The Interplay Between Professional Noticing and Knowledge: The Case of Whole Number Multiplication
Keywords:
mathematics teacher education research, teacher knowledge, teacher notic, whole number multiplication, student invented strategies, prospective teachersAbstract
The aim of this study was to reveal how instruction enriched with reasoning about students’ thought processes and strategies regulates prospective teacher’s noticing skills in the context of whole number multiplication. In addition, we examined the knowledge evidences that emerged during the process supporting the interplay between knowledge and noticing. To this end, noticing skills of one prospective teacher regarding to correct and incorrect student invented strategies were examined before and after his engagement with the instruction enriched with reasoning about students’ thought processes and strategies. The data gathered through the teacher’s written responses and the follow up interviews were analyzed with respect to the dimensions of the Professional Noticing of Children’s Mathematical Thinking framework. Moreover, to explore the knowledge evidences, that underpin the teacher’s noticing skills, data were analyzed through the MTSK model. The findings indicated positive change in the prospective teacher’s noticing skills for both correct and incorrect student invented strategies after his involvement in the intervention. More specifically, effective intervention improves not only the teacher knowledge, but also teacher noticing. Furthermore, the particular type of teacher knowledge, which the prospective teacher gained through the course, provided development of particular noticing skill.
References
Adler, J., & Davis, Z. (2006). Opening another black box: Researching mathematics for teaching in mathematics teacher education. Journal for Research in Mathematics Education, 37(4), 270–296. https://doi.org/10.2307/30034851
Baek, J. M. (2006). Research, reflection, practice: Children’s mathematical understanding and invented strategies for multidigit multiplication. Teaching Children Mathematics, 12(5), 242–247. https://doi.org/10.5951/TCM.12.5.0242
Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics (pp. 83–104). Ablex.
Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(3), 14–46. https://www.aft.org/sites/default/files/media/2014/BallF05.pdf
Blömeke, S., Gustafsson, J. E., & Shavelson, R. J. (2015). Beyond dichotomies: Competence viewed as a continuum. Zeitschrift für Psychologie, 223(1), 3.
Blömeke, S., Hsieh, F., Kaiser, G. & Schmidt, W. H. (Eds.) (2014). International perspectives on teacher knowledge, beliefs and opportunities to learn: TEDS-M results. Springer.
Campbell, P. (1998). What criteria for student-invented algorithms? In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (pp. 49–55). National Council of Teachers of Mathematics.
Carpenter, T. P., Fennema, E., & Franke, M. L. (1992, April). Cognitively guided instruction: Building the primary mathematics curriculum on children’s informal mathematical knowledge. A paper presented at the annual meeting of the American Educational Research Association, San Francisco, CA.
Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A base for reform in primary mathematics instruction. Elementary School Journal, 97, 3–20.
Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Heinemann.
Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3–20. https://doi.org/10.2307/749715
Carrillo-Yañez, J., Climent, N., Montes, M., Contreras, L. C., Flores-Medrano, E., Escudero-Ávila, D., Vasco, D., Rojas, N., Flores, P., Aguilar-González, Á., Ribeiro, M., & Muñoz-Catalán, M. C. (2018). The mathematics teacher’s specialised knowledge (MTSK) model*. Research in Mathematics Education, 20(3), 236–253. https://doi.org/10.1080/14794802.2018.1479981
Carroll, W. M. (1999). Invented computational procedures of students in a standards-based curriculum. Journal of Mathematical Behavior, 18(2), 111–121. https://doi.org/10.1016/s0732-3123(99)00024-3
Carroll, W. M., & Porter, D. (1997). Invented strategies can develop meaningful mathematical procedures. Teaching Children Mathematics, 3(7), 370–375.
Colestock, A., & Sherin, M. G. (2009). Teachers’ sense-making strategies while watching video of mathematics instruction. Journal of Technology and Teacher Education, 17, 7–29.
Dreher, A., & 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. https://doi.org/10.1007/s10649-014-9577-8
Fisher, M. H., Thomas, J., Schack, E. O., Jong, C., & Tassell, J. (2018). Noticing numeracy now! Examining changes in preservice teachers’ noticing, knowledge, and attitudes. Mathematics Education Research Journal, 30(2), 209–232. https://doi.org/10.1007/s13394-017-0228-0
Henry, G. T., Purtell, K. M., Bastian, K. C., Fortner, C. K., Thompson, C. L., Campbell, S. L., & Patterson, K. M. (2014). The effects of teacher entry portals on student achievement. Journal of Teacher Education, 65(1), 7–23.
Hino, K., Stylianides, G. J., Eilerts, K., Lajoie, C., & Pugalee, D. (2017). Topic study group no. 47: Pre-service mathematics education of primary teachers. In G. Kaiser (Ed.), Proceedings of the 13th International Congress on Mathematical Education, ICME-13 Monographs (pp. 593–597). https://doi.org/10.1007/978-3-319-62597-3_74
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.
Jacobs, V. R., Lamb, L. L. C., Philipp, R. A., & Schappelle, B. P. (2011). Deciding how to respond on the basis of children’s understandings. In M. G. Sherin, V. R. Jacobs, Philipp, R. A. (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 97–116). Taylor & Francis.
Jacobs, V. R., & Spangler, D. A. (2017). Research on core practices in K-12 mathematics teaching. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 766–792). National Council of Teachers of Mathematics.
Jong, C., Schack, E. O., Fisher, M. H., Thomas, J., & Dueber, D. (2021). What role does professional noticing play? Examining connections with affect and mathematical knowledge for teaching among preservice teachers. ZDM–Mathematics Education, 53(1), 151–164. https://doi.org/10.1007/s11858-020-01210-5.
Kamii, C., Lewis, B. A., & Livingston, S. J. (1993). Primary arithmetic: Children inventing their own procedures. The Arithmetic Teacher, 41(4), 200–203. https://doi.org/10.5951/AT.41.4.0200
Lampert, M. (2001). Teaching problems and the problems of teaching. Yale University Press.
Merriam, S.B. (1998). Qualitative research and case study applications in education. Jossey-Bass.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all.
Schoenfeld, A. H. (2011). Toward professional development for teachers grounded in a theory of decision making. ZDM Mathematics Education, 43(4), 457–469. https://doi.org/10.1007/s11858-011-0307-8
Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (2011a). Mathematics teacher noticing: Seeing through teachers’ eyes. Taylor & Francis.
Sherin, M., Russ, R., & Colestock, A. (2011b). Accessing mathematics teachers’ in-the-moment noticing. In M. G. Sherin, V. R. Jacobs & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 79–94). Routledge.
Sherin, M. G., Russ, R., Sherin, B. L., & Colestock, A. (2008). Professional vision in action: An exploratory study. Issues in Teacher Education, 17, 27–46.
Sherin, M. G., & Van Es, E. A. (2009). Effects of video club participation on teachers’ professional vision. Journal of Teacher Education, 60(1), 20–37. https://doi.org/10.1177/0022487108328155
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.
Son, J. W., Moseley, J., & Cady, J. (2013). How preservice teachers respond to student-invented strategies on whole number multiplication. 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. Chicago, IL. University of Illinois at Chicago.
Stake, R. E. (1995). The art of case study research. SAGE Publications.
Star, J. R., & Strickland, S. K. (2008). Learning to observe: Using video to improve preservice mathematics teachers’ ability to notice. Journal of Mathematics Teacher Education, 11, 107–125. https://doi.org/10.1007/s10857-007-9063-7
Tekin-Sitrava, R., Kaiser, G., & Işıksal-Bostan, M. (2021). Development of prospective teachers’ noticing skills within initial teacher education. International Journal of Science and Mathematics Education, 20, 1611–1634. https://doi.org/10.1007/s10763-021-10211-z
Van de Walle, J. A., Karp, K. S., & Bay-Williams J. M. (2013). Elementary and middle school mathematics: Teaching developmentally (8th ed.). Pearson Education.
Van Es, E. A., & Sherin, M. G. (2002). Learning to notice: Scaffolding new teachers’ interpretations of classroom interactions. Journal of Technology and Teacher Education,10(4), 571–596.
Yang, X., Kaiser, G., König, J., & Blömeke, S. (2021). Relationship between Chinese mathematics teachers’ knowledge and their professional noticing. International Journal of Science and Mathematics Education, 19(4), 815–837.
Yin, R. K. (2009). Case study methods: Design and methods (4th ed). SAGE Publications.
