Enhancing Noticing: Using a Hypothetical Learning Trajectory to Improve Pre-service Primary Teachers’ Professional Discourse

Pedro Ivars; Ceneida Fernández; Salvador Llinares; Ban Heng Choy

doi:10.29333/ejmste/93421

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Pedro Ivars ¹ ^* , Ceneida Fernández ¹, Salvador Llinares ¹, Ban Heng Choy ²

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¹ Dept. Innovación y Formación Didáctica. University of Alicante, SPAIN² National Institute of Education, Nanyang Technological University, SINGAPORE^* Corresponding Author

Abstract

The aim of this paper is to examine whether the use of a hypothetical learning trajectory as a guide to notice students’ mathematical thinking could improve pre-service teachers professional discourse and enhance pre-service teachers’ noticing. Twenty-nine pre-service primary school teachers participated in a learning environment in which they had to interpret students’ thinking about the fraction concept using a hypothetical learning trajectory as a guide. Results suggest that using a hypothetical learning trajectory as a guide helped pre-service teachers develop a more detailed discourse when interpreting students’ mathematical thinking, enhancing their noticing skill. The enhancement of the skill of noticing, however, was linked to pre-service teachers’ mathematical content knowledge. This research provides teacher educators with resources to help pre-service teachers produce a more detailed professional discourse to attend to the details of students’ answers and their different mathematical levels of thinking in mathematics teacher education programs.

Keywords

fractions
hypothetical learning trajectories
noticing
pre-service primary school teachers learning
professional discourse

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2018, Volume 14, Issue 11, Article No: em1599

https://doi.org/10.29333/ejmste/93421

Publication date: 14 Jul 2018

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Article Downloads: 2815

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References

Battista, M. T. (2012). Cognition-Based assessment and teaching of fractions: Building on students’ reasoning. Portsmouth, N.H. Heinemann.
Blömeke, S., Hoth, J., Döhrmann, M., Busse, A., Kaiser, G., & König, J. (2015). Teacher change during induction: development of beginning primary teachers’ knowledge, beliefs and performance. International Journal of Science and Mathematics Education, 13(2), 287–308. https://doi.org/10.1007/s10763-015-9619-4.
Choy, B. H., Thomas, M. O. J., & Yoon, C. (2017). The FOCUS framework: characterising productive noticing during lesson planning, delivery and review. In E. O. Schack, M. H. Fisher, & J. A. Wilhelm (Eds.), Teacher noticing: bridging and broadening perspectives, contexts, and frameworks (pp. 445-466). Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-319-46753-5_26.
Coles, A. (2013). Using video for professional development: the role of the discussion facilitator. Journal of Mathematics Teacher Education, 16(3), 165-184. https://doi.org/10.1007/s10857-012-9225-0.
Confrey, J., Gianopulos, G., McGowan, W., Shah, M., & Belcher, M. (2017). Scaffolding learner-centered curricular coherence using learning maps and diagnostic assessments designed around mathematics learning trajectories. ZDM Mathematics Education, 49(5), 717-734. https://doi.org/10.1007/s11858-017-0869-1.
Dunekacke, S., Jenßen, L., & Blömeke, S. (2015). Effects of mathematics content knowledge on pre-school teachers’ performance: A video-based assessment of perception and planning abilities in informal learning situations. International Journal of Science and Mathematics Education, 13(2), 267-286. https://doi.org/10.1007/s10763-014-9596-z.
Edgington, C. (2014). Teachers’ use of a learning trajectory as a tool for mathematics lesson planning. In J. J. Lo, K. Leatham, & L. Van Zoest (Eds.), Research trends in mathematics teacher education, (pp.261-284). London: Springer. https://doi.org/10.1007/978-3-319-02562-9_14.
Edgington, C., Wilson, P. H., Sztajn, P., & Webb, J. (2016). Translating learning trajectories into useable tools for teachers. Mathematics Teacher Educator, 5(1), 65-80. https://doi.org/10.5951/mathteaceduc.5.1.0065.
Fernández, C., Llinares, S., & Valls, J. (2012). Learning to notice students’ mathematical thinking through on-line discussions. ZDM. Mathematics Education, 44, 747-759. https://doi.org/10.1007/s11858-012-0425-y.
Goodwin, C. (1994). Professional vision. American anthropologist, 96(3), 606-633. https://doi.org/10.1525/aa.1994.96.3.02a00100.
Ivars, P., & Fernández, C. (2018). The role of writing narratives in developing pre-service primary teachers noticing. In G. Stylianides & K. Hino (Eds.), Research advances in the mathematical education of pre-service elementary Teachers. ICME-13 Monographs, (pp. 245-260). Springer: Cham. https://doi.org/10.1007/978-3-319-68342-3_17.
Ivars, P., Fernández, C., & Llinares, S. (2016). Pre-service teachers' learning to notice students' fractional thinking: The design of a learning environment through a Learning Trajectory. In S. Zehetmeier, B. Rösken-Winter, D. Potari, & M. Ribeiro (Eds.), Proceedings of the Third ERME Topic Conference on Mathematics Teaching, Resources and Teacher Professional Development (pp. 106-115). Berlin, Germany: Humboldt-Universität zu Berlin.
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.
Kahan, J. A., Cooper, D. A., & Bethea, K. A. (2003). The role of mathematics teachers’ content knowledge in their teaching: A framework for research applied to a study of student teachers. Journal of Mathematics Teacher Education, 6(3), 223-252. https://doi.org/10.1023/A:1025175812582.
Kaiser, G., Blömeke, S., Busse, A., Döhrmann, M., & König, J. (2014). Professional knowledge of (prospective) mathematics teachers–its structure and development. 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. 35-50. Vancouver, Canada: PME.
Lamon, S. (1999). Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum Ass. Pub.
Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629-668). Charlotte, NC: Information Age Publishing.
Lee, M. Y., & Choy B. H. (2017). Mathematical teacher noticing: The key to learning from lesson study. In E. Schack, M. Fisher, & J. Wilhelm (Eds.), Teacher Noticing: Bridging and Broadening Perspectives, Contexts, and Frameworks. Research in Mathematics Education (pp. 121-140). Springer International Publishing. https://doi.org/10.1007/978-3-319-46753-5_8.
Levin, D. M., Hammer, D., & Coffey, J. E. (2009). Novice teachers’ attention to student thinking. Journal of Teacher Education, 60(2), 142-154. https://doi.org/10.1177/0022487108330245.
Mason, J. (2002). Researching your own practice: the discipline of noticing. London: Routledge. https://doi.org/10.4324/9780203471876.
Mason, J. (2011). Noticing: Roots and branches. In M. G. Sherin, V. R. Jacobs, & R. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 35–50). New York: Routledge.
Mason, J. (2017). Probing Beneath the Surface of Experience. In E. Schack, M. Fisher, & J. Wilhelm (Eds.), Teacher Noticing: Bridging and Broadening Perspectives, Contexts, and Frameworks. Research in Mathematics Education (pp.1-17). Springer International Publishing. https://doi.org/10.1007/978-3-319-46753-5_1.
McDonough, A., Clarke, B., & Clarke, D. M. (2002). Understanding, assessing and developing children’s mathematical thinking: the power of a one-to-one interview for preservice teachers in providing insights into appropriate pedagogical practices. International Journal of Educational Research, 37(2), 211-226. https://doi.org/10.1016/s0883-0355(02)00061-7.
Mitchell, R. N., & Marin, K. A. (2015). Examining the use of a structured analysis framework to support prospective teacher noticing. Journal of Mathematics Teacher Education, 18(6), 551-575. https://doi.org/10.1007/s10857-014-9294-3.
National Council of Teachers of Mathematics (NCTM). (2014). Principles to actions. Reston, VA: Author.
National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press. https://doi.org/10.17226/9822.
Nickerson, S. D., Lamb, L., & LaRochelle, R. (2017). Challenges in measuring secondary mathematics teachers’ professional noticing of students’ mathematical thinking. In E. Schack, M. Fisher, & J. Wilhelm (Eds.), Teacher Noticing: Bridging and Broadening Perspectives, Contexts, and Frameworks (pp. 381-398). Springer International Publishing. https://doi.org/10.1007/978-3-319-46753-5_22.
Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2015). Developing pre-service teachers' noticing of students' understanding of the derivative concept. International Journal of Science and Mathematics Education, 13(6), 1305-1329. https://doi.org/0.1007/s10763-014-9544-y.
Santagata, R. & Angelici, G. (2010). Studying the impact of the lesson analysis framework on preservice teachers’ abilities on videos of classroom teaching. Journal of Teacher Education, 61(4), 339-349. https://doi.org/10.1177/0022487110369555.
Schack, E., Fisher, M., & Wilhelm, J. (Eds.). (2017). Teacher noticing: Bridging and broadening perspective contexts, and frameworks. Cham, Switzerland: Springer International Publishing. https://doi.org/10.1007/978-3-319-46753-5.
Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (Eds.). (2011). Mathematics teacher noticing: Seeing through teachers’ eyes. New York, NY: Routledge. https://doi.org/10.4324/9780203832714.
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 114-145. https://doi.org/10.2307/749205.
Simon, M. A., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: An elaboration of the Hypothetical Learning Trajectory. Mathematical Thinking and Learning, 6(2), 91-104. https://doi.org/10.1207/s15327833mtl0602_2.
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(4), 335-355. https://doi.org/10.1007/s10857-016-9352-0.
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. https://doi.org/10.1007/s10649-013-9475-5.
Stahnke, R., Schueler, S., & Roesken-Winter, B. (2016). Teachers’ perception, interpretation, and decision-making: a systematic review of empirical mathematics education research. ZDM. Mathematics Education, 48(1-2), 1-27. https://doi.org/10.1007/s11858-016-0775-y.
Steffe, L. P. (2004). On the construction of learning trajectories of children: The case of commensurate fractions. Mathematical Thinking and Learning, 6(2), 129-162. https://doi.org/10.1207/s15327833mtl0602_4.
Steffe, L., & Olive, J. (2009). Children’s fractional knowledge. Springer Science & Business Media. https://doi.org/10.1007/978-1-4419-0591-8.
Sztajn, P., Confrey, J., Wilson, P. H., & Edgington, C. (2012). Learning trajectory based instruction toward a theory of teaching. Educational Researcher, 41(5), 147-156. https://doi.org/10.3102/0013189x12442801.
van Es, 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.
van Es, E. (2011). A framework for learning to notice student thinking. In M. G. Sherin, V.R. Jacobs, & R. Philipp, (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 134-151). New York: Routledge.
van Es, E., Tunney, J., Goldsmith, L. & Seago, N. (2014). A framework for the facilitation of teachers’ analysis of video. Journal of Teacher Education, 65(4), 340-356. https://doi.org/10.1177/0022487114534266.
Walkoe, J. (2015). Exploring teacher noticing of student algebraic thinking in a video club. Journal of Mathematics Teacher Education, 18, 523–550. https://doi.org/10.1007/s10857-014-9289-0.
Weiland, I. S. & Amador, J. M. (2015). Lexical and indexical conversational components that mediate Professional noticing during lesson study. Eurasia Journal of Mathematics, Science & Technology Education, 11(6), 1339-1361. https://doi.org/10.12973/eurasia.2015.1392a.
Wells, G. (1999). Dialogic Inquiry. Toward a Sociocultural Practice and Theory of Education. Cambridge: Cambridge University Press. https://doi.org/10.1017/cbo9780511605895.
Wilson, P. H., Mojica, G. F., & Confrey, J. (2013). Learning trajectories in teacher education: Supporting teachers’ understandings of students’ mathematical thinking. The Journal of Mathematical Behavior, 32(2), 103-121. https://doi.org/10.1016/j.jmathb.2012.12.003.
Wilson, P. H., Sztajn, P., Edgington, C., Webb, J., & Myers, M. (2017). Changes in teachers’ discourse about students in a professional development on learning trajectories. American Educational Research Journal, 54(3), 568-604. https://doi.org/10.3102/0002831217693801.
Wilson, S. M., & Berne, J. (1999). Teacher learning and the acquisition of professional knowledge: An examination of research on contemporary professional development. Review of Research in Education, 24, 173-209. https://doi.org/10.2307/1167270.

How to cite this article

APA

Ivars, P., Fernández, C., Llinares, S., & Choy, B. H. (2018). Enhancing Noticing: Using a Hypothetical Learning Trajectory to Improve Pre-service Primary Teachers’ Professional Discourse. Eurasia Journal of Mathematics, Science and Technology Education, 14(11), em1599. https://doi.org/10.29333/ejmste/93421

Vancouver

Ivars P, Fernández C, Llinares S, Choy BH. Enhancing Noticing: Using a Hypothetical Learning Trajectory to Improve Pre-service Primary Teachers’ Professional Discourse. EURASIA J Math Sci Tech Ed. 2018;14(11):em1599. https://doi.org/10.29333/ejmste/93421

AMA

Ivars P, Fernández C, Llinares S, Choy BH. Enhancing Noticing: Using a Hypothetical Learning Trajectory to Improve Pre-service Primary Teachers’ Professional Discourse. EURASIA J Math Sci Tech Ed. 2018;14(11), em1599. https://doi.org/10.29333/ejmste/93421

Chicago

Ivars, Pedro, Ceneida Fernández, Salvador Llinares, and Ban Heng Choy. "Enhancing Noticing: Using a Hypothetical Learning Trajectory to Improve Pre-service Primary Teachers’ Professional Discourse". Eurasia Journal of Mathematics, Science and Technology Education 2018 14 no. 11 (2018): em1599. https://doi.org/10.29333/ejmste/93421

Harvard

Ivars, P., Fernández, C., Llinares, S., and Choy, B. H. (2018). Enhancing Noticing: Using a Hypothetical Learning Trajectory to Improve Pre-service Primary Teachers’ Professional Discourse. Eurasia Journal of Mathematics, Science and Technology Education, 14(11), em1599. https://doi.org/10.29333/ejmste/93421

MLA

Ivars, Pedro et al. "Enhancing Noticing: Using a Hypothetical Learning Trajectory to Improve Pre-service Primary Teachers’ Professional Discourse". Eurasia Journal of Mathematics, Science and Technology Education, vol. 14, no. 11, 2018, em1599. https://doi.org/10.29333/ejmste/93421