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A Learning Trajectory as a Scaffold for Pre-service Teachers’ Noticing of Students’ Mathematical Understanding

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Abstract

The objective of this research is to understand how a learning trajectory supports pre-service primary teachers in their noticing of students’ mathematical understanding. A total of 95 pre-service primary school teachers used a learning trajectory related to the part-whole meaning of the fraction concept to interpret students’ understanding and provide instructional decisions. The findings indicate that the learning trajectory provided pre-service teachers with a specific language to describe students’ understanding and helped them to notice students’ mathematical understanding. Furthermore, the pre-service primary teachers who produced a more detailed discourse also proposed more suitable activities based on students’ understanding. These findings suggest that the learning trajectory can act as a scaffold to notice students’ mathematical understanding.

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References

  • Ball, D. L. & Cohen, D. (1999). Developing practice, developing practitioners: Toward a practice-based theory of professional education. In L. Darling-Hammond & G. Sykes (Eds.), Teaching as the learning profession (pp. 3–32). San Francisco, CA: Jossey-Bass.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Battista, M. T. (2012). Cognition-based assessment and teaching of fractions: Building on students’ reasoning. Portsmouth, England: N.H. Heinemann.

    Google Scholar 

  • Bobos, G. & Sierpinska, A. (2017). Measurement approach to teaching fractions: A design experiment in a preservice course for elementary teachers. International Journal of Mathematics Teaching and Learning, 18(2), 203–239. Retrieved from http://www.cimt.org.uk/ijmtl/index.php/IJMTL/article/view/65/34.

    Google Scholar 

  • Choy, B. H. (2016). Snapshots of mathematics teacher noticing during task design. Mathematics Education Research Journal, 28(3), 421–440. https://doi.org/10.1007/s13394-016-0173-3.

    Article  Google Scholar 

  • Clarke, D. J. (2013). Contingent conceptions of accomplished practice: The cultural specificity of discourse in and about the mathematics classroom. ZDM Mathematics Education, 45(1), 21–33. https://doi.org/10.1007/s11858-012-0452-8.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Daro, P., Mosher, F. & Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction. CPRE research report #RR-68. Philadelphia, PA: Consortium for Policy Research in Education. https://doi.org/10.12698/cpre.2011.rr68.

    Book  Google Scholar 

  • Edgington, C. P. (2012). Teachers’ uses of a learning trajectory to support attention to students’ mathematical thinking (Doctoral dissertation). Available from ProQuest Dissertations Publishing (UMI No. 3538262).

  • 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). Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-319-02562-9_14.

    Chapter  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Erickson, F. (2011). On noticing teacher noticing. In M. G. Sherin, V. R. Jacobs & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 17–34). New York, NY: Routledge.

    Google Scholar 

  • Fosnot, C. T. & Dolk, M. (2002). Young mathematicians at work, 3: Constructing fractions, decimals, and percents. Portsmouth, England: Heinemann.

    Google Scholar 

  • 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. An international perspective (pp. 245–260). Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-319-68342-3_17.

    Google Scholar 

  • 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.

  • Jacobs, V. R., Lamb, L. C. & Philipp, R. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • Little, J. W. (1993). Teachers’ professional development in a climate of educational reform. Educational Evaluation and Policy Analysis, 15(2), 129–151. https://doi.org/10.3102/01623737015002129.

    Article  Google Scholar 

  • Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1(3), 243–267. https://doi.org/10.1023/A:1009973717476.

    Article  Google Scholar 

  • Mason, J. (2002). Researching your own practice. The discipline of noticing. London, England: Routledge-Falmer. https://doi.org/10.4324/9780203471876.

    Book  Google Scholar 

  • Mason, J. (2011). Noticing: Roots and branches. In M. G. Sherin, V. R. Jacobs & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 35–50). New York, NY: Routledge.

    Google Scholar 

  • Mason, J. (2017). Proving beneath the surface of experience. In E. O. Schack, M. H. Fisher & J. A. Wilhelm (Eds.), Teacher noticing: Bridging and broadening perspectives, contexts, and frameworks (pp. 49–72). Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-319-46753-5_1.

    Chapter  Google Scholar 

  • 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.

    Article  Google Scholar 

  • National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematics success for all. Reston, VA: Author.

    Google Scholar 

  • 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/10.1007/s10763-014-9544-y.

    Article  Google Scholar 

  • Schack, E. O., Fisher, M. H., Thomas, J. N., Eisenhardt, S., Tassell, J. & Yoder, M. (2013). Prospective elementary school teachers’ professional noticing of children’s early numeracy. Journal of Mathematics Teacher Education, 16(5), 379–397. https://doi.org/10.1007/s10857-013-9240-9.

    Article  Google Scholar 

  • Schoenfeld, A. H. (2011). Noticing matters. A lot. Now what. In M. G. Sherin, V. R. Jacobs & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 223–238). New York, NY: Routledge.

    Google Scholar 

  • 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.

    Book  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Smith, T. (2003). Connecting theory and reflective practice through the use of personal theories. In N. Pateman, B. Dougherty & J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 215–222). CRDG, College of Education, University of Hawaii: PME.

  • 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(2), 107–125. https://doi.org/10.1007/s10857-007-9063-7.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Steffe, L. & Olive, J. (2010). Children’s fractional knowledge. Boston, MA: Springer Science & Business Media. https://doi.org/10.1007/978-1-4419-0591-8.

    Book  Google Scholar 

  • Stürmer, K., Könings, K. D. & Seidel, T. (2013). Declarative knowledge and professional vision in teacher education: Effect of courses in teaching and learning. British Journal of Educational Psychology, 83(3), 467–483. https://doi.org/10.1111/j.2044-8279.2012.02075.x.

    Article  Google Scholar 

  • Stürmer, K. & Seidel, T. (2017). A standardized approach for measuring teachers’ professional vision: The observer research tool. In E. O. Schack, M. H. Fisher & J. A. Wilhelm (Eds.), Teacher noticing: Bridging and broadening perspectives, contexts, and frameworks (pp. 359–380). Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-319-46753-5_21.

    Chapter  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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), 571–595.

    Google Scholar 

  • Van Es, E. A. & Sherin, M. G. (2008). Mathematics teachers’ “learning to notice” in the context of a video club. Teaching and Teacher Education, 24(2), 244–276. https://doi.org/10.1016/j.tate.2006.11.005.

    Article  Google Scholar 

  • Van Galen, F., Feijs, E., Figueiredo, N., Gravemeijer, K., van Herpen, E. & Keijzer, R. (2008). Fractions, percentages, decimals and proportions: A learning-teaching trajectory for grade 4, 5 and 6. Rotterdam, Netherlands: Sense Publishers.

    Book  Google Scholar 

  • Wessels, H. (2018). Noticing in pre-service teacher education: Research lessons as a context for reflection on learners’ mathematical reasoning and sense-making. In G. Kaiser, H. Forgasz, M. Graven, A. Kuzniak, E. Simmt & B. Xu (Eds.), Invited lectures from the 13th International Congress on Mathematical Education (pp. 731–748). Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-319-72170-5_41.

    Chapter  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Wilson, P. H., Sztajn, P., Edgington, C. & Myers, M. (2015). Teachers’ uses of a learning trajectory in student-centered instructional practices. Journal of Teacher Education, 66(3), 227–244. https://doi.org/10.1177/0022487115574104.

    Article  Google Scholar 

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Acknowledgments

This research was supported by the project EDU2017-87411-R from Ministerio de Economía y Empresa (Spain) and by the project GV/2018/066 from the Conselleria de Educación, Investigación, Cultura y Deporte de la Generalitat Valenciana, Spain.

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Correspondence to Pedro Ivars.

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Ivars, P., Fernández, C. & Llinares, S. A Learning Trajectory as a Scaffold for Pre-service Teachers’ Noticing of Students’ Mathematical Understanding. Int J of Sci and Math Educ 18, 529–548 (2020). https://doi.org/10.1007/s10763-019-09973-4

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  • DOI: https://doi.org/10.1007/s10763-019-09973-4

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