11/2018 vol. 14
Stats
CC-BY 4.0
Get citation
RESEARCH PAPER
Enhancing Noticing: Using a Hypothetical Learning Trajectory to Improve Pre-service Primary Teachers’ Professional Discourse
Pedro Ivars 1
,
 
Ceneida Fernández 1
,
 
Salvador Llinares 1
,
 
Ban Heng Choy 2
 
 
 
More details
Hide details
1
Dept. Innovación y Formación Didáctica. University of Alicante, SPAIN
 
2
National Institute of Education, Nanyang Technological University, SINGAPORE
 
 
Publication date: 2018-07-14
 
 
EURASIA J. Math., Sci Tech. Ed 2018;14(11):em1599
DOI: https://doi.org/10.29333/ejmste/93421
 
 
Article (PDF)
References (48)
 
KEYWORDS
fractions
hypothetical learning trajectories
noticing
pre-service primary school teachers learning
professional discourse
 
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.
REFERENCES (48)
1.
Battista, M. T. (2012). Cognition-Based assessment and teaching of fractions: Building on students’ reasoning. Portsmouth, N.H. Heinemann.
Google Scholar
 
 
2.
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....
CrossRef
 
Google Scholar
 
 
3.
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-....
CrossRef
 
Google Scholar
 
 
4.
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....
CrossRef
 
Google Scholar
 
 
5.
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....
CrossRef
 
Google Scholar
 
 
6.
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....
CrossRef
 
Google Scholar
 
 
7.
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-....
CrossRef
 
Google Scholar
 
 
8.
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/mathte....
CrossRef
 
Google Scholar
 
 
9.
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....
CrossRef
 
Google Scholar
 
 
10.
Goodwin, C. (1994). Professional vision. American anthropologist, 96(3), 606-633. https://doi.org/10.1525/aa.199....
CrossRef
 
Google Scholar
 
 
11.
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-....
CrossRef
 
Google Scholar
 
 
12.
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.
Google Scholar
 
 
13.
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.
Google Scholar
 
 
14.
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:1025....
CrossRef
 
Google Scholar
 
 
15.
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.
Google Scholar
 
 
16.
Lamon, S. (1999). Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum Ass. Pub.
Google Scholar
 
 
17.
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.
Google Scholar
 
 
18.
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-....
CrossRef
 
Google Scholar
 
 
19.
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/10.1177/002248....
CrossRef
 
Google Scholar
 
 
20.
Mason, J. (2002). Researching your own practice: the discipline of noticing. London: Routledge. https://doi.org/10.4324/978020....
CrossRef
 
Google Scholar
 
 
21.
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.
Google Scholar
 
 
22.
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-....
CrossRef
 
Google Scholar
 
 
23.
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-....
CrossRef
 
Google Scholar
 
 
24.
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....
CrossRef
 
Google Scholar
 
 
25.
National Council of Teachers of Mathematics (NCTM). (2014). Principles to actions. Reston, VA: Author.
Google Scholar
 
 
26.
National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press. https://doi.org/10.17226/9822.
CrossRef
 
Google Scholar
 
 
27.
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-....
CrossRef
 
Google Scholar
 
 
28.
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-....
Google Scholar
 
 
29.
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/10.1177/002248....
CrossRef
 
Google Scholar
 
 
30.
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-....
CrossRef
 
Google Scholar
 
 
31.
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/10.4324/978020....
CrossRef
 
Google Scholar
 
 
32.
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.
CrossRef
 
Google Scholar
 
 
33.
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/s15327....
CrossRef
 
Google Scholar
 
 
34.
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....
CrossRef
 
Google Scholar
 
 
35.
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....
CrossRef
 
Google Scholar
 
 
36.
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....
CrossRef
 
Google Scholar
 
 
37.
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/s15327....
CrossRef
 
Google Scholar
 
 
38.
Steffe, L., & Olive, J. (2009). Children’s fractional knowledge. Springer Science & Business Media. https://doi.org/10.1007/978-1-....
CrossRef
 
Google Scholar
 
 
39.
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/10.3102/001318....
CrossRef
 
Google Scholar
 
 
40.
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.
Google Scholar
 
 
41.
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.
Google Scholar
 
 
42.
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/10.1177/002248....
CrossRef
 
Google Scholar
 
 
43.
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....
CrossRef
 
Google Scholar
 
 
44.
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/euras....
CrossRef
 
Google Scholar
 
 
45.
Wells, G. (1999). Dialogic Inquiry. Toward a Sociocultural Practice and Theory of Education. Cambridge: Cambridge University Press. https://doi.org/10.1017/cbo978....
CrossRef
 
Google Scholar
 
 
46.
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.jmat....
CrossRef
 
Google Scholar
 
 
47.
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/10.3102/000283....
CrossRef
 
Google Scholar
 
 
48.
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/10.2307/116727....
CrossRef
 
Google Scholar
 
 
Share
Send by email
RELATED ARTICLE
Profiles of young students’ understanding of fractions on number lines
Egyptian Fractions and Representation Registers in the Construction of the Fraction Concept
A Test for Understanding Simple Fractions Among 5th Grade Students at the Beginning of Lower Secondary Education
Professional noticing as student-centered: Pre-service teachers’ attending to students’ mathematics in 360 video
Formation of Language Identities in a Bilingual Teaching Intervention on Fractions
Indexes
 
Keywords index
Topics index
Authors index
eISSN:1305-8223
ISSN:1305-8215
Journals System - logo
© 2006-2024 Journal hosting platform by Bentus
Scroll to top