Pre-service Teachers' Learning from Significant Opportunities for Improvement in a Positive Error Culture
Keywords:
errors, professional noticing, discourse, design-based researchAbstract
Many teacher education models involve reflecting on teaching practice for the sake of improving it. Such reflection must be carefully structured to help practitioners identify and act upon significant opportunities for improvement (SOIs). Learning from SOIs requires the cognitive activities of noticing students’ mathematical thinking and its connection to instructional practice, along with an affective disposition to view sub-optimal teaching practices as learning opportunities. We draw upon existing literature and theory related to the notion of developing positive error cultures to identify design principles for helping teachers learn from their own sub-optimal practices rather than becoming discouraged by them. The design principles include experience-based learning, low-stakes settings, collaboration, process reflection, and exploration of disagreements. We then describe a mathematics teacher education environment incorporating the design principles. Examples of pre-service teachers’ work within the environment are analysed for possible patterns of learning from SOIs within a positive error culture. Based on these examples, a four-quadrant model to characterise teachers’ learning from SOIs is proposed. The four quadrants describe various outcomes related to recognising and resolving SOIs.
References
Amador, J. M., Carter, I., & Hudson, R. A. (2016). Analyzing preservice mathematics teachers’ professional noticing. Action in Teacher Education, 38, 371-383. https://doi.org/10.1080/01626620.2015.1119764
Bakker, A., & van Eerde, D. (2015). An introduction to design-based research with an example from statistics education. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education (pp. 429-466). Springer.
Barkatsas, A., & Malone, J. (2005). A typology of mathematics teachers’ beliefs about teaching and learning mathematics and instructional practices. Mathematics Education Research Journal, 17(2), 69-90. https://doi.org/10.1007/BF03217416
Barnhart, T., & van Es, E. (2015). Studying teacher noticing: Examining the relationship among pre-service science teachers’ ability to attend, analyze and respond to student thinking. Teaching and Teacher Education, 45, 83–93. https://doi.org/10.1016/j.tate.2014.09.005
Boaler, J. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. Jossey-Bass.
Borasi, R. (1994). Capitalizing on errors as “springboards for inquiry”: A teaching experiment. Journal for Research in Mathematics Education, 25, 166-208. https://doi.org/10.5951/jresematheduc.25.2.0166
Cannon, M., & Edmondson, A. (2001). Confronting failure: Antecedents and consequences of shared beliefs about failure in organizational work groups. Journal of Organizational Behavior, 22, 161-177. https://doi.org/10.1002/job.85
Corbin, J., & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for developing grounded theory (3rd ed.). SAGE.
Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the educative process. Heath.
Edmondson, A. C. (1999). Psychological safety and learning behavior in work teams. Administrative Science Quarterly, 44, 350–383. https://doi.org/10.2307/2666999
Fennema, E., Carpenter, T. P., Franke, M. L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics education. Journal for Research in Mathematics Education, 27(4), 403–434. https://doi.org/10.5951/jresematheduc.27.4.0403
Fernandez, C., & Yoshida, M. (2004). Lesson study: A Japanese approach to improving mathematics teaching and learning. Routledge.
Finch, A. (2010). Critical incidents and language learning: Sensitivity to initial conditions. System, 38, 422-431. https://doi.org/10.1016/j.system.2010.05.004
Goldsmith, L. T., & Seago, N. (2011). Using classroom artifacts to focus teachers’ noticing: Affordances and opportunities. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 169–186). Routledge.
Groth, R.E. (2011). Improving teaching through lesson study debriefing. Mathematics Teacher, 104, 446-451. https://doi.org/10.5951/MT.104.6.0446
Groth, R.E., Bergner, J., Austin, J.W., Burgess, C.R., & Holdai, V. (2020). Undergraduate research in mathematics education: Using qualitative data about children’s learning to make teaching decisions. Mathematics Teacher Educator, 8(3), 134-151. https://doi.org/10.5951/MTE.2020.0008
Groth, R.E., Bergner, J.A., & Burgess, C.R. (2016). An exploration of prospective teachers' learning of clinical interview techniques. Mathematics Teacher Education and Development, 18(2), 48-71.
Harteis, C., Bauer, J., & Gruber, H. (2008). The culture of learning from mistakes: How employees handle mistakes in everyday work. International Journal of Educational Research, 47, 223-231. https://doi.org/10.1016/j.ijer.2008.07.003
He, C., & Yan, C. (2011). Exploring authenticity of microteaching in pre-service teacher education programmes. Teaching Education, 22(3), 291-302. https://doi.org/10.1080/10476210.2011.590588
Herbel-Eisenmann, B. A., & Breyfogle, M. L. (2005). Questioning our patterns of questioning. Mathematics Teaching in the Middle School, 10, 484-489. https://doi.org/10.5951/MTMS.10.9.0484
Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. https://doi.org/10.5951/jresematheduc.41.2.0169
Karp, K. S., Bush, S. B., & Dougherty, B. J. (2015). 12 math rules that expire in the middle grades. Mathematics Teaching in the Middle School, 21, 208-215. https://doi.org/10.5951/mathteacmiddscho.21.4.0208
Keith, N., & Frese, M. (2005). Self-regulation in error management training: Emotion control and metacognition as mediators of performance effects. Journal of Applied Psychology, 90, 677–691. https://doi.org/10.1037/0021-9010.90.4.677
Keogh, B., & Naylor, S. (2009). Puppets count. Mathematics Teaching, 213, 32-34.
Leatham, K. R. & Peterson, B. E. (2010). Purposefully designing student teaching to focus on students’ mathematical thinking. In J. Luebeck, & J. W. Lott (Eds.), AMTE Monograph 7: Mathematics teaching: Putting research into practice at all levels (pp. 225-239). Association of Mathematics Teacher Educators.
Leitch, R., & Day, C. (2006). Action research and reflective practice: Towards a holistic view. Educational Action Research, 8(1), 179-193. https://doi.org/10.1080/09650790000200108
Lewis, J. (2016). Learning to lead, leading to learn: How facilitators learn to lead lesson study. ZDM Mathematics Education, 48(4), 527-540. doi: 10.1007/s11858-015-0753-9
Lewis, C., & Hurd, J. (2011). Lesson study step-by-step: How teacher learning communities improve instruction. Heinemann.
Lewis, C. C., Perry, R. R., & Hurd, J. (2009). Improving mathematics instruction through lesson study: A theoretical model and a North American Case. Journal of Mathematics Teacher Education, 12, 285-304. https://doi.org/10.1007/s10857-009-9102-7
Matusov, E. (1996). Intersubjectivity without agreement. Mind, Culture, and Activity, 3, 25-45. https://doi.org/10.1207/s15327884mca0301_4
Metcalfe, J. (2017). Learning from errors. Annual Review of Psychology, 68, 465-489. https://doi.org/10.1146/annurev-psych-010416-044022
Minnameier, G. (2006). Learning is painful - on the theory of negative knowledge and a practice and error culture. Book review. Journal of Moral Education, 35, 410-413.
Moyer, P. S., & Milewicz, E. (2002). Learning to question: Categories of questioning used by preservice teachers during diagnostic mathematics interviews. Journal of Mathematics Teacher Education, 5, 293-315. https://doi.org/10.1023/A:1021251912775
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Author.
Perry, B., Howard, P., & Tracey, D. (1999). Head mathematics teachers’ beliefs about the learning and teaching of mathematics. Mathematics Education Research Journal, 11(1), 39–57.
Ricks, T. E. (2011). Process reflection during Japanese Lesson Study experiences by prospective secondary mathematics teachers. Journal of Mathematics Teacher Education, 14, 251-267. https://doi.org/10.1007/s10857-010-9155-7
Santagata, R. (2011). From teacher noticing to a framework for analyzing and improving classroom lessons. In M. G. Sherin, V. R. Jacobs, & R. A. Phillip (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 152-168). Routledge.
Schön, D. A. (1983). The reflective practitioner: How professionals think in action. Basic Books.
Sherin, M. G., Jacobs, V. R., & Philipp, R. A. ( 011). Situating the study of teacher noticing. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 3-14). Routledge.
Shilo, A., & Kramarski, B. (2019). Mathematical-metacognitive discourse: How can it be developed among teachers and their students? Empirical evidence from a videotaped lesson and two case studies. ZDM Mathematics Education, 51, 625-640. https://doi.org/10.1007/s11858-018-01016-6
Sibbald, T. (2009). The relationship between lesson study and self-efficacy. School Science and Mathematics, 109(8), 450-460. https://doi.org/10.1111/j.1949-8594.2009.tb18292.x
Simpson, A., & Haltiwanger, L. (2017). “This is the first time I’ve done this”: Exploring secondary prospective mathematics teachers’ noticing of students’ mathematical thinking. Journal of Mathematics Teacher Education, 20, 335-355. https://doi.org/10.1007/s10857-016-9352-0
Star, J. R., Lynch, K., & Perova, N. (2011). Using video to improve preservice mathematics teachers’ abilities to attend to classroom features. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 117-133). Routledge.
Steuer, G., Rosentritt-Brunn, G., & Dresel, M. (2013). Dealing with errors in mathematics classrooms: Structure and relevance of perceived error climate. Contemporary Educational Psychology, 38, 196-210. https://doi.org/10.1016/j.cedpsych.2013.03.002
Stockero, S. L. (2013). The effects of framing on mathematics student teacher noticing. In M. Martinez & A. Castro Superfine (Eds.), Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 709-716). University of Illinois at Chicago.
Sutton, P. S., & Shouse, A. (2016). Building a culture of collaboration in schools. Phi Delta Kappan, 97(7), 69-73. https://doi.org/10.1177/0031721716641653
Tobias, J. M., & Andreasen, J. B. (2013). Developing multiplicative thinking from additive reasoning. Teaching Children Mathematics, 20(2), 102-109. https://doi.org/10.5951/teacchilmath.20.2.0102
Tjosvold, D., Yu, Z., & Hui, C. (2004). Team learning from mistakes: The contribution of cooperative goals and problem-solving. Journal of Management Studies, 41, 1223-1245. https://doi.org/10.1111/j.1467-6486.2004.00473.x
Tripp, D. H. (1993). Critical incidents in teaching: The development of professional judgment. Routledge.
Tulis, M. (2013). Error management behavior in classrooms: Teachers’ responses to students’ mistakes. Teaching and Teacher Education, 33, 56-68. https://doi.org/10.1016/j.tate.2013.02.003
Van Es, E. A., & Sherin, M. G. (2010). The influence of video clubs on teachers’ thinking and practice. Journal of Mathematics Teacher Education, 13. 155-176. https://doi.org/10.1007/s10857-009-9130-3
Westermann, K., & Rummel, N. (2012). Delaying instruction: Evidence from a study in a university relearning setting. Instructional Science, 40, 673–689. https://doi.org/10.1007/s11251-012-9207-8
Wieman, R., & Hiebert, J. (2018). Learning from mistakes: Not just for students. Teachers College Record. http://www.tcrecord.org; ID Number: 22527.
Woulfin, S. L. (2020). Crystallizing coaching: An examination of the institutionalization of instructional coaching in three educational systems. Teachers College Record, 122(10). https://www-tcrecord-org.proxy-su.researchport.umd.edu/library