. (1999). Error orientation questionnaire (EOQ): Reliability, validity, and different language equivalence. Journal of Organizational Behavior, 20(4), 527–547. https://doi.org/10.1002/(SICI)1099-1379(199907)20:4<527::AID-JOB886>3.0.CO;2-G
[29] Santagata, R. (2004). "Are you joking or are you sleeping?" Cultural beliefs and practices in Italian and U. S. teachers' mistake–handling strategies. Linguistics and Education, 15(1–2), 141–164. https://doi.org/10.1016/j.linged.2004.12.002
[30] 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. https://doi.org/10.1016/j.tate.2005.03.004
[31] Schleppenbach, M., Flevares, L., Sims, L., & Perry, M. (2007). Teachers' responses to student mistakes in Chinese and U.S. mathematics classrooms. The Elementary School Journal, 108(2), 131–147. https://doi.org/10.1086/525551
[32] Steuer, G., & Dresel, M. (2015). A constructive error climate as an element of effective learning environments. Psychological Test and Assessment Modeling, 57(2), 262–275.
[33] Swan, K. (2002). Building learning communities in online courses: The importance of interaction. Education, Communication & Information, 2(1), 23–49. http://dx.doi.org/10.1080/1463631022000005016
[34] Šeďová, K., Šalamounová, Z., Švaříček R., & Sedláček, M. (2020). Getting dialogic teaching into classrooms. Making change possible. Springer. https://doi.org/10.1007/978-981-15-9243-0
[35] Tulis, M. (2013). Error management behavior in classrooms: Teachers' responses to student mistakes. Teaching and Teacher Education, 33, 56–68. https://doi.org/10.1016/j.tate.2013.02.003
[36] Tulis, M., Steuer, G., & Dresel, M. (2016). Learning from errors: A model of individual processes. Frontline Learning Research, 4(2), 12–26. https://doi.org/10.14786/flr.v4i2.168
[37] Tulis, M., Steuer, G., & Dresel, M. (2017). Positive beliefs about errors as an important element of adaptive individual dealing with errors during academic learning. Educational Psychology, 38(2), 139–158. https://doi.org/10.1080/01443410.2017.1384536
[38] Voss, T., Kleickmann, T., Kunter, M., & Hachfeld, A. (2013). Mathematics teachers' beliefs. In M. Kunter, J. Baumert, W. Blum, U. Klusmann, S. Krauss, & M. Neubrand (Eds.), Cognitive activation in the mathematics classroom and professional competence of teachers. Mathematics teacher education (pp. 249–271). Springer. https://doi.org/10.1007/978-1-4614-5149-5_12
[39] Weinert, F. E. (1997). Lernkultur im Wandel. In E. Beck, T. Guldimann, & M. Zutavern (Eds.), Lernkultur im Wandel. Tagungsband drei Schweizerischen Gesellschaft für Lehrerinnen und Lehrerbildung und der Schweizerischen Gesellschaft für Bildungsforschung (pp. 11–29). UVK.
[40] Wiater, W. (2005). Die neue Lernkultur im Widerstreit der Meinungen. In E. M. Lanthaler & R. Meraner (Eds.), Neue Lernkultur im Kindergarten und Schule (pp. 46–62). Pädagogisches Institut.
[41] Wuttke, E., & Seifried, J. (2017). Competence, teacher competence and professional error competence: An introduction. In E. Wuttke & J. Seifried (Eds.), Professional error competence of preservice teachers (pp. 1–14). Springer.
[42] Zander, L., Kreutzmann, M., & Wolter, I. (2014). Constructive handling of mistakes in the classroom: The conjoint power of collaborative networks and self-efficacy beliefs. Zeitschrift für Erziehungswissenschaft, 17(5), 205–223. https://doi.org/10.1007/s11618-014-0558-6