Using a hypothetical learning trajectory to propose instructional activities
Abstract
To decide on how to on a teaching-learning situation has been identified as the most difficult of the three skills that make up the competence of noticing student’s mathematical understanding. In this study 95 pre-service primary school teachers solved a task in which they had to propose a learning objective and activities to support this development of the understanding the meaning of fraction as part-whole using as a reference a hypothetical learning trajectory. Results suggest that the hypothetical learning trajectory helped pre-service teachers propose activities focused on students’ understanding of how to use the mathematical elements that articulate the hypothetical learning trajectory.Keywords
Noticing, Fractions, Hypothetical learning trajectory, Primary teacher education, Instructional activitiesReferences
Barnhart, T. y 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
Battista, M. T. (2012). Cognition-Based Assessment and teaching of fractions: Building on students’ reasoning. Portsmouth: Heinemann.
Chao, T., Murray, E. y Star, J. R. (2016). Helping mathematics teachers develop noticing skills: Utilizing smartphone technology for one-on-one teacher/student interviews. Contemporary Issues in Technology and Teacher Education, 16(1), 22-37.
Choy, B. H. (2013). Productive mathematical noticing: What it is and why it matters. En V. Steinle, L. Ball y C. Bardini (Eds.), Proceedings of the 36th Annual Conference of Mathematics Education Research Group of Australasia (pp. 186-193). Melbourne, Victoria: MERGA.
Clarke, D., Roche, A. y Mitchell, A. (2011). One-to-one student interviews provide powerful insights and clear focus for the teaching of fractions in the middle years. En J. Way y J. Bobis (Eds.), Fractions: Teaching for Understanding (pp. 23-41). Australia: A.A.M.T. Inc.
D’Ambrosio, B. S. y Mewborn, D. S. (1994). Children’s constructions of fractions and their implications for classroom instruction. Journal of Research in Childhood Education, 8(2), 150-161. https://doi.org/10.1080/02568549409594863
Daro, P., Mosher, F. y Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction (CPRE Research Report #RR68). Filadelfia: CPRE. https://doi.org/10.12698/cpre.2011.rr68
Edgington, C., Wilson, P. H., Sztajn, P. y 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
Empson, S. B. (1999). Equal sharing and shared meaning: The development of fraction concepts in a first-grade classroom. Cognition and Instruction, 17(3), 283-342. https://doi.org/10.1207/S1532690XCI1703_3
Empson, S. y Levi, L. (2011). Extending children’s mathematics. Fractions and decimals. Portsmouth, New Hampshire: Heinemann.
Fernández, C., Sánchez-Matamoros, G., Valls, J. y Callejo, M. L. (2018). Noticing students’ mathematical thinking: characterization, development and contexts. Avances de Investigación en Educación Matemática, 13, 39-61. https://doi.org/10.35763/aiem.v0i13.229
Fortuny, J. M. y Rodríguez, R. (2012). Aprender a mirar con sentido: facilitar la interpretación de las interacciones en el aula. Avances de Investigación en Educación Matemática, 1, 23-37. https://doi.org/10.35763/aiem.v1i1.3
Grossman, P., Copton, C., Igra, D., Ronfeldt, M., Shahan, E. y Willianson, P. (2009). Teaching practice: A cross-professional perspective. Teachers College Record, 111(9), 2055-2100.
Gupta, D., Soto, M., Dick, L., Broderick, S. D. y Appelgate, M. (2018). Noticing and deciding the next steps for teaching: A cross-university study with elementary pre-service teachers. En J. Stylianides y K. Hino (Eds.), Research advances in the mathematical education of pre-service elementary teachers (pp. 261-275). Cham, Suiza: Springer. https://doi.org/10.1007/978-3-319-68342-3_18
Ivars, P., Fernández, C. y Llinares, S. (2020). A learning trajectory as a scaffold for pre-service teachers’ noticing of students’ mathematical understanding. International Journal of Science and Mathematics Education, 18(3), 529-548. https://doi.org/10.1007/s10763-019-09973-4
Ivars, P., Fernández, C., Llinares, S. y 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. y Philipp, R. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169-202.
Krupa, E. E., Huey, M., Lesseig, K., Casey, S. y Monson, D. (2017). Investigating secondary preservice teacher noticing of students’ mathematical thinking. En E. O. Schack, M. H. Fishery y J. A. Wilhelm (Eds.), Teacher Noticing: Bridging and broadening perspectives, contexts, and frameworks (pp. 49-72). Cham, Suiza: Springer. https://doi.org/10.1007/978-3-319-46753-5_4
Kurt, G. y Cakiroglu, E. (2009). Middle grade students’ performances in translating among representations of fractions: A Turkish perspective. Learning and Individual Differences, 19(4), 404-410. https://doi.org/10.1016/j.lindif.2009.02.005
Lai, M., Lam, K. M. y Lim, C. P. (2016). Design principles for the blend in blended learning: a collective case study. Teaching in Higher Education, 21(6), 716-729. https://doi.org/10.1080/13562517.2016.1183611
Larson, C. N. (1988). Teaching fraction terms to primary students. En M. J. Behr, C. B. Lacampagne y M. M. Wheeler (Eds.), Proceedings of the 10th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 100-106). Dekalb, IL: PME-NA.
Levin, D. M., Hammer, D. y 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. Londres: Routledge. https://doi.org/10.4324/9780203471876
Mason, J. (2016). Perception, interpretation and decision making: understanding gaps between competence and performance –a commentary. ZDM Mathematics Education, 48(1-2), 219-226. https://doi.org/10.1007/s11858-016-0764-1
National Council of Teachers of Mathematics (2014). Principles to actions: Ensuring mathematics success for all. Reston, VA: NCTM.
Ni, Y. (2001). Semantic domains of rational numbers and the acquisition of fraction equivalence. Contemporary Educational Psychology, 26(3), 400-417. https://doi.org/10.1006/ceps.2000.1072
Sánchez-Matamoros, G., Fernández, C. y 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
Sánchez-Matamoros, G., Moreno, M., Prez-Tyteca, P. y Callejo, M. L. (2018). Trayectoria de aprendizaje de la longitud y su medida como instrumento conceptual usado por futuros maestros de Educación Infantil. Revista Latinoamericana de Investigación en Matemática Educativa, 21(2), 203-228. https://doi.org/10.12802/relime.18.2124
Schack, E. O., Fisher, M. H. y Wilhelm, J. A. (Eds.) (2017). Teacher Noticing: Bridging and broadening perspectives, contexts, and frameworks. Cham, Suiza: Springer. https://doi.org/10.1007/978-3-319-46753-5
Schoenfeld, A. H. (2011). How we think: A theory of goal-oriented decision making and its educational applications. Nueva York: Routledge. https://doi.org/10.4324/9780203843000
Sherin, M. G., Jacobs, V. R. y Philipp, R. A. (Eds.) (2011). Mathematics teacher noticing: Seeing through teachers’ eyes. Nueva York: Routledge. https://doi.org/10.4324/9780203832714
Simon, M. A. y 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
Smith, T. (2003). Connecting theory and reflective practice through the use of personal theories. En N. Pateman, B. Dougherty y J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education (vol. 4, pp. 215-222). Honolulu, HI: PME.
Smith, M. S. y Stein, M. K. (2011). 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA: NCTM.
Son, J. W. y Crespo, S. (2009). Prospective teachers’ reasoning and response to a student’s non-traditional strategy when dividing fractions. Journal of Mathematics Teacher Education, 12(4), 235-261. https://doi.org/10.1007/s10857-009-9112-5
Son, J. W. y Sinclair, N. (2010). How prospective teachers interpret and respond to student geometric errors. School Science and Mathematics, 110(1), 31-46. https://doi.org/10.1111/j.1949-8594.2009.00005.x
Stahnke, R., Schueler, S. y 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. y Olive, J. (2010). Children’s fractional knowledge. Nueva York: Springer. https://doi.org/10.1007/978-1-4419-0591-8
Sztajn, P., Confrey, J., Wilson, P. H. y Edgington, C. (2012). Learning trajectory based instruction toward a theory of teaching. Educational Researcher, 41(5), 147-156. https://doi.org/10.3102/0013189x12442801
Sztajn, P., Edgington, C., Wilson, P. H., Webb, J. y Myers, M. (2019). The Learning Trajectory Based Instruction Project. En P. Sztajn y P. Holt Wilson (Eds.), Learning Trajectories for teachers: Designing effective professional development for math instruction (pp. 15-47). Nueva York: Teachers’ College Press.
Sztajn, P. y Wilson, P. H. (2019). Learning trajectories for teachers: Designing effective professional development for math instruction. Nueva York: Teachers’ College Press.
Timinsky, A. M., Land, T. J., Drake, C., Zambak, V. S. y Simpson, A. (2014). Preservice elementary mathematics teachers’ emerging ability to write problems to build on children’s mathematics. En J. J. Lo, K. R. Leatham y L. R. Van Zoest (Eds.), Research trends in mathematics teacher education (pp. 193-218). Cham, Suiza: Springer. https://doi.org/10.1007/978-3-319-02562-9_11
van Es, E. (2011). A framework for learning to notice student thinking. En M. G. Sherin, V. R. Jacobs y R. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 134-151). Nueva York: Routledge.
Wager, A. A. (2014). Noticing children’s participation: Insights into teacher positionality toward equitable mathematics pedagogy. Journal for Research in Mathematics Education, 45(3), 312-350. https://doi.org/10.5951/jresematheduc.45.3.0312
Published
Downloads
Copyright (c) 2020 Autor
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.